Minimal matrices and the corresponding minimal curves on flag manifolds in low dimension

In general C*-algebras, elements with minimal norm in some equivalence class are introduced and characterized. We study the set of minimal hermitian matrices, in the case where the C*-algebra consists of 3 × 3 complex matrices, and the quotient is taken by the subalgebra of diagonal matrices. We thoroughly study the set of minimal matrices particularly because of its relation to the geometric problem of finding minimal curves in flag manifolds. For the flag manifold of 'four mutually orthogonal complex lines' in C4, it is shown that there are infinitely many minimal curves joining arbitrarily close points. In the case of the flag manifold of 'three mutually orthogonal complex lines' in C3, we show that the phenomenon of multiple minimal curves joining arbitrarily close points does not occur.Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Mata Lorenzo, Luis E.. Universidad Simón Bolivar; VenezuelaFil: Mendoza, Alberto. Universidad Simón Bolivar; VenezuelaFil: Recht, Lázaro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Simón Bolivar; VenezuelaFil: Varela, Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentin

Minimal hermitian matrices with fixed entries outside the diagonal

We survey some results concerning the problem of finding the complex hermitian matrix or matrices of least supremum norm with variable diagonal. Some qualitative general results are given and more specific descriptions are shown for the 3 x 3 case. We also comment some results and examples concerning this approximation problem.Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Mata Lorenzo, Luis E.. Universidad Simón Bolívar; VenezuelaFil: Mendoza, Alberto.. Universidad Simón Bolívar; VenezuelaFil: Recht, Lázaro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Simón Bolívar; VenezuelaFil: Varela, Alejandro. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin

A note on the codimension of a map

AbstractJ. Mather proved that for smooth proper mappings, infinitesimal stability is equivalent to local infinitesimal stability. The Reduction Theorem extends that result to mappings of arbitrary codimensions, for it says that the (global) codimension of the mapping is the sum of all the multigerm codimensions. This formula seems to give a more manageable expression for the codimension than the original definition

A uniqueness theorem for the unitary part of a reflection

AbstractThe following uniqueness theorem is proved: Let ε be a reflection in the C∗-algebra A (i.e., ε2 = 1). If ε = μϱ is the polar decomposition of ε (μ > 0, ϱ unitary), then ϱ is the only unitary reflection in A satisfying: for every self-adjoint projection p that commutes with ε, 1.(i) ϱ commutes with p,2.(ii) the “reduction” pϱp, of ϱ to the C∗-algebra , minimizes the distance from the reduction pεp of ε to the set of unitary reflections in

Natural Variational Problems in the Grassmann Manifold of a C*-Algebra with Trace

AbstractWe consider a family of natural variational problems in the Grassmannian of a C*-algebra with trace which can be considered as slightly degenerate Finsler metrics. We show that all these problems have as solutions the standard geodesics and that the short standard geodesics are absolute minima of the functionals restricted to a special class of curves

Metric geometry in homogeneous spaces of the unitary group of a C∗-algebra: Part I—minimal curves

AbstractWe study the metric geometry of homogeneous spaces P of the unitary group of a C∗-algebra A modulo the unitary group of a C∗-subalgebra B, where the invariant Finsler metric is induced by the quotient norm of A/B.The main results include the following. In the von Neumann algebra context, for each tangent vector X at a point ρ∈P, there is a geodesic γ(t), γ̇(0)=X, which is obtained by the action on ρ of a 1-parameter group in the unitary group of A. This geodesic is minimizing up to length π/2

Integrando escalas y métodos LTER para comprender la dinámica global de un espacio protegido de montaña: el Parque Nacional de Ordesa y Monte Perdido.

