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Every synchronising permutation group is primitive and of one of three types: affine, almost simple, or diagonal. We exhibit the first known example of a synchronising diagonal type group. More precisely, we show that $\mathrm{PSL}(2,q)\times \mathrm{PSL}(2,q)$ acting in its diagonal action on $\mathrm{PSL}(2,q)$ is separating, and hence synchronising, for $q=13$ and $q=17$. Furthermore, we show that such groups are non-spreading for all prime powers $q$.

H. L. Skala (1992) gave the first elegant first-order axiom system for hyperbolic geometry by replacing Menger’s axiom involving projectivities with the theorems of Pappus and Desargues for the hyperbolic plane. In so doing, Skala showed that hyperbolic geometry is incidence geometry. We improve upon Skala’s formulation by doing away with Pappus and Desargues altogether, by substituting for them two simpler axioms.

PREMISE Plant domestication can be detected when transport, use, and manipulation of propagules impact reproductive functionality, especially in species with self-incompatible breeding systems. METHODS Evidence for human-caused founder effect in the Four Corners potato (Solanum jamesii Torr.) was examined by conducting 526 controlled matings between archaeological and non-archaeological populations from field-collected tubers grown in a greenhouse. Specimens from 24 major herbaria and collection records from >160 populations were examined to determine which produced fruits. RESULTS Archaeological populations did not produce any fruits when self-crossed or outcrossed between individuals from the same source. A weak ability to self- or outcross within populations was observed in non-archaeological populations. Outcrossing between archaeological and non-archaeological populations, however, produced fully formed, seed-containing fruits, especially with a non-archaeological pollen source. Fruit formation was observed in 51 of 162 occurrences, with minimal evidence of constraint by monsoonal drought, lack of pollinators, or spatial separation of suitable partners. Some archaeological populations (especially those along ancient trade routes) had records of fruit production (Chaco Canyon), while others (those in northern Arizona, western Colorado, and southern Utah) did not. CONCLUSIONS The present study suggests that archaeological populations could have different origins at different times-some descending directly from large gene pools to the south and others derived from gardens already established around occupations. The latter experienced a chain of founder events, which presumably would further reduce genetic diversity and mating capability. Consequently, some archaeological populations lack the genetic ability to sexually reproduce, likely as the result of human-caused founder effect.

Potato has about 100 related wild Solanum species growing naturally in the Americas. The US Potato Genebank aims to keep samples useful for research and breeding to improve the crop, often in the form of botanical seed families. A key component of genebank efficiency is assessing diversity within and among populations, and DNA marker sequence diversity is a powerful proxy for trait diversity. We previously reported on three factors which can cause under-estimation of heterozygosity: ascertainment, allele frequency, and ploidy bias. We here report, using GBS data for four diploid potato species, that average percent of apparent heterozygosity increases as data is more complete—the maximum difference was 2% heterozygotes when only a few individuals are called, to 36% when nearly all individuals were called. However, there was evidence that estimates of average heterozygosity based only on loci for which every individual has data can also be biased upward. Implausibly high levels of heterozygosity suggest non-segregating non-homologous SNPs, which occurred as 5–9% of all loci with complete data. We propose that best estimates of average heterozygosity in unselected seedlings should be based on loci with data for all samples after eliminating those loci that appear to be artificially fixed as heterozygous, which reduces observed heterozygote frequency by 16–26%. On that basis, the wild species examined have similar heterozygosity to the cultivated phureja.

A partial linear space is a pair $(\mathcal{P},\mathcal{L})$ where $\mathcal{P}$ is a non-empty set of points and $\mathcal{L}$ is a collection of subsets of $\mathcal{P}$ called lines such that any two distinct points are contained in at most one line, and every line contains at least two points. A partial linear space is proper when it is not a linear space or a graph. A group of automorphisms $G$ of a proper partial linear space acts transitively on ordered pairs of distinct collinear points and ordered pairs of distinct non-collinear points precisely when $G$ is transitive of rank 3 on points. In this paper, we classify the finite proper partial linear spaces that admit rank 3 affine primitive automorphism groups, except for certain families of small groups, including subgroups of $A\Gamma L_1(q)$. Up to these exceptions, this completes the classification of the finite proper partial linear spaces admitting rank 3 primitive automorphism groups. We also provide a more detailed version of the classification of the rank 3 affine primitive permutation groups, which may be of independent interest., Comment: In this version, we have removed the assumption $V\leq H$ from 18.1 (old 13.2) and we have a new elementary proof of 10.10 (old 13.1). We have also reorganised some of the sections and made minor revisions throughout. 69 pages, 1 figure

