Over-Specification of Small Cardinalities in Referential Communication

The paper presents experimental evidence for the over-specification of small cardinalities in referential communication. The first experiment shows that when presented with a small set (2, 3, or 4) of unique objects, the speaker includes a numeral denoting a small cardinality into the description of given objects, although it is over-informative for the hearer (e.g., “three stars”). On the contrary, when presented with a large set of unique objects, the speaker does not include a numeral denoting a large cardinality into their description, so she produces a bare plural (e.g., “stars”). The effect of small cardinalities resembles the effect of over-specifying color in referential communication, which has been extensively studied in recent years (cf. Tarenskeen et al., 2015; Rubio-Fernández, 2016, among many others). This suggests that, like color, small cardinalities are absolute and salient. The second experiment demonstrates that when presented with an identical small set of monochrome objects, the speaker over-specifies a small cardinality to a much greater extent than a color. This suggests that small cardinalities are even more salient than color. The third experiment reveals that when slides are flashed on the screen one by one, highlighted objects of small cardinalities are still over-specified. We argue that a plausible explanation for the salience of small cardinalities is a subitizing effect, which is the human capacity to instantaneously grasp small cardinalities.

Cardinal estimates involving the weak Lindelöf game

We show that if X is a first-countable Urysohn space where player II has a winning strategy in the game $$G^{\omega _1}_1({\mathcal {O}}, {\mathcal {O}}_D)$$ G 1 ω 1 ( O , O D ) (the weak Lindelöf game of length $$\omega _1$$ ω 1 ) then X has cardinality at most continuum. This may be considered a partial answer to an old question of Bell, Ginsburg and Woods. It is also the best result of this kind since there are Hausdorff first-countable spaces of arbitrarily large cardinality where player II has a winning strategy even in the weak Lindelöf game of countable length. We also tackle the problem of finding a bound on the cardinality of a first-countable space where player II has a winning strategy in the game $$G^{\omega _1}_{fin}({\mathcal {O}}, {\mathcal {O}}_D)$$ G fin ω 1 ( O , O D ) , providing some partial answers to it. We finish by constructing an example of a compact space where player II does not have a winning strategy in the weak Lindelöf game of length $$\omega _1$$ ω 1 .

Performance Analysis of a Novel 2-D Code in the Network Access Segment

In this paper, the performance of one-coincidence frequency hopping code/quadratic congruence code (OCFHC/QCC) is analysed and compared with the existing codes. The impact of variation in code weight, code length and number of wavelengths on error probability is evaluated. It is observed that for the same code weight, code length and number of available wavelengths, error probability of OCFHC/QCC is better than the existing codes under certain conditions. The large cardinality, simple design methodology and good performance suggest its use in local area network, multi-code keying and multi-rate multimedia applications.

A NEW SUM–PRODUCT ESTIMATE IN PRIME FIELDS

We obtain a new sum–product estimate in prime fields for sets of large cardinality. In particular, we show that if$A\subseteq \mathbb{F}_{p}$satisfies$|A|\leq p^{64/117}$then$\max \{|A\pm A|,|AA|\}\gtrsim |A|^{39/32}.$Our argument builds on and improves some recent results of Shakan and Shkredov [‘Breaking the 6/5 threshold for sums and products modulo a prime’, Preprint, 2018,arXiv:1806.07091v1] which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy$E^{+}(P)$of some subset$P\subseteq A+A$. Our main novelty comes from reducing the estimation of$E^{+}(P)$to a point–plane incidence bound of Rudnev [‘On the number of incidences between points and planes in three dimensions’,Combinatorica 38(1) (2017), 219–254] rather than a point–line incidence bound used by Shakan and Shkredov.

NILPOTENCY IN UNCOUNTABLE GROUPS

The main purpose of this paper is to investigate the behaviour of uncountable groups of cardinality $\aleph$ in which all proper subgroups of cardinality $\aleph$ are nilpotent. It is proved that such a group $G$ is nilpotent, provided that $G$ has no infinite simple homomorphic images and either $\aleph$ has cofinality strictly larger than $\aleph _{0}$ or the generalized continuum hypothesis is assumed to hold. Furthermore, groups whose proper subgroups of large cardinality are soluble are studied in the last part of the paper.