Log-majorization type inequalities

Several inequalities have been established in the context of Hilbert spaces operators or operator algebras. Our discussion will be limited to matrices. Important inequalities in mathematics and other sciences, such as Golden-Thompson inequality or von Neumann trace inequality, and extensions, are revisited. Our main goal is to emphasize the link between majorization theory and other relevant inequalities.Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), Centre for Mathematics of the University of Coimbra (CMUC), Center for Research and Development in Mathematics and Applications (CIDMA) , CMAT-UTADpublishe

On the Courant-Fischer theory for Krein spaces

http://www.sciencedirect.com/science/article/B6V0R-4V462G8-2/2/25c16be9e99d2fbaa89b7c1a6a47e95

Analogs of Cauchy–Poincaré and Fan–Pall interlacing theorems for J-Hermitian and J-normal matrices

AbstractThe interlacing theorem of Cauchy–Poincaré states that the eigenvalues of a principal submatrix A0 of a Hermitian matrix A interlace the eigenvalues of A. Fan and Pall obtained an analog of this theorem for normal matrices. In this note we investigate analogs of Cauchy–Poincaré and Fan–Pall interlacing theorems for J-Hermitian and J-normal matrices. The corresponding inverse spectral problems are also considered

Fundamental properties of Tsallis relative entropy

Fundamental properties for the Tsallis relative entropy in both classical and quantum systems are studied. As one of our main results, we give the parametric extension of the trace inequality between the quantum relative entropy and the minus of the trace of the relative operator entropy given by Hiai and Petz. The monotonicity of the quantum Tsallis relative entropy for the trace preserving completely positive linear map is also shown without the assumption that the density operators are invertible. The generalized Tsallis relative entropy is defined and its subadditivity is shown by its joint convexity. Moreover, the generalized Peierls-Bogoliubov inequality is also proven

World Health Organization cardiovascular disease risk charts: revised models to estimate risk in 21 global regions

BACKGROUND: To help adapt cardiovascular disease risk prediction approaches to low-income and middle-income countries, WHO has convened an effort to develop, evaluate, and illustrate revised risk models. Here, we report the derivation, validation, and illustration of the revised WHO cardiovascular disease risk prediction charts that have been adapted to the circumstances of 21 global regions. METHODS: In this model revision initiative, we derived 10-year risk prediction models for fatal and non-fatal cardiovascular disease (ie, myocardial infarction and stroke) using individual participant data from the Emerging Risk Factors Collaboration. Models included information on age, smoking status, systolic blood pressure, history of diabetes, and total cholesterol. For derivation, we included participants aged 40-80 years without a known baseline history of cardiovascular disease, who were followed up until the first myocardial infarction, fatal coronary heart disease, or stroke event. We recalibrated models using age-specific and sex-specific incidences and risk factor values available from 21 global regions. For external validation, we analysed individual participant data from studies distinct from those used in model derivation. We illustrated models by analysing data on a further 123 743 individuals from surveys in 79 countries collected with the WHO STEPwise Approach to Surveillance. FINDINGS: Our risk model derivation involved 376 177 individuals from 85 cohorts, and 19 333 incident cardiovascular events recorded during 10 years of follow-up. The derived risk prediction models discriminated well in external validation cohorts (19 cohorts, 1 096 061 individuals, 25 950 cardiovascular disease events), with Harrell's C indices ranging from 0·685 (95% CI 0·629-0·741) to 0·833 (0·783-0·882). For a given risk factor profile, we found substantial variation across global regions in the estimated 10-year predicted risk. For example, estimated cardiovascular disease risk for a 60-year-old male smoker without diabetes and with systolic blood pressure of 140 mm Hg and total cholesterol of 5 mmol/L ranged from 11% in Andean Latin America to 30% in central Asia. When applied to data from 79 countries (mostly low-income and middle-income countries), the proportion of individuals aged 40-64 years estimated to be at greater than 20% risk ranged from less than 1% in Uganda to more than 16% in Egypt. INTERPRETATION: We have derived, calibrated, and validated new WHO risk prediction models to estimate cardiovascular disease risk in 21 Global Burden of Disease regions. The widespread use of these models could enhance the accuracy, practicability, and sustainability of efforts to reduce the burden of cardiovascular disease worldwide. FUNDING: World Health Organization, British Heart Foundation (BHF), BHF Cambridge Centre for Research Excellence, UK Medical Research Council, and National Institute for Health Research

Three observations on the determinantal range

Let A, C [set membership, variant] Mn, the algebra of n × n complex matrices. The set of complex numbersis the C-determinantal range of A. In this note, it is proved that [Delta]C(A) is an elliptical disc for A, C [set membership, variant] M2. A necessary and sufficient condition for [Delta]C(A) to be a line segment is given when A and C are normal matrices with pairwise distinct eigenvalues. The linear operators L that satisfy the linear preserver property [Delta]C(A) = [Delta]C(L(A)), for all A, C [set membership, variant] Mn, are characterized.http://www.sciencedirect.com/science/article/B6V0R-4D0Y71M-2/1/682f05b81ddaa6914f85e6ba5e02f63

Flat portions on the boundary of the indefinite numerical range of 3×3 matrices

We focus on complex 3×3 matrices whose indefinite numerical ranges have a flat portion on the boundary. The results here obtained are parallel to those of Keeler, Rodman and Spitkovsky for the classical numerical range.http://www.sciencedirect.com/science/article/B6V0R-4S09585-1/1/cfd4cc3f5f7e4af2bc2e55b75b3a397