Matrices commuting with a given normal tropical matrix

Consider the space $M_n^{nor}$ of square normal matrices $X=(x_{ij})$ over $\mathbb{R}\cup\{-\infty\}$, i.e., $-\infty\le x_{ij}\le0$ and $x_{ii}=0$. Endow $M_n^{nor}$ with the tropical sum $\oplus$ and multiplication $\odot$. Fix a real matrix $A\in M_n^{nor}$ and consider the set $\Omega(A)$ of matrices in $M_n^{nor}$ which commute with $A$. We prove that $\Omega(A)$ is a finite union of alcoved polytopes; in particular, $\Omega(A)$ is a finite union of convex sets. The set $\Omega^A(A)$ of $X$ such that $A\odot X=X\odot A=A$ is also a finite union of alcoved polytopes. The same is true for the set $\Omega'(A)$ of $X$ such that $A\odot X=X\odot A=X$. A topology is given to $M_n^{nor}$. Then, the set $\Omega^{A}(A)$ is a neighborhood of the identity matrix $I$. If $A$ is strictly normal, then $\Omega'(A)$ is a neighborhood of the zero matrix. In one case, $\Omega(A)$ is a neighborhood of $A$. We give an upper bound for the dimension of $\Omega'(A)$. We explore the relationship between the polyhedral complexes $span A$, $span X$ and $span (AX)$, when $A$ and $X$ commute. Two matrices, denoted $\underline{A}$ and $\bar{A}$, arise from $A$, in connection with $\Omega(A)$. The geometric meaning of them is given in detail, for one example. We produce examples of matrices which commute, in any dimension.Comment: Journal versio

An algorithm to describe the solution set of any tropical linear system A x=B x

An algorithm to give an explicit description of all the solutions to any tropical linear system A x=B x is presented. The given system is converted into a finite (rather small) number p of pairs (S,T) of classical linear systems: a system S of equations and a system T of inequalities. The notion, introduced here, that makes p small, is called compatibility. The particular feature of both S and T is that each item (equation or inequality) is bivariate, i.e., it involves exactly two variables; one variable with coefficient 1 and the other one with -1. S is solved by Gaussian elimination. We explain how to solve T by a method similar to Gaussian elimination. To achieve this, we introduce the notion of sub-special matrix. The procedure applied to T is, therefore, called sub-specialization

Filariasis mamaria. A propósito de un caso

Introducción: La filariasis con afectación mamaria es una enfermedad endémica de áreas tropicales y subtropicales de África, Asia, el Pacífico y América, que afecta a unos 120 millones de personas. Aunque es una patología rara en España, dado el aumento de pacientes procedentes de dichos países, debemos conocerla para saber diagnosticarla y tratarla adecuadamente. Objetivo de reportar el caso: Dar a conocer la filariasis con afectación mamaria, sus manifestaciones clínicas y radiológicas principales, a través de un caso clínico de nuestras consultas de ginecología. Descripción del caso: Paciente de 43 años, procedente de Guinea Ecuatorial que acude a la consulta de Ginecología por mastalgia bilateral y aumento del volumen de las mamas de semanas de evolución. La exploración es anodina por lo que se solicita mamografía bilateral en la que describen calcificaciones compatibles con filariasis. A pesar de que el resto de pruebas fueron negativas, dada la alta sospecha clínica y radiológica se diagnosticó de filariasis mamaria. Conclusiones: A pesar de la actual campaña mundial para eliminar la filariasis, el aumento de migración global incrementa la probabilidad de padecer casos importados de filariasis mamaria. Por tanto, el conocimiento de las diferentes parasitosis es imprescindible para realizar un buen diagnóstico diferencial con otras entidades clínicamente similares, e instaurar el tratamiento más adecuado. Introduction: Breast filariasis is an endemic disease from tropical and subtropical areas of Africa, Asia, the Pacific and America affecting about 120 million people. Although it is a rare pathology in Spain, given the increase in patients from the referred countries, we must be aware of it in order to know how to diagnose and treat it properly. Objective of reporting the case: To raise awareness of filariasis with breast involvement and its main clinical and radiological manifestations, through a clinical case of our gynecology consultations. Case report: 43-year-old patient from Equatorial Guinea who comes to the Gynecology consultation for bilateral mastalgia and breast enlargement of weeks of evolution. The examination is anodyne, so bilateral mammography is requested in wich calcifications compatible with filariasis are described. Despite the rest of the tests are negative, given the high clinical and radiological suspicion, the diagnosis of breast filariasis is made. Conclusions: Despite the current worldwide campaign to eliminate filariasis, the increase in global migration enhances the probability of suffering from imported cases of breast filariasis Therefore, the knowledge of the different parasitoses is essential to make a correct differential diagnosis with other clinically similar entities, and to establish the most appropriate treatment