Los espacios protegidos, por el hecho de albergar una gran geo-biodiversidad y asegurar una baja intervención humana, constituyen lugares muy adecuados para el seguimiento de organismos y procesos a escala ecológica, así como para la obtención de series temporales largas a escala geológica. En el marco de la red LTER-España, el Parque Nacional de Ordesa y Monte Perdido (PNOMP) y el Instituto Pirenaico de Ecología-CSIC están impulsando estudios para la detección de cambios a distintas escalas mediante variados métodos y aproximaciones. Destacamos aquí los más consolidados, entre los que se encuentran los análisis de registros de sedimentos en lagos, espeleotemas en cuevas, la dinámica de uno de los pocos glaciares activos de la Península ibérica, el análisis físico-químico de aguas corrientes e ibones de alta montaña, el registro del cambio climático actual en árboles longevos, la afección que éste ejerce sobre masas actuales de pinos en el límite superior del bosque y de abetales en zonas húmedas, la matorralización de algunos pastos y los procesos mecanicistas que subyacen, la reorganización de la diversidad florística en pastos tras el abandono paulatino o drástico de la ganadería, la biodiversidad de las comunidades alpinas y la dinámica poblacional de especies amenazadas o indicadoras de hábitats o de motores de cambio global. Los seguimientos ecológicos actuales muestran que tanto el cambio climático como el de usos del suelo están teniendo una considerable trascendencia en la fisionomía y la estructura de algunos de los ambientes más icónicos y frecuentes del parque (deterioro del glaciar, termofilización de la flora en cumbres alpinas, densificación del bosque en su límite superior, pérdida de productividad en algunos pastos supraforestales, etc.). También sugieren una importante variabilidad espacial en los procesos (por ej. en el PNOMP conviven pastos matorralizados y pastos muy estables), y evidencian que los cambios observados no siempre siguen los paradigmas establecidos (por ej., las especies amenazadas mantienen dinámicas poblacionales estables). La integración de resultados parciales proporcionados por cada aproximación relativiza la importancia de las percepciones que cada estudio destaca por separado, y permite medir los cambios actuales en el marco de referencia de los cambios a escala geológica.Predecir la resistencia y resiliencia de los ecosistemas o las poblaciones de seres vivos para enfrentarse a los futuros cambios ambientales es complicado, no sólo por la falta de conocimientos disponibles sino también porque las respuestas que observamos no siempre son tan rápidas o lineales como se espera. La modelización constituye una herramienta cada vez más utilizada, pero requiere de evidencias reales para validar sus pronósticos, por lo que la observación de los procesos que actúan en el PNOMP ha de incluir un esfuerzo continuado de monitorización multiescalar y multidisciplinar de los distintos componentes de la geo, hidro-, crio- y biosfera, sin olvidar el componente humano. Entender la complejidad supone conectar las interacciones que existen entre todos los sistemas y ponderar sus efectos según las escalas de trabajo

Familial hypercholesterolaemia in children and adolescents from 48 countries: a cross-sectional study

Background: Approximately 450 000 children are born with familial hypercholesterolaemia worldwide every year, yet only 2·1% of adults with familial hypercholesterolaemia were diagnosed before age 18 years via current diagnostic approaches, which are derived from observations in adults. We aimed to characterise children and adolescents with heterozygous familial hypercholesterolaemia (HeFH) and understand current approaches to the identification and management of familial hypercholesterolaemia to inform future public health strategies. Methods: For this cross-sectional study, we assessed children and adolescents younger than 18 years with a clinical or genetic diagnosis of HeFH at the time of entry into the Familial Hypercholesterolaemia Studies Collaboration (FHSC) registry between Oct 1, 2015, and Jan 31, 2021. Data in the registry were collected from 55 regional or national registries in 48 countries. Diagnoses relying on self-reported history of familial hypercholesterolaemia and suspected secondary hypercholesterolaemia were excluded from the registry; people with untreated LDL cholesterol (LDL-C) of at least 13·0 mmol/L were excluded from this study. Data were assessed overall and by WHO region, World Bank country income status, age, diagnostic criteria, and index-case status. The main outcome of this study was to assess current identification and management of children and adolescents with familial hypercholesterolaemia. Findings: Of 63 093 individuals in the FHSC registry, 11 848 (18·8%) were children or adolescents younger than 18 years with HeFH and were included in this study; 5756 (50·2%) of 11 476 included individuals were female and 5720 (49·8%) were male. Sex data were missing for 372 (3·1%) of 11 848 individuals. Median age at registry entry was 9·6 years (IQR 5·8-13·2). 10 099 (89·9%) of 11 235 included individuals had a final genetically confirmed diagnosis of familial hypercholesterolaemia and 1136 (10·1%) had a clinical diagnosis. Genetically confirmed diagnosis data or clinical diagnosis data were missing for 613 (5·2%) of 11 848 individuals. Genetic diagnosis was more common in children and adolescents from high-income countries (9427 [92·4%] of 10 202) than in children and adolescents from non-high-income countries (199 [48·0%] of 415). 3414 (31·6%) of 10 804 children or adolescents were index cases. Familial-hypercholesterolaemia-related physical signs, cardiovascular risk factors, and cardiovascular disease were uncommon, but were more common in non-high-income countries. 7557 (72·4%) of 10 428 included children or adolescents were not taking lipid-lowering medication (LLM) and had a median LDL-C of 5·00 mmol/L (IQR 4·05-6·08). Compared with genetic diagnosis, the use of unadapted clinical criteria intended for use in adults and reliant on more extreme phenotypes could result in 50-75% of children and adolescents with familial hypercholesterolaemia not being identified. Interpretation: Clinical characteristics observed in adults with familial hypercholesterolaemia are uncommon in children and adolescents with familial hypercholesterolaemia, hence detection in this age group relies on measurement of LDL-C and genetic confirmation. Where genetic testing is unavailable, increased availability and use of LDL-C measurements in the first few years of life could help reduce the current gap between prevalence and detection, enabling increased use of combination LLM to reach recommended LDL-C targets early in life