Potato has about 100 related wild Solanum species growing naturally in the Americas. Solanum jamesii (jam), native to the southwest USA and Mexico, has many valuable traits for breeding, but making hybrids is extremely difficult. We investigated the approach of using the bridge species Solanum verrucosum (ver). A survey of all ver populations in the US Potato Genebank identified the best females. The standard bridge crossing technique using emasculation of ver and “mentor” double pollination after first pollinating with jam was very inefficient for most ver because: 1) emasculation depressed seedset in ver females, 2) despite very careful emasculation many accidental ver selfs resulted, and 3) mentor pollination produced many unwanted hybrids with the mentor pollen parent. We therefore produced populations of ver by backcrossing five generations into tuberosum cytoplasm. The BC5 lines are very vigorous in growth and flowering, but having very low male fertility and complete self-incompatibility they can be mass crossed with jam pollen without emasculation. Even with no mentor pollen, most pistils of verBC5 pollinated with jam produce a small fruit with an average of less than one normal seed. Resulting seedlings were confirmed ver-jam hybrids by DNA markers.

Ostrom and Wagner (1959) proved that if the automorphism group $G$ of a finite projective plane $\pi$ acts $2$-transitively on the points of $\pi$, then $\pi$ is isomorphic to the Desarguesian projective plane and $G$ is isomorphic to $\mathrm{P\Gamma L}(3,q)$ (for some prime-power $q$). In the more general case of a finite rank $2$ irreducible spherical building, also known as a \emph{generalized polygon}, the theorem of Fong and Seitz (1973) gave a classification of the \emph{Moufang} examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-transitive generalized quadrangles up to duality. Recently, the authors made progress toward this conjecture by classifying those finite generalized quadrangles which have an automorphism group $G$ acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to $G$ being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines., Comment: Sections 3-6 rewritten and restructured to fill in gaps in, streamline, and avoid repetition of arguments. To appear in Transactions of the American Mathematical Society

Potato cultivars are propagated by tubers which are sensitive to damage by freezing. Potato has about 100 related wild Solanum species growing naturally in the Americas. When tubers of a spectrum of these species were slowly cooled, most were killed at a few degrees below 0 C. Only tubers of Solanum jamesii, native to the southwest USA, remained firm, sprouted and grew after one week exposure to freezing temperatures as cold as -15C. Differential Thermal Analysis was performed to detect low temperature exotherms (LTEs) in jamesii and similar-sized tubers of cultivars Russet Burbank and Snowden. LTEs, indicating supercooling, were detected only in the jamesii tubers. Survival of potato tubers at freezing temperatures could be useful in several ways. Even non-lethal cold exposure of potato tubers at harvest reduces quality for eating and for seed, but storage at freezing temperatures without damage could help preserve tuber dormancy and limit diseases.

Useful genetic diversity found in wild potato germplasm is important for progress in potato breeding and science. An effective way to maximize its utilization is through characterization of ex situ collections. For example, marker-based core subsets from large collections are an effective strategy to accelerate evaluation, utilize germplasm, and enhance cost-effectiveness. This project used AFLP markers to create a core subset for the US Potato Genebank’s collection of S. demissum. This species from Mexico and Guatemala is known for excellent levels of late blight resistance and frost tolerance. A total of 149 accessions with diverse geographic origin were characterized with 1403 AFLP markers; these markers were effective on determining genetic associations among the accessions. For the core subset, the AFLP loci were considered as traits, with the presence of the marker as the required state to include in a core. The results of the analyses identified that 10 of the accessions of S. demissum captured 83% of all the markers detected in the whole collection. The inclusion of 28 additional accessions, which had between 1 to 5 unique markers, resulted in a core set of 38 accessions that captured 96% of all the marker diversity. We expect this can benefit the genebank and the potato community by opening opportunities for research, trait screenings, and marker association studies.

Crop genebanks keep a sample of the diversity in wild and cultivated germplasm. Knowing the structure of diversity within and among populations informs best decisions on efficient collecting, preservation and use. The US Potato Genebank has over 90 Solanum species, ranging in representation from only a few populations to several hundred. In a previous study we showed that populations of three species captured nearly all AFLP bands in polymorphic loci when represented by 100 populations that had been added over time to the genebank. In the current study we add analysis of two additional species and propose a simple metric, L100, to compare the expected marker diversity capture of 100 populations among different species. L100 detects when there is less population overlap and more rare markers, indicating that more populations are needed to capture the available diversity in a species.