Update of EULAR recommendations for the treatment of systemic sclerosis

The aim was to update the 2009 European League against Rheumatism (EULAR) recommendations for the treatment of systemic sclerosis (SSc), with attention to new therapeutic questions. Update of the previous treatment recommendations was performed according to EULAR standard operating procedures. The task force consisted of 32 SSc clinical experts from Europe and the USA, 2 patients nominated by the pan-European patient association for SSc (Federation of European Scleroderma Associations (FESCA)), a clinical epidemiologist and 2 research fellows. All centres from the EULAR Scleroderma Trials and Research group were invited to submit and select clinical questions concerning SSc treatment using a Delphi approach. Accordingly, 46 clinical questions addressing 26 different interventions were selected for systematic literature review. The new recommendations were based on the available evidence and developed in a consensus meeting with clinical experts and patients. The procedure resulted in 16 recommendations being developed (instead of 14 in 2009) that address treatment of several SSc-related organ complications: Raynaud's phenomenon (RP), digital ulcers (DUs), pulmonary arterial hypertension (PAH), skin and lung disease, scleroderma renal crisis and gastrointestinal involvement. Compared with the 2009 recommendations, the 2016 recommendations include phosphodiesterase type 5 (PDE-5) inhibitors for the treatment of SSc-related RP and DUs, riociguat, new aspects for endothelin receptor antagonists, prostacyclin analogues and PDE-5 inhibitors for SSc-related PAH. New recommendations regarding the use of fluoxetine for SSc-related RP and haematopoietic stem cell transplantation for selected patients with rapidly progressive SSc were also added. In addition, several comments regarding other treatments addressed in clinical questions and suggestions for the SSc research agenda were formulated. These updated data-derived and consensus-derived recommendations will help rheumatologists to manage patients with SSc in an evidence-based way. These recommendations also give directions for future clinical research in SSc

Spread of a SARS-CoV-2 variant through Europe in the summer of 2020.

Following its emergence in late 2019, the spread of SARS-CoV-21,2 has been tracked by phylogenetic analysis of viral genome sequences in unprecedented detail3–5. Although the virus spread globally in early 2020 before borders closed, intercontinental travel has since been greatly reduced. However, travel within Europe resumed in the summer of 2020. Here we report on a SARS-CoV-2 variant, 20E (EU1), that was identified in Spain in early summer 2020 and subsequently spread across Europe. We find no evidence that this variant has increased transmissibility, but instead demonstrate how rising incidence in Spain, resumption of travel, and lack of effective screening and containment may explain the variant’s success. Despite travel restrictions, we estimate that 20E (EU1) was introduced hundreds of times to European countries by summertime travellers, which is likely to have undermined local efforts to minimize infection with SARS-CoV-2. Our results illustrate how a variant can rapidly become dominant even in the absence of a substantial transmission advantage in favourable epidemiological settings. Genomic surveillance is critical for understanding how travel can affect transmission of SARS-CoV-2, and thus for informing future containment strategies as travel resumes. © 2021, The Author(s), under exclusive licence to Springer Nature Limited

Seguimiento de las guías españolas para el manejo del asma por el médico de atención primaria: un estudio observacional ambispectivo

