Twistor construction of asymptotically hyperbolic Einstein--Weyl spaces

Starting from a real analytic conformal Cartan connection on a real analytic surface $S$, we construct a complex surface $T$ containing a family of pairs of projective lines. Using the structure on $S$ we also construct a complex $3$-space $Z$, such that $Z$ is a twistor space of a self-dual conformal $4$-fold and $T$ is a quotient of $Z$ by a holomorphic local $\mathbb{C}^*$ action. We prove that $T$ is a minitwistor space of an asymptotically hyperbolic Einstein-Weyl space with $S$ as an asymptotic boundary

Klein coverings of genus 2 curves

We investigate the geometry of \'etale $4:1$ coverings of smooth complex genus 2 curves with the monodromy group isomorphic to the Klein four-group. There are two cases, isotropic and non-isotropic depending on the values of the Weil pairing restricted to the group defining the covering. We recall from our previous work \cite{bo} the results concerning the non-isotropic case and fully describe the isotropic case. We show that the necessary information to construct the Klein coverings is encoded in the 6 points on $\mathbb{P}^1$ defining the genus 2 curve. The main result of the paper is the fact that, in both cases the Prym map associated to these coverings is injective. Additionally, we provide a concrete description of the closure of the image of the Prym map inside the corresponding moduli space of polarised abelian varieties.Comment: The proof of injectivity of the Prym map in the non-isotropic case has been corrected. To appear in Trans. AM

S6S^6 and the geometry of nearly K\"ahler 66-manifolds

We review results on and around the almost complex structure on $S^6$, both from a classical and a modern point of view. These notes have been prepared for the Workshop "(Non)-existence of complex structures on $S^6$" (\emph{Erste Marburger Arbeitsgemeinschaft Mathematik -- MAM-1}), held in Marburg in March 2017.Comment: 12 page

C-projective symmetries of submanifolds in quaternionic geometry

The generalized Feix--Kaledin construction shows that c-projective $2n$-manifolds with curvature of type $(1,1)$ are precisely the submanifolds of quaternionic $4n$-manifolds which are fixed points set of a special type of quaternionic $S^1$ action $v$. In this paper, we consider this construction in the presence of infinitesimal symmetries of the two geometries. First, we prove that the submaximally symmetric c-projective model with type $(1,1)$ curvature is a submanifold of a submaximally symmetric quaternionic model, and show how this fits into the construction. We give conditions for when the c-projective symmetries extend from the fixed points set of $v$ to quaternionic symmetries, and we study the quaternionic symmetries of the Calabi-- and Eguchi-Hanson hyperk\"ahler structures, showing that in some cases all quaternionic symmetries are obtained in this way

Zarys poglądów społeczno-etycznych Szymona Budnego

Simon Budny is the one of the most significant Antitrinitarian cogitators. He enunciated his views on the grounds of the Bible contents. By the incisive interpretation, he found out the rationale for the Republic of Nobles polity and for some of the values, which should exist in the society. In this area he emphasized such qualities as: religious tolerance, justice, charity and diligence.Szymon Budny jest jednym z bardziej znaczących myślicieli ariańskich. Swoje poglądy społeczno-etyczne sformułował na podstawie tekstów z Pisma Świętego. Umiejętnie je interpretując, znalazł uzasadnienie dla ówczesnego ustroju Rzeczypospolitej szlacheckiej, a także dla pewnych wartości, jakie powinny funkcjonować w społeczeństwie. W tym obszarze akcentował takie przymioty, jak: tolerancja religijna, sprawiedliwość, miłosierdzie i pracowitość

Is determination of transition zone volume by transrectal ultrasound in patients with clinically benign prostatic enlargement sufficiently reliable in the clinical setting?

INTRODUCTION: Controversies exist regarding the accuracy of transrectal ultrasound (TRUS) determination of transition zone volume (TZV) when compared with enucleated adenoma weight. AIM: To determine the accuracy and reliability of measurements of the TZV by TRUS, by comparing preoperative radiological findings with the enucleated prostate adenoma volume, measured by fluid displacement volumetry (FDV), after retropubic prostatectomy performed by the Millin method, and, moreover, to evaluate changes in the surgical capsule size in the intermediate postoperative period. MATERIAL AND METHODS: We measured TZV preoperatively using TRUS and postoperatively with FDV in 112 patients who underwent retroperitoneal prostatectomy for benign prostatic hyperplasia (BPH). RESULTS: The TRUS volume correlated well with specimen volumes (r = 0.945, p < 0.0001). The median (quartile 1, quartile 3) absolute error was 7.35 ml (4.15 ml, 9.28 ml) and the median percent error was 9.12% (4.75%, 14.98%). Percent error, but not absolute error, was significantly related to TRUS TZV (p < 0.001 and 0.217, respectively). Adenomas > 80 cc were associated with lower percent error. The median volume of the residual prostate tissue measured 3.5 years after prostatectomy was 92.65 cc (65.75 cc, 109.58 cc), whereas the median volume of the surgical capsule, depending on the equation used for its calculation, was 24.80 cc (16.25 cc, 37.37 cc) and 31.43 cc (23.14 cc, 43.32 cc). CONCLUSIONS: The TRUS TZV correlated well with values determined by FDV. It can be reliably used in clinical management of BPH