Periodic orbits 1-5 of quadratic polynomials on a new coordinate plane

While iterating the quadratic polynomial f_{c}(x)=x^{2}+c the degree of the iterates grows very rapidly, and therefore solving the equations corresponding to periodic orbits becomes very difficult even for periodic orbits with a low period. In this work we present a new iteration model by introducing a change of variables into an (u,v)-plane, which changes situation drastically. As an excellent example of this we can compare equations of orbits period four on (x,c)- and (u,v)-planes. In the latter case, this equation is of degree two with respect to u and it can be solved explicitly. In former case the corresponding equation ((((x^{2}+c)^{2}+c)^{2}+c)^{2}+c-x)/((x^{2}+c)^{2}+c-x)=0 is of degree 12 and it is thus much more difficult to solve

Systematic revision and cladistic analysis of the genus Metabilibitia Roewer, 1912 (Opiliones, Cosmetidae, Cosmetinae)

O gênero Metalibitia Roewer, 1912 é revisado pela primeira vez e uma análise cladística foi feita para testar seu monofiletismo e suas relações internas, além disso, a chave de identificação das espécies e a descrição de uma espécie nova são apresentadas. Uma nova classificação é proposta baseada nos resultados da análise cladística. São reconhecidas nove espécies válidas das 11 espécies nominais, sendo duas sinonímias propostas: Metalibitia maculata (Roewer, 1914) e Metalibitia arenosa (Mello-Leitão, 1939) em Metalibitia argentina (Sørensen, 1884) e uma nova espécie é descrita, Metalibitia/ sp. n. (localidade-tipo: município de Abunã, Porto Velho, Rondônia, Brasil). A matriz de dados utilizada na análise cladística possui dez espécies de Metalibitia e oito terminais como grupo externo, compreende 65 caracteres baseados na morfologia masculina, sendo 18 pertencentes às estruturas morfológicas externas presentes no escudo dorsal e tergitos livres, dos quais dois são caracteres referentes à coloração; 20 referentes à genitália masculina; 21 pertencentes às pernas; cinco pertencentes ao pedipalpo e um referente à quelícera. A análise resultou em duas árvores igualmente parcimoniosas com 198 passos (C.I.= 0,50; R.I.= 0,56), sendo aqui apresentada a árvore de consenso com 204 passos (C.I.= 0,49; R.I.= 0,54), cuja otimização aplicada foi ACCTRAN. Metalibitia constitui um grupo monofilético, sustentado por três sinapomorfias de genitália masculina não homoplásticas e não-ambíguas: espinhos apicais retos na placa ventral do pênis, presença de prolongamento membranoso na região lateral mediana da placa ventral do pênis e inserção da glande na região mediana da placa ventral. Há a formação de dois clados no gênero e dois subclados, estes diferenciados principalmente pela quantidade de tubérculos do trocanter IV.The genus Metalibitia Roewer, 1912 is revised for the first time and a cladistic analysis was performed to test its monophyletism and its internal relationships, a dicotomic key of species was presented and a new species was described. A new classification based on cladistic analysis is proposed. Nine species are considered as valid of the 11 nominal species currently recognized, being synonymyzed Metalibitia maculate (Roewer, 1914) and Metalibitia arenosa (Mello-Leitão, 1939) as Metalibitia argentina (Sørensen, 1884) and a new species is described (type-locality: Abunã, Porto Velho, Rondônia, Brazil). The cladistic analysis was performed with 10 species of Metalibitia and outgroup is composed of eight species. The character matrix comprises 65 morphological characters from males, being 18 belonging to external morphology features on dorsal scute and free tergites, including two features regarding to color, 20 to male genital, 21 to legs, 5 to pedipalp and one to chelicerae. The analysis resulted in two trees equally parsimonious with 198 steps (C.I.= 0.50; R.I.= 0.56), the strict consensus tree with 204 steps (C.I.= 0.49; R.I.= 0.54), and the ACCTRAN otimization was applied. Metalibitia is a monophyletic group, supported for three synapomorphies of the male genitalia non-homoplastics and unambiguous: straight apical setae on the ventral plate of the penis, membranous prolongation on region median lateral on ventral plate of the penis and insert of the penis glans on median region of the ventral plate. Two main groups and two subclades were obtained in genus, whose mainly separation is based on the tubercles of the trochanter IV

Four new species of Cosmetus from Panama, with comments on the systematics of the genus (Opiliones: Cosmetidae)

ABSTRACT Four new sympatric species of Cosmetus Perty, 1833 are described from "Reserva Natural Privada Burbayar, Provinciar Panamá, Panamá" (male holotypes deposited in MIUP). Cosmetus balboa sp. nov. can be distinguished by the combinations of following features: smooth ocularium, larger distal tubercle on pedipalpal femur, coxa I with large ventral tubercle directed upwards, coxa IV with one large dorsoproximal tubercle, two geminate (from base) dorsoapical tubercles with blunt apex. Cosmetus burbayar sp. nov., can be distinguished from other species of the genus by the irregular and discontinuous shape of its yellow spot, extending from lateral anterior to posterior margins and invading prosoma, areas I-III and free tergites. Cosmetus pollera sp. nov. can be distinguished from congeners by two small yellow and two large pairs of spots on prosoma and two other spots on posterior margin of dorsal scutum. Cosmetus tamboritos sp. nov. is distinguished from other species of the genus by the following combinations of characters: having two retrolateral apical tubercles on bulla, one being double size of other; coxa IV lacks patches of a cluster of four tubercles on dorsolateral proximal region and two pointed tubercles fused at their apices; and femur IV with bifid retrolateral apical tubercle. The penis of Cosmetus arietinus (Mello-Leitão, 1940) and C. variolosus Mello-Leitão, 1942 are described for the first time. A table with the main diagnostic features of Cosmetus species is given. We suggest that the spine of area III, sexually dimorphic chelicerae and posterior legs, and pigmentation of dorsal scutum are good diagnosti c features at species level

A new species of Imeri (Opiliones: Stygnidae: Heterostygninae) from the Brazilian Amazon

A new species of Stygnidae Simon, 1879, Imeri ajuba sp. nov., is described based on males and females from Coari, Amazonas, Brazil. The new species, differs from the only other species of the genus, Imerilomanhungae Pinto-da-Rocha & Tourinho, 2012, in having the following characteristics: three white marks on posterior margin of dorsal scutum, one white mark on free tergite I, three pairs of tubercles on area I, one pair of tubercles on area II, femur IV with ventromesal row of five tubercles, truncus of penis with six pairs of setae on lateral and ventral sides and ventral plate with two pairs of distal setae on dorsal portion

Catalogue of Opiliones (Arachnida) types deposited in the Arachnida and Myriapoda collection of the Instituto Butantan, São Paulo, Brazil

Coronato-Ribeiro, Amanda, Pinto-Da-Rocha, Ricardo, Rheims, Cristina Anne (2013): Catalogue of Opiliones (Arachnida) types deposited in the Arachnida and Myriapoda collection of the Instituto Butantan, São Paulo, Brazil. Zootaxa 3637 (5): 501-520, DOI: 10.11646/zootaxa.3637.5.