Solanum jamesii is a wild potato found in the US southwest. There is ample evidence that this potato was used by ancestral Puebloans as a food source, where some researchers think it was used as a starvation food while others consider it to be regular food source. Currently this potato is being grown by Native Americans, notably the Navajo, as a specialty food as well as a food crop. There are several attributes to this potato that make it especially suitable for development as our climate changes and food needs become more demanding, including its drought tolerance and ability to be crossed with other wild potato species and cultivars.

We correct some lemmas in Section 3 and Section 4 in ’New non-existence proofs for ovoids of Hermitian polar spaces and hyperbolic quadrics’, Ann. Comb., 21(1):25–42, 2017.

AFLP markers were used to build a core subset in 144 accessions of the United States Potato Genebank (USPG) ex situ collection of the species formerly known as Solanum phureja (now reclassified as Solanum tuberosum L. subsp. andigenum). A core subset aims to sample the minimum number possible of non-redundant germplasm units capturing the maximum diversity of the entire collection. Characterization and trait evaluation of a core subset is expected to be easier and more effective when compared to the entire germplasm collection. In this study 1534 polymorphic AFLP markers were generated and demonstrated to be useful to determine genetic relationships in the materials used. All the accessions were clearly discriminated from each other with genetic similarity levels ranging from 62 to 89%, and no duplicate germplasm samples were detected. To create the core subset, the presence of the AFLP marker was considered as the genetic trait to retain and the selection of accessions was determined by their rankings on the number of most markers added to the core. The results of this selection process revealed that the 9 top-ranked accessions of phureja, when combined, achieved the benchmark of capturing 85% of the markers detected in the complete collection. Further selection to increase the percentage of marker capture added 17 accessions with unique marker contributions which resulted in a final core subset of 26 accessions capturing 96% of the marker diversity. A Principal Component analysis determined that spatial distribution of diversity in the core was a proportional sampling of the entire diversity of the collection validating the selecting approach used. A review of the public database for valuable traits for the USPG phureja germplasm revealed that the core subset also included accessions with desirable pest/disease resistances and stress tolerances (in 25 out of the 30 traits reported in the database). The USPG expects this core subset would create opportunities for facilitating evaluation and research by using a reduced set of germplasm units with nearly all the diversity of the entire set.

Genebanks aim to maximize the preservation, classification, evaluation and distribution of germplasm in their collections to researchers and breeders. The wild is the source of most of the diversity available to genebanks, so diversity in the wild and how to best capture it also deserves study. The ultimate in valuable information from the wild would be the discovery of a large robust population at a single location on public land that is very easy to visit and contains a large proportion of the total genetic diversity detected in the species, which we have termed a “MegaPopulation”. We here report such a population of the wild potato species Solanum fendleri in the USA (also known as S. stoloniferum) on Mount Lemmon at the top of the Santa Catalina mountains near Tucson, Arizona.

Let $p$ be an odd prime. We construct a $p$-group $P$ of nilpotency class two, rank seven and exponent $p$, such that $\text{Aut}(P)$ induces $N_{\text{GL}(7,p)}(G_{2}(p))=Z(\text{GL}(7,p))G_{2}(p)$ on the Frattini quotient $P/\unicode[STIX]{x1D6F7}(P)$. The constructed group $P$ is the smallest $p$-group with these properties, having order $p^{14}$, and when $p=3$ our construction gives two nonisomorphic $p$-groups. To show that $P$ satisfies the specified properties, we study the action of $G_{2}(q)$ on the octonion algebra over $\mathbb{F}_{q}$, for each power $q$ of $p$, and explore the reducibility of the exterior square of each irreducible seven-dimensional $\mathbb{F}_{q}[G_{2}(q)]$-module.

