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Studies of long-term trends in phenology often rely on climatic averages or accumulated heat, overlooking climate variability. Here we test the hypothesis that unusual weather conditions are critical in driving adult insect phenology. First, we generate phenological estimates for Lepidoptera (moths and butterflies) across the Eastern USA, and over a 70 year period, using natural history collections data. Next, we assemble a set of predictors, including the number of unusually warm and cold days prior to, and during, the adult flight period. We then use phylogenetically informed linear mixed effects models to evaluate effects of unusual weather events, climate context, species traits, and their interactions on flight onset, offset and duration. We find increasing numbers of both warm and cold days were strong effects, dramatically increasing flight duration. This strong effect on duration is likely driven by differential onset and termination dynamics. For flight onset, impact of unusual climate conditions is dependent on climatic context, but for flight cessation, more unusually cold days always lead to later termination particularly for multivoltine species. These results show that understanding phenological responses under global change must account for unusual weather events, especially given they are predicted to increase in frequency and severity. Butterfly phenology appears more strongly influenced by extreme weather events, such as unusually hot or cold days, than by mean temperatures over the year. [ABSTRACT FROM AUTHOR]

All finite simple groups of Lie type of rank n over a field of size q, with the possible exception of the Ree groups 2G2(q), have presentations with at most 49 relations and bit-length O(log n + log q). Moreover, An and Sn have presentations with 3 generators, 7 relations and bit- length O (log n), while SL(n, q) has a presentation with 6 generators, 25 relations and bit-length O(logn + logq). [ABSTRACT FROM AUTHOR]

For a finite group G let Γ(G) denote the graph defined on the non-identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. In this paper it is shown that the graph Γ(G) contains a Hamiltonian cycle for many finite groups G. [ABSTRACT FROM PUBLISHER]

Let G ≔ G2( K) be a simple algebraic group of type G2 defined over an algebraically closed field K of characteristic p > 0. Let σ denote a standard Frobenius automorphism of G such that Gσ ≅ G2( q) with q ⩾ 4. In this paper we find all reductive subgroups of G and quasi-simple subgroups of Gσ in the defining characteristic. Our results extend the complete reducibility results of [Liebeck and Seitz, Mem. Amer. Math. Soc. 121: 580, 1996, Theorem 1]. [ABSTRACT FROM AUTHOR]

If G is a group and H is a subgroup of G we write for the lattice of over-groups of H in G. It is an open question whether or not every finite lattice is isomorphic to some finite group interval lattice , where G is finite. We prove that no DΔ( m1, ... , mt)-lattice and few M-lattices have the form , where G is an alternating or symmetric group of prime degree. [ABSTRACT FROM AUTHOR]

We show that a connected split reductive group G over a field of characteristic 0 is determined up to isomorphism by specifying a maximal torus T of G, the set of isomorphism classes of irreducible representations of G, and the character homomorphism from the Grothendieck ring of G to that of T. More precisely, we determine all isomorphisms compatible with the specified data. [ABSTRACT FROM AUTHOR]

Let G be a finite Coxeter group. Using previous results on Weyl groups, and covering the cases of non-crystallographic groups, we show that G has an involution model if and only if all of its irreducible factors are of type A n, B n, D2 n+1, H3, or I2( n). [ABSTRACT FROM AUTHOR]

Let G denote a finite group of permutations of a finite set . Given an orbit function : and a class function : G and their extensions to subsets of and G, consider the quantity m,, which is the maximal value of ()( CG()) over all nontrivial subsets of . Denote by the set of non-empty subsets of which satisfy m()( CG() = m,. Our main result, Theorem 1, states that if contains a unique maximal or minimal element with respect to inclusion, then g = for each g G and CG() is a normal subgroup of G. This result may have potential future applications. As an example of an application of Theorem 1 we prove that if either G is a simple group or it is transitive on , T is a normal subset of G not containing 1, is a G-invariant subset of and , then (see Theorem 6). [ABSTRACT FROM AUTHOR]

In Part I it was shown that if G is a p-group of class k, generated by elements of orders , then a necessary condition for the capability of G is that r > 1 and . It was also shown that when G is the k-nilpotent product of the cyclic groups generated by those elements and k = p = 2 or k < p, then the given conditions are also sufficient. We make a correction related to the small class case, and extend the sufficiency result to k = p for an arbitrary prime p. [ABSTRACT FROM AUTHOR]

