A characterization of positive linear maps and criteria of entanglement for quantum states

Let $H$ and $K$ be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from ${\mathcal B}(H)$ into ${\mathcal B}(K)$ is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is give which shows that a state $\rho$ on $H\otimes K$ is separable if and only if $(\Phi\otimes I)\rho\geq 0$ for all positive finite rank elementary operators $\Phi$. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.Comment: 20 page

Is America coming apart? Socioeconomic segregation in neighborhoods, schools, workplaces, and social networks, 1970–2020

As income inequality in the United States has reached an all-time high, commentators from across the political spectrum warn about the social implications of these economic changes. America, they fear, is “coming apart” as the gap between the rich and poor grows into a fault line. This paper provides a comprehensive review of empirical scholarship in sociology, education, demography, and economics in order to address the question: How have five decades of growing economic inequality shaped America's social landscape? We find that growing levels of income inequality have been accompanied by increasing socioeconomic segregation across (1) friendship networks and romantic partners, (2) residential neighborhoods, (3) K-12 and university education, and (4) workplaces and the labor market. The trends documented in this review give substance to commentators' concerns: compared to the 1970s, rich and poor Americans today are less likely to know one another and to share the same social spaces. The United States is a nation divided.Published versio

Mathematical models for immunology:current state of the art and future research directions

The advances in genetics and biochemistry that have taken place over the last 10 years led to significant advances in experimental and clinical immunology. In turn, this has led to the development of new mathematical models to investigate qualitatively and quantitatively various open questions in immunology. In this study we present a review of some research areas in mathematical immunology that evolved over the last 10 years. To this end, we take a step-by-step approach in discussing a range of models derived to study the dynamics of both the innate and immune responses at the molecular, cellular and tissue scales. To emphasise the use of mathematics in modelling in this area, we also review some of the mathematical tools used to investigate these models. Finally, we discuss some future trends in both experimental immunology and mathematical immunology for the upcoming years