Let G be a compact, connected, abelian group with dual group Γ. The set E ⊂ has zero discrete harmonic density (z.d.h.d.) if for every open U ⊂ G and μ ∈ Md(G) there exists ν ∈ Md(U) with = on E. I0 sets in the duals of these groups have z.d.h.d. We give properties of such sets, exhibit non-Sidon sets having z.d.h.d., and prove union theorems. In particular, we prove that unions of I0 sets have z.d.h.d. and provide a new approach to two long-standing problems involving Sidon sets.

In order to protect and sustainably manage fishery resource species, it is essential to understand their movements and habitat use. To detect the hypothesised migration of maturing veined squid Loligo forbesi from the west coast of Scotland (UK) to the North Sea and identify possible inshore-offshore movements, we analysed seasonal, spatial and environmental patterns in abundance and size distribution, based on commercial fishery landings data and trawl survey data from Scottish coastal waters (International Council for the Exploration of the Sea, ICES areas IVa, IVb and VIa). A geographic information system (GIS) was used to build monthly contour maps of abundance. Generalised additive mixed models (GAMM) were used to quantify patterns in size distribution and abundance. In most years, there was no evidence of movement from the West to the East coast of Scotland. Evidence of inshore-offshore movements during the life-cycle of the cohort that recruits in autumn (winter breeders) was found instead. The winter breeding cohort appears to spawn in inshore waters and some evidence suggests that the spawning grounds of the summer breeders are also inshore. Across seasons, higher abundance of L. forbesi can generally be found in the north of Scotland at intermediate water depths and in warmer waters.

The organization of the primary visual cortex (VI) of the common marmoset (Callithrix jacchus) was studied both physiologically and by means of transneuronal labelling of geniculocortical afferents. We addressed the question whether monocular deprivation (MD) could stabilize segregation into ocular dominance (OD) columns, which are not seen in normal adult marmosets but are present in juvenile animals (Spatz, 1979, 1989). Properties of neurons in normal marmosets closely resembled those of other New-World and Old-World monkeys and orderly tangential progressions of preferred orientation were observed. However, in contrast to species that display well-defined OD columns, neurons of layer 4 in VI of normal adult marmosets received balanced inputs from the two eyes. Early MD (even though followed by prolonged binocular experience into adulthood) resulted in a reduction of cell size in laminae of the lateral geniculate nucleus with input from the deprived eye and a dramatic overall shift in ocular dominance towards the non-deprived eye in the cortex. However, isolated clusters of cells dominated by the deprived eye were found in both layers 4 and 6. Injection of lectin-conjugated horseradish peroxidase (WGA-HRP) into the deprived eye revealed elongated patches of terminal label, about 350 μm wide, in flat-mounted sections through layer 4. Afferent segregation was sharper and more regular in the region of VI representing parafoveal visual space than in that representing the fovea. Our findings support the notion that all Old-World and New-World monkeys possess the capacity for segregation of geniculocortical afferents into OD columns.

In the dual object of an infinite compact, connected group, every infinite Sidon set contains an infinite subset on which full interpolation can be performed using very small classes of measures (discrete measures on arbitrarily small sets or nonnegative discrete measures). In particular, the Figà-Talamanca–Rider subset of an infinite product of compact, connected, simple Lie groups has these kinds of interpolation. This substantially improves previous interpolation results.

We identify the support of a tempered distribution by evaluation of a sequence of test functions against the Fourier transform of the distribution. This improves previous results by removing the restriction that the distribution's Fourier transform be in and be of polynomial growth. We use an apparently new technical lemma that implies that certain bounded approximate identities for are also topological approximate identities for elements of the space of Schwartz functions.

PM(E) denotes the set of pseudomeasures on with support in the closed set E ⊆ . Then y ∈ is not in E if and only if there is a neighbourhood W of y with uniformly for w ∈ W and S ∈ PM(E) with ‖S‖PM ≤ 1. This improves previous results by adding “uniformly” and its scope. The proof uses the fact that squashing the central spike of the Fejer kernel leads to A-norm convergence.

A subset $E$ of the locally compact abelian group $\Gamma$ is “$\varepsilon$-Kronecker” if every continuous function from $E$ to the unit circle can be uniformly approximated on $E$ by a character with error less than $\varepsilon$. The set $E\subset \Gamma$ is $I_0$ if every bounded function on $E$ can be interpolated by the Fourier Stieltjes transform of a discrete measure on the dual group.

We show that products (sums) of $\varepsilon$-Kronecker sets can be all of the group if the number of terms is sufficiently large, but are shown to be $U_0$ sets (sets of uniqueness in the weak sense) if the number is small. Results about cluster points of products are extended from Hadamard to $\varepsilon$-Kronecker sets. One consequence of that is that finite unions of translates of a fixed $\varepsilon$-Kronecker set are $I_0$.

A subset $E$ of the locally compact abelian group $\Gamma$ is “$\varepsilon$-Kronecker” if every continuous function from $E$ to the unit circle can be uniformly approximated on $E$ by a character with error less than $\varepsilon$. The set $E\subset \Gamma$ is $I_0$ if every bounded function on $E$ can be interpolated by the Fourier–Stieltjes transform of a discrete measure on the dual group.

We show that if $\varepsilon\,{<}\,\sqrt2$ then an $\varepsilon$-Kronecker set is $I_0$, but this is not true for at least one $\sqrt 2$-Kronecker set. $\varepsilon$-Kronecker sets in ${\mathbb Z}$ need not be finite unions of Hadamard sets. As with Sidon sets, $\varepsilon$-Kronecker sets with $\varepsilon\,{<}\,2$ do not contain arbitrarily long arithmetic progressions or large squares. When $\varepsilon\,{<}\,\sqrt 2$ they can contain only a bounded number of pairs with common differences and their step length tends to infinity. Related results and examples are given to show the sharpness of these results.

If the subset $E$ of ${\bb T}$ supports a synthesizable $p$-pseudofunction $T$, then the restriction $A_p(E)$ to $E$ of the Herz algebra on ${\bb T}$ is not Arens regular. The proof is direct, by brute force, and does not show the existence of Day points for $A_p(E)$.

Let X and Y be metrizable compact spaces and μ and v be nonzero continuous measures on X and Y, respectively. Then there is no bounded operator from the space of bimeasures BM(X, Y) onto the closed subspace of BM(X, Y) generated by L1 (μ X v); in particular, if X and Fare nondiscrete locally compact groups, then there is no bounded projection from BM(X, Y) onto the closed subspace of BM(X, Y) generated by L1 (X X Y).

We show that if μ is a measure on the LCA group G whose Gelfand transform vanishes off the set Σ of symmetric maximal ideals, then μ µ M0(G), that is, then the Fourier-Steiltjes transform of μ vanishes at infinity. This result is then used to show μ µ L1(G)½.

This is proved: if B is a commutative Banach algebra with identity, then the non-invertible elements of B are weakly dense in B if and only if the maximal ideal space of B is infinite.