A better understanding on how genetic diversity is structured at natural habitats can be helpful for exploration and acquisition of plant germplasm. Historically, studies have relied on DNA markers to elucidate potato genetic diversity. Current advances in genomics are broadening applications allowing the identification of markers linked to genomic regions under selection. Those markers, known as adaptive markers, unlock additional ways to value and organize germplasm diversity. For example, conservation priorities could be given to germplasm units containing markers associated to unique geographic identity, and/or linked to traits of tolerance to abiotic stresses. This study investigated if adaptive marker loci were possible to be identified in a large AFLP marker dataset of ninety-four populations of the wild potato species S. fendleri. These populations originated from six different mountain ranges in southern Arizona, USA. A total of 2094 polymorphic AFLP markers were used to conduct genetic diversity analyses of populations and mountain ranges. Adaptive markers were detected using Bayesian methods which distinguished marker loci departing significantly from frequencies expected under neutral models of genetic differentiation. This identified 16 AFLP loci that were considered to be adaptive. To contrast diversity parameters generated with each set of markers, analyses that included all the 2094 AFLP markers, and only the 16 adaptive markers were conducted. The results showed that both were efficient for establishing genetic associations among populations and mountain ranges. However, adaptive markers were better on revealing geographic patterns and identity which would suggest these markers were linked to selection at the natural sites. An additional test to determine if adaptive markers associated to climate variables found two loci associated to specific climate variables in populations from different regions but sharing similar environmental structure. The distribution of adaptive markers among populations revealed that only two were needed to build a core subset able to keep all the markers. This preliminary assessment shows that adaptive genetic diversity could offer an additional way to measure diversity in potato germplasm and to set up options for conservation and research.

The Potato Introduction Station in Sturgeon Bay Wisconsin has maintained herbarium specimens as part of its operations. These specimens mostly were grown from seeds in the field plots or greenhouses, but also include original specimens made during germplasm collecting expeditions. The herbarium of the Station (herbarium acronym PTIS) has been transferred from the potato genebank in Sturgeon Bay Wisconsin to the Wisconsin State Herbarium (acronym WIS) in Birge Hall, University of Wisconsin-Madison where it is now fully integrated and curated and funded by WIS; PTIS is discontinued and is no longer maintained as a separate herbarium. Separately, maps and taxonomic literature that were used by the potato taxonomist in Madison were transferred to the Potato Station in Sturgeon Bay.

Ploidy manipulation and the resulting need for rapid ploidy screening can be important in potato research and breeding programs. We tested three predictors of ploidy, particularly seeking the quickest, simplest, and most reliable: Chloroplast number per guard cell (C#), guard cell length (GC), and pollen diameter (P), with a total of seven variations in methods of preparation. Time required for each preparation was assessed, and a panel of inexperienced volunteers compared these methods for accuracy using a standard set of coded samples of known ploidy. The common method of counting C# with iodine stain took longer and was no more accurate than observing C# or GC in tap water. GC from tape impressions of the underside of leaves was reliable and has the advantage of permanent slides for later reference. We recommend GC, whether in water, stained, or as tape impressions. GC is significantly different in diploids and tetraploids, but the distributions do overlap, so experience and care in selecting a representative sample of cells contributes to accuracy. The standard measurement of P after staining with aceto-carmine was faster to prep and just as reliable as epidermal methods for some technicians, even with no previous experience. Pursuit of ultra-simplified methods led us to measure P in tap water. Diameters of pollen in plain water are significantly larger, but only for living pollen, suggesting this method might also be developed into a rapid and reliable way to estimate pollen viability.

In this paper we show that if $\theta$ is a $T$-design of an association scheme $(\Omega, \mathcal{R})$, and the Krein parameters $q_{i,j}^h$ vanish for some $h \not \in T$ and all $i, j \not \in T$ ($i, j, h \neq 0$), then $\theta$ consists of precisely half of the vertices of $(\Omega, \mathcal{R})$ or it is a $T'$-design, where $|T'|>|T|$. We then apply this result to various problems in finite geometry. In particular, we show for the first time that nontrivial $m$-ovoids of generalised octagons of order $(s, s^2)$ do not exist. We give short proofs of similar results for (i) partial geometries with certain order conditions; (ii) thick generalised quadrangles of order $(s,s^2)$; (iii) the dual polar spaces $\mathsf{DQ}(2d, q)$, $\mathsf{DW}(2d-1,q)$ and $\mathsf{DH}(2d-1,q^2)$, for $d \ge 3$; (iv) the Penttila-Williford scheme. In the process of (iv), we also consider a natural generalisation of the Penttila-Williford scheme in $\mathsf{Q}^-(2n-1, q)$, $n\geqslant 3$., Comment: This paper builds on part of the doctoral work of the second author under the supervision of the first. The second author acknowledges the support of an Australian Government Research Training Program Scholarship and Australian Research Council Discovery Project DP200101951