Random planar trees and the Jacobian conjecture

We develop a probabilistic approach to the celebrated Jacobian conjecture, which states that any Keller map (i.e. any polynomial mapping $F\colon \mathbb{C}^n \to \mathbb{C}^n$ whose Jacobian determinant is a nonzero constant) has a compositional inverse which is also a polynomial. The Jacobian conjecture may be formulated in terms of a problem involving labellings of rooted trees; we give a new probabilistic derivation of this formulation using multi-type branching processes. Thereafter, we develop a simple and novel approach to the Jacobian conjecture in terms of a problem about shuffling subtrees of $d$-Catalan trees, i.e. planar $d$-ary trees. We also show that, if one can construct a certain Markov chain on large $d$-Catalan trees which updates its value by randomly shuffling certain nearby subtrees, and in such a way that the stationary distribution of this chain is uniform, then the Jacobian conjecture is true. Finally, we show that the subtree shuffling conjecture is true in a certain asymptotic sense, and thereafter use our machinery to prove an approximate version of the Jacobian conjecture, stating that inverses of Keller maps have small power series coefficients for their high degree terms.Comment: 36 pages, 4 figures. Section 2.5 added, Section 3 expanded, further minor edit

External Iliac Vein Aneurysm Treated Via Balloon-Assisted Aneurysmorrhaphy with a Contemporary Review of the Literature

Isolated external iliac vein aneurysm is exceedingly rare, not well-described in the literature, and presents several potential surgical approaches. Herein, we describe the case of a 72-year-old woman who presented with incidentally found 4.3 cm x 3.4 cm x 5.6 cm right external iliac vein aneurysm after undergoing magnetic resonance imaging for orthopedic work-up. She was treated via parallel supra- and infra-inguinal incisions and novel combination of primary aneurysmorrhaphy with intraluminal balloon mandrel-assisted closure. The patient was discharged on postoperative day two, and 6-month follow-up ultrasound showed a normal caliber vessel with normal compressibility, suggesting this technique is safe and effective for appropriately selected patients

Caracterização química de frutos de seleções avançadas de pitaia produzidas no cerrado.

A pitaya é uma fruta exótica que tem ocupado um nicho crescente no mercado de frutas devido a ser uma excelente fonte de vitaminas, minerais,fibras e proteína vegetal. Neste sentido, a pitaya tem sido uma ótima opção para essa diversificação de produtos, pois, além de serem muito atrativos visualmente, seus frutos possuem inúmeras qualidades nutricionais e funcionais, auxiliando no bom funcionamento do sistema digestivo e na redução da resistência à insulina, além de possuir características anti-inflamatórias e antioxidantes (COELHO et al., 2020; IBRAHIM et al., 2018). Com o intuito de disponibilizar para os produtores rurais materiais mais adaptados às condições edafoclimáticas brasileiras, a Embrapa Cerrados desenvolveu seis seleções avançadas de pitaya de diferentes espécies. O programa de melhoramento daspitaya da Embrapa Cerrados iniciado na década de 1990, deu origem aseis seleções superiores de quatroespécies diferentes de pitaya, as quais se apresentaram adaptadas para o cultivo na região do Cerrado Brasileiro (FALEIRO;JUNQUEIRA, 2021). De modo a fornecer aos produtores rurais maiores informações sobre essas variedades, estudos visando avaliar as características químicas de frutos dessas seleções avançadas são de extrema importância, para que, com base nessas informações, o produtor rural possa escolher a variedade que mais se encaixe na demanda do seu mercado consumidor. Neste sentido, objetivou-se avaliar as características químicas dos frutos de seis seleções avançadas de pitaya desenvolvidas pela Embrapa Cerrados

Bronchial artery revascularization in lung transplantation: a systematic review and meta-analysis.

Background: Bronchial artery revascularization (BAR) during lung transplantation has been hypothesized to improve early tracheal healing and delay the onset of bronchiolitis obliterans syndrome (BOS). We aimed to assess the outcomes of BAR after lung transplantation. Methods: Electronic search in Ovid Medline, Cumulative Index of Nursing and Allied Health Literature (CINAHL), Scopus, and Cochrane Controlled Trials Register (CCTR) databases was performed to identify all relevant studies published about lung transplantation with BAR. Studies discussing lung transplantation utilizing BAR were included while those without outcome data such as BOS and survival were excluded. Cohort-level data were extracted and pooled for analysis. A binary outcome meta-analysis of proportions with logit transformation was conducted. Newcastle-Ottawa scale was used for risk of bias assessment. Results: Seven studies were selected for the analysis comprising 143 patients. Mean patient age was 47 (95% CI: 40-55) years. Sixty-one percent (48-72%) were male. Seventy-three percent (65-79%) of patients underwent double lung transplant while 27% (21-25%) underwent single lung transplant. In patients with postoperative angiography, successful BAR was demonstrated in 93% (82-97%) of all assessed conduits. The 30-day/in-hospital mortality was 6% (3-11%). Seventy-nine percent (63-89%) of patients were free from rejection at three months. Eighty-three percent (29-98%) of patients were free from signs of airway ischemia at three and six months. Pooled survival at one year and five years was 87% (78-92%) and 71% (46-87%), respectively, with a mean follow-up time of 21 (3-38) months. Pooled freedom from bronchiolitis obliterans was 86% (77-91%) at two years. Conclusions: While this systematic review and meta-analysis is limited by the available surgeons, institutions, and papers discussing a highly specialized technique, it does show that BAR is a viable technique to minimize BOS and early anastomotic intervention following lung transplantation

Endovascular Intervention for Tracheo-Innominate Fistula: A Systematic Review and Meta-analysis

Introduction: Fistula formation between the trachea and the innominate artery is a life-threatening complication rarely seen with existing or previous tracheostomy. Fatal upon rupture, swift diagnosis and immediate intervention are paramount for survival. We aim to identify feasibility and outcomes of endovascular intervention for trachea-innominate fistula (TIF). Methods: Patient-level data of reported individuals above the age of 14 that underwent endovascular intervention for TIF was extracted and analyzed. Identification of 25 patients from 27 studies was accomplished by electronic database search of Cochrane Central Register of Controlled Trials, Cumulative Index to Nursing and Allied Health Literature, Ovid Medline, and Scopus. Survival data was evaluated by Kaplan-Meier analysis. Results: Median patient age was 39.0 years [IQR 16.0, 47.5]. Median time to TIF presentation following tracheostomy was 2.2 months [0.5, 42.5]. 84.6% (22/27) exhibited tracheal hemorrhage at presentation. Covered stent graft placement was performed in 96.3% (26/27) and coil embolization in 3.8% (1/27). Repeat endovascular intervention was necessary in 18.5% (5/27) and rescue sternotomy was required in 11.1% (3/27). Overall mortality was 29.6% (8/27) with a median follow-up time of 5 months [1.2, 11.5]. Discussion: Endovascular intervention may be an effective method of TIF repair at presentation. As an alternative to conventional surgical repair, endovascular intervention may be an appropriate method for TIF repair particularly in patients unfit for open sternotomy repair

Estimating Averages of Order Statistics of Bivariate Functions

© 2016, Springer Science+Business Media New York. We prove uniform estimates for the expected value of averages of order statistics of bivariate functions in terms of their largest values by a direct analysis. As an application, uniform estimates for the expected value of averages of order statistics of sequences of independent random variables in terms of Orlicz norms are obtained. In the case where the bivariate functions are matrices, we provide a “minimal” probability space which allows us to C-embed certain Orlicz spaces ℓMn into ℓ1cn3, with c, C> 0 being absolute constants