A generic C1C^1 map has no absolutely continuous invariant probability measure

Let $M$ be a smooth compact manifold (maybe with boundary, maybe disconnected) of any dimension $d \ge 1$. We consider the set of $C^1$ maps $f:M\to M$ which have no absolutely continuous (with respect to Lebesgue) invariant probability measure. We show that this is a residual (dense $G_\delta) set in the$C^1$ topology. In the course of the proof, we need a generalization of the usual Rokhlin tower lemma to non-invariant measures. That result may be of independent interest.Comment: 12 page

Pulmonary talc granulomatosis mimicking malignant disease 30 years after last exposure: a case report

<p>Abstract</p> <p>Introduction</p> <p>Pulmonary talc granulomatosis is a rare disorder characterized by the development of foreign body granuloma secondary to talc exposure. Previous case reports have documented the illness in current intravenous drug users who inject medications intended for oral use. We present a rare case of the disease in a patient with a distant history of heroin abuse who presented initially with history and imaging findings highly suggestive of malignancy.</p> <p>Case presentation</p> <p>A 53-year-old man reported a 4-month history of increasing dyspnea and weight loss. He had a long history of smoking and admission chest X-ray revealed a density in the right hemithorax. Computed tomography confirmed a probable mass with further speculated opacities in both lung fields suspicious for malignant spread. Biopsies obtained using endobronchial ultrasound-guided aspiration returned negative for malignancy and showed bronchial epithelial cells with foreign body giant cell reaction and polarizable birefringent talc crystals.</p> <p>Conclusion</p> <p>This case demonstrates a rare presentation of talc granulomatosis three decades after the last likely exposure. The history and imaging findings in a chronic smoker were initially strongly suggestive of malignant disease, and we recommend that talc-induced lung disease is considered in any patient with multiple scattered pulmonary lesions and a history of intravenous drug use. Confirmation of the disease by biopsy is essential, but unfortunately there are few successful proven management options for patients with worsening disease.</p

A Note on Tropical Triangles in the Plane

We define transversal tropical triangles (affine and projective) and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically independent and they tropically span a classical hexagon whose sides have slopes ∞, 0, 1. Using this classical hexagon, we determine a parameter space for transversal tropical triangles. The coordinates of the vertices of a transversal tropical triangle determine a tropically regular matrix. Triangulations of the tropical plane are obtained

Dispersion of sound velocity in certain organic liquids /

Prepared for the National Science Foundation Russian Science Translation-Dictionary Project, Columbia University, July, 1953--page 5.Translated from Doklady Akademii Nauk SSSR, 92, 285-88 (1953)--title page."February 1954."Includes bibliographical references (page 4).Mode of access: Internet