Compactification of a map which is mapped to itself

We prove that if $T: X \to X$ is a selfmap of a set $X$ such that \bigcap \{T^{n}X: n\in N}\} is a one-point set, then the set $X$ can be endowed with a compact Hausdorff topology so that $T$ is continuous.Comment: 5 page

Rotor interaction in the annulus billiard

Introducing the rotor interaction in the integrable system of the annulus billiard produces a variety of dynamical phenomena, from integrability to ergodicity

Home parenteral nutrition a life-saving therapy in a primary intestinal lymphangiectasia patient affecting the entire GI tract - 3 year follow-up case report

Introduction and Importance: Primary intestinal lymphangiectasia (PIL) is a rare protein-losing gastroenteropathy of unknown etiology, characterized by impaired lymphatic vessels drainage. The pathological changes in PIL result in usually localized or diffuse dilatation of intestinal lacteals, leading to leakage of lymphatic fluid rich of proteins, lymphocytes, and immunoglobulins into the intestinal lumen. PIL may be asymptomatic or mildly symptomatic in moderate forms of the disease. In some patients, though, the outcome may be poor or even life-threatening. This case report demonstrates the severity of protein malnutrition, in some cases, and the extent of GI tract affected, requiring to start PN early and the need for its continuation as home parenteral nutrition (HPN). Case presentation: We present a case of 39-year-old male with Factor V Leiden deficiency, who presented initially with symptoms of malnutrition and anasarca. The diagnosis was confirmed by histopathological findings pathognomonic for PIL from biopsies of the stomach, small intestine and colon. Clinical discussion: The patient was started on low fat, high protein parenteral nutrition from the beginning of the treatment and required a long-term HPN for 3 years, because trials of tapering off and discontinuation of PN led to worsening of the biochemical results and recurrence of symptoms. Patient gradually improved and stabilized with persistent nutritional support. Conclusions: The presented case report shows the magnitude of nutritional support (HPN) needed for severe PIL patients. HPN offers PIL patients with poor outcome and life-threatening complications a chance to improve and lead a normal life

Topological chaos: what may this mean ?

We confront existing definitions of chaos with the state of the art in topological dynamics. The article does not propose any new definition of chaos but, starting from several topological properties that can be reasonably called chaotic, tries to sketch a theoretical view of chaos. Among the main ideas in this article are the distinction between overall chaos and partial chaos, and the fact that some dynamical properties may be considered more chaotic than others

Evaluation of the Acute Oral Toxicity Class of Trinuclear Chromium(III) Glycinate Complex in Rat

Chromium(III) is considered as an essential element playing a role in carbohydrate and lipid metabolism, and various chemical forms of this element are widely used in dietary supplements. A new trinuclear chromium(III) glycinate complex [Cr3O(NH2CH2CO2)6(H2O)3]+NO3− (CrGly), an analogue of Cr3 (trinuclear Cr(III) propionate complex) has been synthesized as a potential source of supplementary Cr. In this study, we evaluated the acute toxicity class of CrGly in Wistar rats applying the OECD 423 procedure. Male and female Wistar rats (n = 12, 6 ♀ and 6 ♂) were given by gavage either a single dose of CrGly 2,000 mg/kg body mass (equals to 300 mg Cr(III)/kg body mass; in aqueous solution) or equivalent volumes of distilled water and fed ad libitum commercial Labofeed B diet, and observed carefully for 14 days, then sacrificed to collect blood and internal organs for biochemical and histologic examination. No death cases were detected. No abnormalities in animal behavior, body mass gains, gross organ histology, or blood morphology and biochemistry were observed. The results demonstrate that LD50 of CrGly is greater than 2,000 mg/kg when administrated orally to rat; thus, this compound appears to belong to the fifth category in the GHS system or the fourth class (“unclassified”) in the EU classification system