Objetivo Evaluar el grado de seguimiento de las recomendaciones de las versiones de la Guía española para el manejo del asma (GEMA 2009 y 2015) y su repercusión en el control de la enfermedad. Material y métodos Estudio observacional y ambispectivo realizado entre septiembre del 2015 y abril del 2016, en el que participaron 314 médicos de atención primaria y 2.864 pacientes. Resultados Utilizando datos retrospectivos, 81 de los 314 médicos (25, 8% [IC del 95%, 21, 3 a 30, 9]) comunicaron seguir las recomendaciones de la GEMA 2009. Al inicio del estudio, 88 de los 314 médicos (28, 0% [IC del 95%, 23, 4 a 33, 2]) seguían las recomendaciones de la GEMA 2015. El tener un asma mal controlada (OR 0, 19, IC del 95%, 0, 13 a 0, 28) y presentar un asma persistente grave al inicio del estudio (OR 0, 20, IC del 95%, 0, 12 a 0, 34) se asociaron negativamente con tener un asma bien controlada al final del seguimiento. Por el contrario, el seguimiento de las recomendaciones de la GEMA 2015 se asoció de manera positiva con una mayor posibilidad de que el paciente tuviera un asma bien controlada al final del periodo de seguimiento (OR 1, 70, IC del 95%, 1, 40 a 2, 06). Conclusiones El escaso seguimiento de las guías clínicas para el manejo del asma constituye un problema común entre los médicos de atención primaria. Un seguimiento de estas guías se asocia con un control mejor del asma. Existe la necesidad de actuaciones que puedan mejorar el seguimiento por parte de los médicos de atención primaria de las guías para el manejo del asma. Objective: To assess the degree of compliance with the recommendations of the 2009 and 2015 versions of the Spanish guidelines for managing asthma (Guía Española para el Manejo del Asma [GEMA]) and the effect of this compliance on controlling the disease. Material and methods: We conducted an observational ambispective study between September 2015 and April 2016 in which 314 primary care physicians and 2864 patients participated. Results: Using retrospective data, we found that 81 of the 314 physicians (25.8%; 95% CI 21.3–30.9) stated that they complied with the GEMA2009 recommendations. At the start of the study, 88 of the 314 physicians (28.0%; 95% CI 23.4–33.2) complied with the GEMA2015 recommendations. Poorly controlled asthma (OR, 0.19; 95% CI 0.13–0.28) and persistent severe asthma at the start of the study (OR, 0.20; 95% CI 0.12–0.34) were negatively associated with having well-controlled asthma by the end of the follow-up. In contrast, compliance with the GEMA2015 recommendations was positively associated with a greater likelihood that the patient would have well-controlled asthma by the end of the follow-up (OR, 1.70; 95% CI 1.40–2.06). Conclusions: Low compliance with the clinical guidelines for managing asthma is a common problem among primary care physicians. Compliance with these guidelines is associated with better asthma control. Actions need to be taken to improve primary care physician compliance with the asthma management guidelines

Comparación de distintas estrategias para la predicción de muerte a corto plazo en el paciente anciano infectado

Objective. The aim of this study was to determine the utility of a post hoc lactate added to SIRS and qSOFA score to predict 30-day mortality in older non-severely dependent patients attended for infection in the Emergency Department (ED). Methods. We performed an analytical, observational, prospective cohort study including patients of 75 years of age or older, without severe functional dependence, attended for an infectious disease in 69 Spanish ED for 2-day three seasonal periods. Demographic, clinical and analytical data were collected. The primary outcome was 30-day mortality after the index event. Results. We included 739 patients with a mean age of 84.9 (SD 6.0) years; 375 (50.7%) were women. Ninety-one (12.3%) died within 30 days. The AUC was 0.637 (IC 95% 0.587-0.688; p= 2 and 0.698 (IC 95% 0.635- 0.761; p= 2. Comparing receiver operating characteristic (ROC) there was a better accuracy of qSOFA vs SIRS (p=0.041). Both scales improve the prognosis accuracy with lactate inclusion. The AUC was 0.705 (IC95% 0.652-0.758; p<0.001) for SIRS plus lactate and 0.755 (IC95% 0.696-0.814; p<0.001) for qSOFA plus lactate, showing a trend to statistical significance for the second strategy (p=0.0727). Charlson index not added prognosis accuracy to SIRS (p=0.2269) or qSOFA (p=0.2573). Conclusions. Lactate added to SIRS and qSOFA score improve the accuracy of SIRS and qSOFA to predict short-term mortality in older non-severely dependent patients attended for infection. There is not effect in adding Charlson index

The asymptotic values of a polynomial function on the real plane.

Let a polynomial function f of two real variables be given. We prove the existence of a finite number of unbounded regions of the real plane along which the tangent planes to the graph of f tend to horizontal position, when moving away from the origin. The real limit values of this function on these regions are called asymptotic values. We also define the real critical values at infinity of f and prove the theorem of local trivial fibration at infinity, away from these values