The paper studies the dual algebras of localization Roe algebras over proper metric spaces and develops a localization version of Paschke duality for $K$ -homology. It is shown that the localization $K$ -homology groups are isomorphic to Kasparov's $K$ -homology groups for the Rips complex of proper metric spaces with bounded geometry. It follows that the obstruction groups to the coarse Baum–Connes conjecture can also be derived from the dual localization algebras.

Let R be a commutative Noetherian ring. Let [Pscr ](R) (respectively, [Iscr ](R)) be the category of all finite R-modules of finite projective (respectively, injective) dimension. Sharp [9] constructed a category equivalence between [Iscr ](R) and [Pscr ](R) for certain Cohen–Macaulay local rings R. Thus many properties about finite modules of finite projective dimension can be connected with those of finite injective dimension through this equivalence.

The paper proves that a set which contains spheres centered at all points of a set of Hausdorff dimension greater than 1 must have positive Lebesgue measure. It also proves the corresponding result for circles, provided that the set of centers has Hausdorff dimension greater than 3/2.

The paper provides a new look at compensated compactness and paracommutators. It is shown that there is a one-to-one correspondence between compensated quantities and paracommutators, that the H1-regularity of compensated quantities is equivalent to bounded mean oscillation boundedness of paracommutators, that the weak convergence of compensated quantities is equivalent to vanishing mean oscillation compactness of paracommutators, that the Schatten–von Neumann Sp-property is a natural generalization of vanishing mean oscillation compactness, that the theory of paracommutators provides a ready-made tool for Sp-regularity of compensated quantities, and that the cut-off phenomenon of paracommutators exactly coincides with the vanishing moment properties of the compensated quantities. Some further examples are given.

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.

The 2-modular and 3-modular decomposition numbers for the sporadic simple O'Nan group O'N and its triple cover are determined. The results are obtained with the help of the computer algebra packages MeatAxe and GAP. Permutation modules and tensor products of modules have been condensed. The result relies on the construction of a 153-dimensional representation for 3·O'N over GF(4) and a 154-dimensional representation for O'N over GF(3).

let $l$ denote a right-invariant sub-laplacian on an exponential, hence solvable lie group $g$, endowed with a left-invariant haar measure. depending on the structure of $g$, and possibly also that of $l, l$ may admit differentiable $l^p$-functional calculi, or may be of holomorphic $l^p$-type for a given $p{\ne} 2$. ‘holomorphic $l^p$-type’ means that every $l^p$-spectral multiplier for $l$ is necessarily holomorphic in a complex neighbourhood of some non-isolated point of the $l^2$-spectrum of $l$. this can in fact only arise if the group algebra $l^1(g)$ is non-symmetric.

assume that $p{\ne} 2$. for a point $\ell$ in the dual $\frak{g}^*$ of the lie algebra $\frak{g}$ of $g$, denote by $\omega(\ell){=}\ad^*(g)\ell$ the corresponding coadjoint orbit. it is proved that every sub-laplacian on $g$ is of holomorphic $l^p$-type, provided that there exists a point $\ell{\in} \frak{g}^*$ satisfying boidol's condition (which is equivalent to the non-symmetry of $l^1(g)$), such that the restriction of $\omega(\ell)$ to the nilradical of $\frak{g}$ is closed. this work improves on results in previous work by christ and müller and ludwig and müller in twofold ways: on the one hand, no restriction is imposed on the structure of the exponential group $g$, and on the other hand, for the case $p{>}1$, the conditions need to hold for a single coadjoint orbit only, and not for an open set of orbits.

it seems likely that the condition that the restriction of $\omega(\ell)$ to the nilradical of $\frak{g}$ is closed could be replaced by the weaker condition that the orbit $\omega(\ell)$ itself is closed. this would then prove one implication of a conjecture by ludwig and müller, according to which there exists a sub-laplacian of holomorphic $l^1$ (or, more generally, $l^p$) type on $g$ if and only if there exists a point $\ell{\in} \frak{g}^*$ whose orbit is closed and which satisfies boidol's condition.

Let [Escr ] denote the space of all entire functions, equipped with the topology of local uniform convergence (the compact-open topology). MacLane [15] constructed an entire function f whose sequence of derivatives (f, f′, f'', …) is dense in [Escr ]; his construction is succinctly presented in a much later note by Blair and Rubel [2], who unwittingly rederived it (see also [3]). We shall call such a function f a universal entire function. In this note we show that analogous universal functions exist in the space [Hscr ]N of functions harmonic on ℝN, where N[ges ]2. We also study the permissible growth rates of universal functions in [Hscr ]N and show that the set of all such functions is very large.

For purposes of comparison, we first review relevant facts about universal entire functions. The function constructed by MacLane is of exponential type 1. Duyos Ruiz [7] observed that a universal entire function cannot be of exponential type less than 1. G. Herzog [11] refined MacLane's growth estimate by proving the existence of a universal entire function f such that [mid ]f(z)[mid ]=O(rer) as [mid ]z[mid ]=r→∞. Finally, Grosse–Erdmann [10] proved the following sharp result.

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.

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 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.