Association of BMI, lipid-lowering medication, and age with prevalence of type 2 diabetes in adults with heterozygous familial hypercholesterolaemia: a worldwide cross-sectional study

Background: Statins are the cornerstone treatment for patients with heterozygous familial hypercholesterolaemia but research suggests it could increase the risk of type 2 diabetes in the general population. A low prevalence of type 2 diabetes was reported in some familial hypercholesterolaemia cohorts, raising the question of whether these patients are protected against type 2 diabetes. Obesity is a well known risk factor for the development of type 2 diabetes. We aimed to investigate the associations of known key determinants of type 2 diabetes with its prevalence in people with heterozygous familial hypercholesterolaemia. Methods: This worldwide cross-sectional study used individual-level data from the EAS FHSC registry and included adults older than 18 years with a clinical or genetic diagnosis of heterozygous familial hypercholesterolaemia who had data available on age, BMI, and diabetes status. Those with known or suspected homozygous familial hypercholesterolaemia and type 1 diabetes were excluded. The main outcome was prevalence of type 2 diabetes overall and by WHO region, and in relation to obesity (BMI ≥30·0 kg/m2) and lipid-lowering medication as predictors. The study population was divided into 12 risk categories based on age (tertiles), obesity, and receiving statins, and the risk of type 2 diabetes was investigated using logistic regression. Findings: Among 46 683 adults with individual-level data in the FHSC registry, 24 784 with heterozygous familial hypercholesterolaemia were included in the analysis from 44 countries. 19 818 (80%) had a genetically confirmed diagnosis of heterozygous familial hypercholesterolaemia. Type 2 diabetes prevalence in the total population was 5·7% (1415 of 24 784), with 4·1% (817 of 19 818) in the genetically diagnosed cohort. Higher prevalence of type 2 diabetes was observed in the Eastern Mediterranean (58 [29·9%] of 194), South-East Asia and Western Pacific (214 [12·0%] of 1785), and the Americas (166 [8·5%] of 1955) than in Europe (excluding the Netherlands; 527 [8·0%] of 6579). Advancing age, a higher BMI category (obesity and overweight), and use of lipid-lowering medication were associated with a higher risk of type 2 diabetes, independent of sex and LDL cholesterol. Among the 12 risk categories, the probability of developing type 2 diabetes was higher in people in the highest risk category (aged 55–98 years, with obesity, and receiving statins; OR 74·42 [95% CI 47·04–117·73]) than in those in the lowest risk category (aged 18–38 years, without obesity, and not receiving statins). Those who did not have obesity, even if they were in the upper age tertile and receiving statins, had lower risk of type 2 diabetes (OR 24·42 [15·57–38·31]). The corresponding results in the genetically diagnosed cohort were OR 65·04 (40·67–104·02) for those with obesity in the highest risk category and OR 20·07 (12·73–31·65) for those without obesity. Interpretation: Adults with heterozygous familial hypercholesterolaemia in most WHO regions have a higher type 2 diabetes prevalence than in Europe. Obesity markedly increases the risk of diabetes associated with age and use of statins in these patients. Our results suggest that heterozygous familial hypercholesterolaemia does not protect against type 2 diabetes, hence managing obesity is essential to reduce type 2 diabetes in this patient population. Funding: Pfizer, Amgen, MSD, Sanofi-Aventis, Daiichi-Sankyo, and Regeneron