Let (fn) and (gn) be two sequences of random variables adapted to an increasing sequence of σ-algebras (ℱn) such that the conditional distributions of fn and gn given ℱn coincide. Suppose further that the sequence (gn) is conditionally independent. Then it is known that where the number C is a universal constant. The aim of this paper is to extend this result to certain classes of Orlicz and rearrangement invariant spaces. This paper includes fairly general techniques for obtaining rearrangement invariant inequalities from Orlicz norm inequalities.

This paper investigates the natural problem of performing the multifractal analysis of heterogeneous sums of Dirac masseswhere (xn)n≥0 is a sequence of points in [0, 1]d and (wn)n≥0 is a positive sequence of weights such that Σn≥0wn < ∞. We consider the case where the points xn are roughly uniformly distributed in [0, 1]d, and the weights wn depend on a random self-similar measure μ, a parameter ρ ∈ (0, 1], and a sequence of positive radii (λn)n≥1 converging to 0 in the following wayThe measure ν has a rich multiscale structure. The computation of its multifractal spectrum is related to heterogeneous ubiquity properties of the system {(xn,λn)n with respect to μ.

We discuss the minimal-to-maximal correspondence between surfaces and show that, under this correspondence, a congruence class of minimal surfaces in 3 determines an 2-family of congruence classes of maximal surfaces in 3. It is proved that further identifications among these classes may exist, depending upon the subgroup of automorphisms preserving the Hopf differential on the underlying Riemann surface. The space of maximal congruence classes inherits a quotient topology from 2. In the case of the Scherk minimal surface, the subgroup has order two and the quotient space is topologically a disc with boundary. Other classical examples are discussed: for the Enneper minimal surface, one obtains a non-Hausdorff space; for the minimal catenoid, a closed interval.

Recent studies by Debye (1), Bloch (2), Stoner (3) and Guggenheim (4) expressing somewhat divergent views suggest that there is still some confusion and uncertainty in the method of application of the thermodynamic argument to magnetic phenomena. The first effective contribution to the subject was made by Larmor (5) over fifty years ago, and developing this further he finally proposed in 1929 a sufficient basis for a general theory of the whole range of phenomena. At about the same time (1927), and apparently in ignorance of Larmor's earlier work, Debye(1) also gave a brief outline of the thermodynamic aspects of the subject, starting, however, from much more tentative magnetic ideas and formulae which are now known to be without adequate physical justification. In 1933, Bloch (2) elaborated the argument still further, using, however, basic ideas more akin to those employed by Larmor, but derived by a very specialized argument. Then in 1935, using the same fundamental equation as Debye, Stoner revived the discussion with a systematic survey of the theoretical results bearing on the thermodynamic aspects of the subject, attempting later (6) to justify the basic magnetic ideas by a discussion which he now admits is not entirely satisfactory.

For a locally compact group G, let B(G) denote its Fourier–Stieltjes algebra. Any continuous, piecewise affine map α: Y ⊂ H → G induces a completely bounded algebra homomorphism jα: B(G) → B(H) [14, 15] and we prove that jα is w* – w* continuous if and only if α is an open map. This extends one of the main results in [3], due to M.B. Bekka, E. Kaniuth, A.T. Lau and G. Schlichting. Several classical theorems regarding isomorphisms and extensions of homomorphisms on group algebras of abelian groups are extended to the setting of Fourier–Stieltjes algebras of amenable groups. As applications, when G is amenable we provide complete characterizations of those maps between Fourier–Stieltjes algebras that are either associated to a piecewise affine mapping, or are completely bounded and w* – w* continuous.

1. Let A be a summability method given by the sequence-to-sequence transformation

We suppose throughout that, for each n

converges; this is a much weaker assumption than the regularity of A. Then we define

We also suppose throughout that the sequence {sk} is formed by taking the partial sums of the series Σak; that is to say that

Let A' denote the summability method given by the series-to-sequence transformation

Following Lorent and Zeller (4), (5), we describe A, A' as dual summability methods. We recall that formally,

We study the size of the range of the derivatives of a smooth function between Banach spaces. We establish conditions on a pair of Banach spaces $X$ and $Y$ to ensure the existence of a $C^p$ smooth (Fréchet smooth or a continuous Gâteaux smooth) function $f$ from $X$ onto $Y$ such that $f$ vanishes outside a bounded set and all the derivatives of $f$ are surjections. In particular we deduce the following results. For the Gâteaux case, when $X$ and $Y$ are separable and $X$ is infinite-dimensional, there exists a continuous Gâteaux smooth function $f$ from $X$ to $Y$, with bounded support, so that $f^{\prime}(X) = {\cal L}(X,Y)$. In the Fréchet case, we get that if a Banach space $X$ has a Fréchet smooth bump and ${\rm dens} X = {\rm dens} {\cal L}(X,Y)$, then there is a Fréchet smooth function $f: X \rightarrow Y$ with bounded support so that $f^{\prime}(X) = {\cal L}(X,Y)$. Moreover, we see that if $X$ has a $C^p$ smooth bump with bounded derivatives and ${\rm dens} X = {\rm dens} {\cal L}^m_s(X;Y)$ then there exists another $C^p$ smooth function $f: X \rightarrow Y$ so that $f^{(k)} (X) = {\cal L}^k_s(X;Y)$ for all $k = 0, 1,\ldots, m$. As an application, we show that every bounded starlike body on a separable Banach space $X$ with a (Fréchet or Gâteaux) smooth bump can be uniformly approximated by smooth bounded starlike bodies whose cones of tangent hyperplanes fill the dual space $X^*$. In the non-separable case, we prove that $X$ has such property if $X$ has smooth partitions of unity.

