Partial isometries, duality, and determinantal point processes

A determinantal point process (DPP) is an ensemble of random nonnegative-integer-valued Radon measures [Formula: see text] on a space [Formula: see text] with measure [Formula: see text], whose correlation functions are all given by determinants specified by an integral kernel [Formula: see text] called the correlation kernel. We consider a pair of Hilbert spaces, [Formula: see text], which are assumed to be realized as [Formula: see text]-spaces, [Formula: see text], [Formula: see text], and introduce a bounded linear operator [Formula: see text] and its adjoint [Formula: see text]. We show that if [Formula: see text] is a partial isometry of locally Hilbert–Schmidt class, then we have a unique DPP [Formula: see text] associated with [Formula: see text]. In addition, if [Formula: see text] is also of locally Hilbert–Schmidt class, then we have a unique pair of DPPs, [Formula: see text], [Formula: see text]. We also give a practical framework which makes [Formula: see text] and [Formula: see text] satisfy the above conditions. Our framework to construct pairs of DPPs implies useful duality relations between DPPs making pairs. For a correlation kernel of a given DPP our formula can provide plural different expressions, which reveal different aspects of the DPP. In order to demonstrate these advantages of our framework as well as to show that the class of DPPs obtained by this method is large enough to study universal structures in a variety of DPPs, we report plenty of examples of DPPs in one-, two- and higher-dimensional spaces [Formula: see text], where several types of weak convergence from finite DPPs to infinite DPPs are given. One-parameter ([Formula: see text]) series of infinite DPPs on [Formula: see text] and [Formula: see text] are discussed, which we call the Euclidean and the Heisenberg families of DPPs, respectively, following the terminologies of Zelditch.

Space curves, X-ranks and cuspidal projections

Let $$X\subset \mathbb {P}^3$$ X ⊂ P 3 be an integral and non-degenerate curve. We say that $$q\in \mathbb {P}^3\setminus X$$ q ∈ P 3 \ X has X-rank 3 if there is no line $$L\subset \mathbb {P}^3$$ L ⊂ P 3 such that $$q\in L$$ q ∈ L and $$\#(L\cap X)\ge 2$$ # ( L ∩ X ) ≥ 2 . We prove that for all hyperelliptic curves of genus $$g\ge 5$$ g ≥ 5 there is a degree $$g+3$$ g + 3 embedding $$X\subset \mathbb {P}^3$$ X ⊂ P 3 with exactly $$2g+2$$ 2 g + 2 points with X-rank 3 and another embedding without points with X-rank 3 but with exactly $$2g+2$$ 2 g + 2 points $$q\in \mathbb {P}^3$$ q ∈ P 3 such that there is a unique pair of points of X spanning a line containing q. We also prove the non-existence of points of X-rank 3 for general curves of bidegree (a, b) in a smooth quadric (except in known exceptional cases) and we give lower bounds for the number of pairs of points of X spanning a line containing a fixed $$q\in \mathbb {P}^3\setminus X$$ q ∈ P 3 \ X . For all integers $$g\ge 5$$ g ≥ 5 , $$x\ge 0$$ x ≥ 0 we prove the existence of a nodal hyperelliptic curve X with geometric genus g, exactly x nodes, $$\deg (X) = x+g+3$$ deg ( X ) = x + g + 3 and having at least $$x+2g+2$$ x + 2 g + 2 points of X-rank 3.

A Corpus-Based Comparison of the Pragmatic Use of Qian and Hou to Examine the Applicability of Space–Time Metaphor Hypothesis in Early Child Mandarin

The Universal Space–Time Mapping Hypothesis suggests that temporal expression is based on spatial metaphor for all human beings. This study examines its applicability in the Chinese language using the data elicited from the Early Childhood Mandarin Corpus (ECMC) (Li and Tse, 2011), which collected the utterances produced by 168 Mandarin-speaking preschoolers in a semistructured play context. The unique pair of Chinese words, qian (前/before/front) and hou (后/after/back), which can be used to express either time (before/after) or space (front/back) in daily communication, was the unit of analysis. The results indicated that: (1) there was a significant age effect in the production of “qian/hou,” indicating that the period before the age of 4.5 may be critical for the development of temporal and spatial expression; (2) the pair was produced to express time (before/after) much earlier than space (front/back), indicating that the expression of time might not necessarily be based on the spatial metaphor; and (3) the pair was used more frequently to express time (before/after) than space (front/back) by the preschoolers, thus challenging the hypothesis.

A new genus of Liocranidae (Arachnida: Araneae) from Tajikistan

A new genus, Platnickgen. n., with three new species, P. shablyaisp. n. (♂, type species), P. astanasp. n. (♀) and P. sangloksp. n. (♀), are described from Tajikistan. The male of the type species has a unique pair of longitudinal ventral postgastral scuta. Females have such scuta also, but they are much shorter. The new genus is placed in Liocranidae Simon, 1897. A discussion on the subfamilies of Liocranidae and comments on the family-group names are provided.

The Rare Richardsitas Betsch (Collembola, Symphypleona, Sminthuridae): A New Species from Australia with Comments on the Genus and on the Sminthurinae

Richardsitas Betsch is a small genus of Sminthurinae with only two species described so far, both from Madagascar. It resembles other Sminthurinae with long antennae, especially Temeritas Richards. Here we provide the first record of Richardsitas from Australia, Richardsitas subferoleum sp. nov., which is similar to R. najtae Betsch and R. griveaudi Betsch in males’ large abdomen chaetotaxy and presence of tenent-hairs on tibiotarsi II–III, but lacks mucronal chaeta and has 28 segments on the fourth antennal segment plus a unique pair of sensilla on the second. We also provide an updated genus diagnosis to Richardsitas, a key to its species, a discussion of the affinities of Temeritas and Richardsitas to other Sminthurinae, and an updated key to this subfamily.

Bisectors determining unique pairs of points in the bidisk

Bisectors are equidistant hypersurfaces between two points and are basic objects in a metric geometry. They play an important part in understanding the action of subgroups of isometries on a metric space. In many metric geometries (spherical, Euclidean, hyperbolic, complex hyperbolic, to name a few) bisectors do not uniquely determine a pair of points, in the following sense: completely different sets of points share a common bisector. The above examples of this non-uniqueness are all rank [Formula: see text] symmetric spaces. However, generically, bisectors in the usual [Formula: see text] metric are such for a unique pair of points in the rank [Formula: see text] geometry [Formula: see text]. This result indicates the striking assertion that non-uniqueness of bisectors holds for “most” geometries.