In 1878, Jordan showed that a finite subgroup of GL( n, ℂ) must possess an abelian normal subgroup whose index is bounded by a function of n alone. We will give the optimal bound for all n; for n ⩾ 71, it is ( n + 1)!, afforded by the symmetric group S n+1. We prove a ‘replacement theorem’ that enables us to study linear groups by breaking them down into individual primitive constituents and we give detailed information about the structure of the groups that achieve the optimal bounds, for every degree n. Our proof relies on known lower bounds for the degrees of faithful representations of each quasisimple group, depending on the classification of finite simple groups, through the use of the bounds for primitive groups that the author has previously obtained. [ABSTRACT FROM AUTHOR]

The article examines fields, values and character extensions in finite groups. It is stated that the number of irreducible F-valued characters of a finite group G does not need to coincide with the number of conjugacy classes of F-elements of G. It is indicated that in order to get the full Theorem A, it is needed to compare the existence of non-trivial F-valued characters with the existence of non-trivial F-elements of finite groups.

Guralnick, R. P. (2005). Combined molecular and morphological approaches to documenting regional biodiversity and ecological patterns in problematic taxa: a case study in the bivalve group Cyclocalyx from western North America. — Zoologica Scripta, 34, 469 –482. Genetic and morphological data were used to examine regional biodiversity in a taxonomically poorly known group of freshwater bivalves, Cyclocalyx. Cyclocalyx 16S rRNA genes were sampled from multiple habitats across western North America, focusing on an elevation gradient from sea level up to 3400 m. From this sampling and previous reference datasets, a phylogenetic hypothesis was constructed and lineages delineated, leading to an estimate of regional biodiversity. Morphometric analyses of shell outlines were used to test whether lineages from the gene tree were significantly different in shape from one another, and whether shell shape can be used to predict lineage affiliation. Finally, broad habitat variables were plotted onto the trees and examined in an evolutionary context. The results uncovered many unnamed, genetically and morphologically distinct lineages, remarkable range extensions, and an unexpected lack of abundance of some lineages. The results also show patterns with basal lineages restricted to high-elevation lakes and more nested lineages at lower elevations or with very wide tolerances. This work challenges almost all previous concepts of regional biodiversity of this lineage, while providing a foundation for future hypothesis testing about the ecology and evolution of the group. It also documents an approach to documenting regional biodiversity that can be applied to other taxa. [ABSTRACT FROM AUTHOR]

A group is called capable if it is a central factor group. We consider the capability of nilpotent products of cyclic groups, and obtain a generalization of a theorem of Baer for the small class case. The approach is also used to obtain some recent results on the capability of certain nilpotent groups of class 2. We also establish a necessary condition for the capability of an arbitrary p-group of class k, and some further results. [ABSTRACT FROM AUTHOR]

Let $G$ be a finite group. It is proved that if the probability that two randomly chosen elements of $G$ generate a soluble group is greater than $\frac{11}{30}$ then $G$ itself is soluble. The bound is sharp, since two elements of the alternating group $A_5$ generate $A_5$ with probability $\frac{11}{30}$. Similar probabilistic statements are proved concerning nilpotency and the property of having odd order. It is also proved that there is a number $\kappa$, strictly between $0$ and $1$, with the following property. Let $\cal X$ be any class of finite groups which is closed for subgroups, quotient groups and extensions. If the probability that two randomly chosen elements of $G$ generate a group in $\cal X$ is greater than $\kappa$ then $G$ is in $\cal X$. The proofs use the classification of the finite simple groups and also some of the detailed information now available concerning maximal subgroups of finite almost simple groups. 1991 Mathematics Subject Classification: 20F16, 20D06, 20D08, 60B99. [ABSTRACT FROM AUTHOR]

Let G be a finite transitive group of rank r . We give a short proof that the proportion of derangements in G is at most 1 − 1 / r and we classify the permutation groups attaining this bound. [ABSTRACT FROM AUTHOR]

Aim: Quantifying the phylogenetic diversity of temperate trees is essential for understanding the processes that have shaped the modern distribution of temperate broadleaf forest and other major forest biomes. Here, we focus on Fagales, an iconic member of forests worldwide, to uncover global diversity and endemism patterns and investigate the distribution of root nodule symbiosis (RNS), an important morphological specialisation in this clade, as a key factor behind these patterns. Location: Global. Taxon: Fagales. Methods: We combined phylogenetic data covering 60.2% of living species, fine‐scale distribution models covering 90% of species, and nodulation data covering all species to investigate the distribution of species richness and phylogenetic diversity at fine spatial scales compared to the distribution of RNS. We identify abiotic environmental factors associated with RNS and with Fagales diversity in general. Results: We find the highest species richness in temperate east Asia, eastern North America, and equatorial montane regions of Asia and Central America. By contrast, relative phylogenetic diversity (RPD) is highest at higher latitudes, where RNS also predominates. We found a strong spatial structuring of regionalisations of Fagales floras, reflecting distinct Northern and Southern Hemisphere floras (except a unique Afro‐Boreal region), each with distinct RNS‐environment relationships. Main Conclusions: Although species richness and phylogenetic regionalisation for Fagales accord well with traditional biogeographic concepts for temperate forests, this is not the case for RPD. RNS is almost universal in the highest RPD regions, which may reflect ecological filtering promoting RNS in these regions. Our results highlight the utility of global‐scale, clade‐specific spatial phylogenetics and its utility for understanding drivers of diversity in species‐rich clades. [ABSTRACT FROM AUTHOR]