It is well known that in an even dimensional Riemannian space there always exists a particular scalar density which is such that its associated Euler–Lagrange tensor vanishes identically. In this note we show that each of these scalar densities can be expressed as the ordinary divergence of a contravariant vector density. Furthermore, there exists an entire class of contravariant vector densities which enjoy this property. These results are discussed with particular reference to the general theory of relativity.

The form of Levinson's relation is obtained for the case where the free Hamiltonian contains discrete eigenstates. The relevance of this to the ambiguous solutions of Low equations and dispersion relations is discussed.

Goussarov and the author independently proved that two knots in S3 have the same values of finite type invariants of degree <n if and only if they are Cn-equivalent, which means that they are equivalent up to modification by a kind of geometric commutator of class n. This property does not generalize to links with more than one component.

In this paper, we study the case of Brunnian links, which are links whose proper sublinks are trivial. We prove that if n ≥ 1, then an (n+1)-component Brunnian link L is Cn-equivalent to an unlink. We also prove that if n ≥ 2, then L can not be distinguished from an unlink by any Goussarov–Vassiliev finite type invariant of degree <2n.

Cochran, Orr and Teichner introduced $L^2$-eta-invariants to detect highly non-trivial examples of non slice knots. Using a recent theorem by Lück and Schick we show that their metabelian $L^2$-eta-invariants can be viewed as the limit of finite dimensional unitary representations. We recall a ribbon obstruction theorem proved by the author using finite dimensional unitary eta-invariants. We show that if for a knot $K$ this ribbon obstruction vanishes then the metabelian $L^2$-eta-invariant vanishes too. The converse has been shown by the author not to be true.

Let V be a nonsingular cubic surface defined over the finite field Fq. It is well known that the number of points on V satisfies #V(Fq) = q2 + nq + 1 where −2 ≤ n ≤ 7 and that n = 6 is impossible; see for example [1], Table 1. Serre has asked if these bounds are best possible for each q. In this paper I shall show that this is so, with three exceptions:

We prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez--Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for 7F6 truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.

The main purpose of this paper is to introduce a method to “stabilise” certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group G, namely Hom(ℤn , G). We show that this stabilised space of homomorphisms decomposes after suspending once with “summands” which can be reassembled, in a sense to be made precise below, into the individual spaces Hom(ℤn , G) after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilised space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilised space also allows the description of the additive reduced homology of the individual spaces Hom(ℤn , G), with the order of the Weyl group inverted.

Painlevé differential equations have been an important topic of research in complex differential equations during the last century, and the last two decades in particular, with many applications not only to pure mathematics but also to physics and engineering. In this paper, we prove that any transcendental solution of the second Painlevé equation $w''\,{=}\,2w^{3}+zw+\alpha$ is of order at least $3/2$, and that any transcendental solution of the fourth Painlevé equation $2ww''\,{=}\,(w')^{2}+3w^{4}+8zw^{3}+4(z^{2}-\alpha )w^{2}+2\beta$ is of order at least $2$.

When a familiar notion is modelled in a certain topos E, the natural problem arises to what extent theorems concerning its models in usual set theory remain valid for its models in E, or how suitable properties of E affect the validity of certain of these theorems. Problems of this type have in particular been studied by Banaschewski[2], Bhutani[5], and Ebrahimi[6, 7], dealing with abelian groups in a localic topos and universal algebra in an arbitrary Grothendieck topos. This paper is concerned with Boolean algebras, specifically with injectivity and related topics for the category of Boolean algebras in the topos of sheaves on a locale and with properties of the initial Boolean algebra in .

As a mathematical area generalized inversion1 was inaugurated in 1955 by R. Penrose [128]. Since then there have appeared about 2000 articles and 15 books2 on generalized inverses of matrices and linear operators. See the annotated bibliography by Nashed and Rall [121] for the period up to 1976.

1. Introduction. Let A[λ] be the irreducible invariant matrix of a general matrix of order n × n, corresponding to a partition (λ) = (λ1, λ2, …, λr) of some integer m. The problem to be discussed here is that of determining the canonical form of A[λ] when that of A is known.

A Haken manifold is a compact, orientable, irreducible 3-manifold which contains a properly embedded 2-sided, incompressible surface of positive genus. These manifolds are important in connection with the work of Haken, Waldhausen and the more recent work of Thurston (8). Thus it is interesting to investigate criteria for testing incompressible surfaces on 3-manifolds.

Amongst the various modifications of the general theory of relativity (GRT), the scalar-tensor theory of Brans and Dicke (BDT) is treated most seriously ([2], [17], [13]). The BDT is consistent with observations as long as the coupling parameter ω between the scalar and tensor components of gravitation is about equal to or greater than 500 [18]. However, there are no a priori theoretical reasons for excluding other values of ω. In the limit ω → ∞, the BDT reduces to the GRT for a constant BDT-scalar field Φ