Factorization in a torus and Riemann-Hilbert problems

A new concept of meromorphic $\Sigma$-factorization, for H\"{o}lder continuous functions defined on a contour $\Gamma$ that is the pullback of $\dot{\mathbb{R}}$ (or the unit circle) in a Riemann surface $\Sigma$ of genus 1, is introduced and studied, and its relations with holomorphic $\Sigma$-factorization are discussed. It is applied to study and solve some scalar Riemann-Hilbert problems in $\Sigma$ and vectorial Riemann-Hilbert problems in $\mathbb{C}$, including Wiener-Hopf matrix factorization, as well as to study some properties of a class of Toeplitz operators with $2 \times 2$ matrix symbols.Comment: accepted for publication in Journal of Mathematical Analysis and Application

Toeplitz operators and Wiener-Hopf factorisation: an introduction

Wiener-Hopf factorisation plays an important role in the theory of Toeplitz operators. We consider here Toeplitz operators in the Hardy spaces $H^p$ of the upper half-plane and we review how their Fredholm properties can be studied in terms of a Wiener-Hopf factorisation of their symbols, obtaining necessary and sufficient conditions for the operator to be Fredholm or invertible, as well as formulae for their inverses or one-sided inverses when these exist. The results are applied to a class of singular integral equations in $L^p(\mathbb R)$.Comment: 26 page

Phase Diagram, Scalar-Pseudoscalar Meson Behavior and Restoration of Symmetries in (2+1) Polyakov-Nambu-Jona-Lasinio Model

We explore the phase diagram and the modification of mesonic observables in a hot and dense medium using the (2+1) Polyakov-Nambu-Jona-Lasinio model. We present the phase diagram in the ($T,\,\mu_B$)-plane, with its isentropic trajectories, paying special attention to the chiral critical end point (CEP). Chiral and deconfinement transitions are examined. The modifications of mesonic observables in the medium are explored as a tool to analyze the effective restoration of chiral symmetry for different regions of the phase diagram. It is shown that the meson masses, namely that of the kaons, change abruptly near the CEP, which can be relevant for its experimental search.Comment: 25 pages, 11 figures, 2 table

Bryophytes from Jalapão, state of Tocantins, northern Brazil

Although occupying almost half the area of Brazil, the northern region of the country remains poorly known bryologically. In an attempt to help remedy this problem, a collecting trip was made to Jalapão Desert, located in the state of Tocantins. A total of 22 taxa were found, 21 are new occurrences to the state and three are new to the northern region of Brazil. Bryum caespiticium Hedw. is listed for the second time from Brazil. A list of the taxa with references of descriptions and illustrations are provided.Embora ocupe quase metade da área do pais, a região norte do Brasil, continua sendo pouco conhecida do ponto de vista briológico. A fim de contribuir com o conhecimento sobre a brioflora da area, uma viajem de coleta foi feita a região do Jalapão, localizado no estado do Tocantins. Um total de 22 taxa foram encontrados, 21 novas ocorrências para o estado e três novas para a região norte. Bryum caespiticium Hedw. é citado pela segunda vez para o Brasil. É fornecida uma lista dos taxa com referências de descrições e ilustrações