For many taxa, determining conservation status and priority is impeded by lack of adequate taxonomic and range data. For these problematic groups, we propose combining molecular techniques with careful geographic sampling to evaluate the validity, extent and phylogenetic relatedness of the proposed units of diversity. We employed such a strategy to document monophyletic lineages, range extents and phylogenetic relatedness for talus snails (genus Sonorella) in the Pinaleño Mountains of Arizona, an isolated range that has the most vertical relief of any of the sky islands in Arizona. Three of the four species found in the Pinaleño Mountains have been considered candidate species for protection under the Endangered Species Act. Further, one of these taxa, Sonorella macrophallus, is of particular concern and was protected under an USFS conservation agreement until 2004, due to its presumed endemicity to a narrow portion of one canyon. We collected a large dataset of 12S and COI mitochondrial DNA, and subsamples of reproductive morphology from specimens collected throughout the Pinaleños and from adjacent ranges (e.g. the Huachucas, Chiricahuas and Santa Catalinas). We generated a phylogeny based on the mitochondrial data, and matched clades with named species utilizing reproductive morphology. Our results show that both S. macrophallus and Sonorella imitator are relatively widespread across the Pinaleños while Sonorella grahamensis and Sonorella christenseni are restricted to very small areas. These results dramatically change our previous knowledge about range extents, especially for S. macrophallus. Given these results, land managers may need to reassess the status of all four Sonorella species. Finally, all Sonorella species from the Pinaleños are more closely related to each other than to other taxa on other ranges. This result strongly suggests that diversification of the four Sonorella species in the Pinaleños occurred in situ. [ABSTRACT FROM AUTHOR]

We prove that, for a positive integer n and subgroup H of automorphisms of a cyclic group Z of order n, there is up to isomorphism a unique connected circulant digraph based on Z admitting an arc-transitive action of Z ⋊ H. We refine the Kovács–Li classification of arc-transitive circulants to determine those digraphs with automorphism group larger than Z ⋊ H. As an application we construct, for each prime power q, a digraph with q – 1 vertices and automorphism group equal to the semilinear group ΓL(1, q), thus proving that ΓL(1, q) is 2-closed in the sense of Wielandt. [ABSTRACT FROM AUTHOR]

Let G be a permutation group acting on a finite set Ω. An uncovering-by-bases (or UBB) for G is a set of bases for G such that any r-subset of Ω is disjoint from at least one base in , where , for d the minimum degree of G. The single-orbit conjecture asserts that for any finite permutation group G, there exists a UBB for G contained in a single orbit of G on its irredundant bases. We prove a case of this conjecture, for when G is k-transitive and has a base of size k + 1. Furthermore, in the more restricted case when G is primitive and has a base of size 2, we show how to construct a UBB of minimum possible size. [ABSTRACT FROM AUTHOR]

Let GL and U denote the finite general linear and unitary groups extended by the transpose inverse automorphism, respectively, where q is a power of the prime p. Let n be odd, and let χ be an irreducible character of either of these groups which is an extension of a real-valued character of GL or U. Let yτ be an element of GL or U such that ( yτ)2 is regular unipotent in GL or U, respectively. We show that is prime to p and otherwise. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way. [ABSTRACT FROM AUTHOR]

Pálfy proved that given a solvable group G and a set of prime divisors of character degrees of G of cardinality at least 3, there exist two different primes such that pq divides some character degree. The solvability hypothesis cannot be removed from Pálfy's theorem, but we show that the same conclusion holds for arbitrary finite groups if . [ABSTRACT FROM AUTHOR]

In this paper we classify irreducible representations of quasi-simple groups with cyclic Sylow p-subgroup P = 〈 g〉 such that ( g) has at least one eigenvalue of multiplicity 1. [ABSTRACT FROM AUTHOR]

We establish linear bounds for the ρ-σ conjecture for irreducible character degrees and conjugacy class sizes of any finite group. [ABSTRACT FROM AUTHOR]

Various combinatorial equivalents are given to the Lebesgue state in archimedean lattice-ordered groups with order-unit. The proofs use piecewise linear functions on polyhedra with rational vertices. [ABSTRACT FROM AUTHOR]

