Laws of large numbers for eigenvectors and eigenvalues associated to random subspaces in a tensor product

Given two positive integers $n$ and $k$ and a parameter $t\in (0,1)$, we choose at random a vector subspace $V_{n}\subset \mathbb{C}^{k}\otimes\mathbb{C}^{n}$ of dimension $N\sim tnk$. We show that the set of $k$-tuples of singular values of all unit vectors in $V_n$ fills asymptotically (as $n$ tends to infinity) a deterministic convex set $K_{k,t}$ that we describe using a new norm in $\R^k$. Our proof relies on free probability, random matrix theory, complex analysis and matrix analysis techniques. The main result result comes together with a law of large numbers for the singular value decomposition of the eigenvectors corresponding to large eigenvalues of a random truncation of a matrix with high eigenvalue degeneracy.Comment: v3 changes: minor typographic improvements; accepted versio

Scarcity in the age of abundance: paradox and remedies

Purpose This paper aims to examine the concepts of scarcity and abundance from an interdisciplinary perspective. It argues that the idea of economics, as the study of human behavior with regard to scarce resources and unlimited wants, leads to competition, confrontation and conflict, whereas the nature of humans is that of kindness, cooperation and sharing. Design/methodology/approach The paper uses content analysis of texts from multiple disciplines, using both deductive and inductive logic, to study the situation of scarcity and abundance, explore reasons and offer remedies. The paper consists of seven sections, five of which are the main discussion. Section 2 looks at the concepts of scarcity and abundance. Section 3 presents reasons for scarcity: economic and behavioral. Section 4 discusses the civilization perspective of poverty and wealth. Section 5 briefly looks at sustainable development and good governance. Section 6 argues in favor of simplicity and spirituality as remedies for the problem of scarcity, and Section 7 concludes the paper. Findings The paper demonstrates the relativity of the concepts of scarcity and abundance, points out the paradox of the modern consumer economies and argues that simplicity, spirituality and universal values are necessary to remedy the ills of overconsumption/overproduction, waste and inequality. Originality/value This paper offers spirituality and ethics-based remedies for the negative consequences of neo-classical economics and social Darwinism