Let $E$ be an elliptic curve defined over a number field $F$. The paper concerns the structure of the $p^{\backslash}\infty$-Selmer group of $E$ over $p$-adic Lie extensions $F_{\backslash}\infty$ of $F$ which are obtained by adjoining to $F$ the $p$-division points of an abelian variety $A$ defined over $F$. The main focus of the paper is the calculation of the Gal$(F_{\backslash}\infty/F)$-Euler characteristic of the $p^{\backslash}\infty$-Selmer group of $E$. The main theory is illustrated with the example of an elliptic curve of conductor 294.

A certain sequence of weight $1/2$ modular forms arises in the theory of Borcherds products for modular forms for $\textrm{SL}_{2}(\Z)$. Zagier proved a family of identities between the coefficients of these weight $1/2$ forms and a similar sequence of weight $3/2$ modular forms, which interpolate traces of singular moduli. We obtain the analogous results for modular forms arising from Borcherds products for Hilbert modular forms.

It is shown that the division ring of quotients of a skew polynomial ring of automorphism type, if infinite-dimensional over its centre k and satisfying suitable hypotheses, contains the group algebra of a free group of large rank (usually at least [mid ]k[mid ]). The result applies, in particular, to the skew polynomial rings constructed from rational function fields, and affirmatively settles the conjectures of Makar–Limanov and Lichtman in this case.

In this paper we continue our study of the Canonical Element Conjecture (henceforth C.E.C.) via the dualizing complex. Throughout the work (A, m, k) will denote a noetherian complete local ring A of dimension n, m its maximal ideal and k=A/m. Since A is complete, we can find a complete local Gorenstein ring (R, mR, k) (complete intersection) such that dim R=dim A and A=R/I. Let Ω denote the canonical module of A, that is, Ω=HomR (A, R), which may be identified with the annihilator of I in R, an ideal of R. When A is a domain, we change notation and denote I by P; in this case P is a height 0 prime ideal of R.

Finding polynomial solutions of Pell's equation is of interest as such solutions sometimes allow the fundamental units to be determined in an infinite class of real quadratic fields.

In the paper, for each triple of positive integers $(c, h, f)$ satisfying $c^2-fh^2 = 1$ , where $(c, h)$ are the smallest pair of integers satisfying this equation, several sets of polynomials $(c(t), h(t), f(t))$ that satisfy $c(t)^2-f(t)h(t)^2 = 1$ and $(c(0), h(0), f(0)) = (c, h, f)$ are derived. Moreover, it is shown that the pair $(c(t), h(t))$ constitute the fundamental polynomial solution to the Pell equation above.

The continued fraction expansion of $\sqrt{f(t)}$ is given in certain general cases (for example when the continued fraction expansion of $\sqrt{f}$ has odd period length, or has even period length, or has period length $\equiv 2 \mod 4$ and the middle quotient has a particular form, etc.). Some applications to the determination of the fundamental unit in real quadratic fields is also discussed.

Let $G$ be a finite group and suppose that $G = AB$, where $A$ and $B$ are abelian subgroups. By a theorem of Ito, the derived subgroup $G'$ is known to be abelian. If either of the subgroups $A$ or $B$ is cyclic, then more can be said. The paper shows, for example, that $G'{/}(G' \cap A)$ is isomorphic to a subgroup of $B$ in this case.

Let K be a kernel on Rn, that is, K is a non-negative, unbounded L1 function that is radially symmetric and decreasing. We define the convolution K [midast ] F by

and note from Lp-capacity theory [11, Theorem 3] that, if F ∈ Lp, p > 1, then K [midast ] F exists as a finite Lebesgue integral outside a set A ⊂ Rn with CK,p(A) = 0. For a Borel set A,

where

We define the Poisson kernel for Rn+1+ = {(x, y) [ratio ] x ∈ Rn, y > 0} by

and set

Thus u is the Poisson integral of the potential f = K [midast ] F, and we write

We are concerned here with the limiting behaviour of such harmonic functions at boundary points of Rn+1+, and in particular with the tangential boundary behaviour of these functions, outside exceptional sets of capacity zero or Hausdorff content zero.

The autonomous composition operator is the nonlinear map which takes a pair of functions into its composite function. The composition operator often appears in problems of nonlinear analysis and to analyse such problems it is often important to know whether the composition operator is continuous or differentiable. A fairly large number of papers in the literature have been devoted to the study of composition operators. For fullscale references, we refer the reader to the extensive monographs of Appell and Zabrejko [1] and Runst and Sickel [8]. To exemplify a typical situation, we consider the semilinear Dirichlet boundary value problem

formula here

where Ω denotes a sufficiently regular bounded open subset of ℝN, and h0 a map of ℝ to ℝ, and where u is the unknown of the problem. We assume that we know that a certain function u0 belonging to a certain function space [Xscr ] solves (1.1). Then if we wish to know whether by perturbing h0 in a certain function space, say [Yscr ], the solutions u depend on h continuously, with differentiability, with analyticity or bifurcate, we could set G[h, u] ≡ Δu+h ∘ u, recast problem (1.1) into the abstract form

formula here

and study the solution set of equation (1.2) around the pair (h0, u0) by means of the implicit function theorem or by local bifurcation theorems in a Banach space setting.

The paper considers the possible cardinalities of maximal cofinitary groups and the cofinalities of permutation groups on the set of natural numbers. These two cardinals are compared with some other well-known cardinal invariants which are related to covering properties in several forcing models. An application of combinatorial group theory to a construction of a forcing partially ordered set is presented.