We discuss finiteness properties of a profinite group G whose probabilistic zeta function PG( s) is rational. In particular we prove that if PG( s) is rational and G has a finite number of non-alternating and non-abelian composition factors in a given composition series, then G/Frat( G) is finite. [ABSTRACT FROM AUTHOR]

In the Kourovka Notebook, Deaconescu asks if |Aut G| ≥ φ(| G|) for all finite groups G, where φ denotes the Euler totient function; and whether G is cyclic whenever |Aut G| = φ(| G|). In an earlier paper we have answered both questions in the negative, and shown that |Aut G|/φ(| G|) can be made arbitrarily small. Here we show that these results remain true if G is restricted to being perfect, or soluble. The problem remains open when G is supersoluble, or nilpotent. [ABSTRACT FROM AUTHOR]

Presents a study on graphs associated with certain subsets of character degrees. Definitions of the graphs; Properties of the graphs; Results of the graphs.

The complex forces that shape butterfly wings have long been a subject of experimental and comparative research. Butterflies use their wings for flight, camouflage, mate recognition, warning, and mimicry. However, general patterns and correlations among wing shape and size evolution are still poorly understood. We collected geometric morphometric measurements from over 1400 digitized museum specimens of Papilio swallowtails and combined them with phylogenetic data to test two hypotheses: 1) forewing shape and size evolve independently of hindwing shape and size and 2) wing size evolves more quickly than wing shape. We also determined the major axes of wing shape variation and discovered that most shape variability occurs in hindwing tails and adjacent areas. We conclude that forewing shape and size are functionally and biomechanically constrained, whereas hindwings are more labile, perhaps in response to disruptive selective pressure for Batesian mimicry or against predation. The development of a significant, re-usable, digitized data resource will enable further investigation on tradeoffs between flight performance and ecological selective pressures, along with the degree to which intraspecific, local-scale selection may explain macroevolutionary patterns. [Batesian mimicry; Lepidoptera; geometric morphometrics; museum specimens.] [ABSTRACT FROM AUTHOR]

The time is ripe for a comprehensive mission to explore and document Earth's species. This calls for a campaign to educate and inspire the next generation of professional and citizen species explorers, investments in cyber-infrastructure and collections to meet the unique needs of the producers and consumers of taxonomic information, and the formation and coordination of a multi-institutional, international, transdisciplinary community of researchers, scholars and engineers with the shared objective of creating a comprehensive inventory of species and detailed map of the biosphere. We conclude that an ambitious goal to describe 10 million species in less than 50 years is attainable based on the strength of 250 years of progress, worldwide collections, existing experts, technological innovation and collaborative teamwork. Existing digitization projects are overcoming obstacles of the past, facilitating collaboration and mobilizing literature, data, images and specimens through cyber technologies. Charting the biosphere is enormously complex, yet necessary expertise can be found through partnerships with engineers, information scientists, sociologists, ecologists, climate scientists, conservation biologists, industrial project managers and taxon specialists, from agrostologists to zoophytologists. Benefits to society of the proposed mission would be profound, immediate and enduring, from detection of early responses of flora and fauna to climate change to opening access to evolutionary designs for solutions to countless practical problems. The impacts on the biodiversity, environmental and evolutionary sciences would be transformative, from ecosystem models calibrated in detail to comprehensive understanding of the origin and evolution of life over its 3.8 billion year history. The resultant cyber-enabled taxonomy, or cybertaxonomy, would open access to biodiversity data to developing nations, assure access to reliable data about species, and change how scientists and citizens alike access, use and think about biological diversity information. [ABSTRACT FROM AUTHOR]

Let G be a finite non-abelian group satisfying Φ( G) = 1 and denote by U the nilpotent residual of G. In this paper, we prove that if G is of odd order then , and if G is of even order not divisible by a Mersenne or a Fermat prime then . These results are best possible and the assumption Φ( G) = 1 cannot be omitted. [ABSTRACT FROM AUTHOR]

We study Coxeter groups from which there Is a natural map onto a symmetric group. Such groups have natural quotient groups related to presentations of the symmetric group on an arbitrary set T of transpositions. These quotients, denoted here by CY(T), are a special case of the generalized Coxeter groups defined in, and also arise in the computation of certain invariants of surfaces. We use a surprising action of Sn on the kernel of the surjection CY(T)→Sn to show that this kernel embeds in the direct product of n copies of the free group π1(T), except when T is the full set of transpositions in S4. As a result, we show that each group CY(T) either is virtually Abelian or contains a non-Abelian free subgroup. [ABSTRACT FROM AUTHOR]