A Theory of Congruences and Birkhoff’s Theorem for Matroids

A congruence is defined for a matroid. This leads to suitable versions of the algebraic isomorphism theorems for matroids. As an application of the congruence theory for matroids, a version of Birkhoff’s Theorem for matroids is given which shows that every nontrivial matroid is a subdirect product of subdirectly irreducible matroids.

Extreme black hole anabasis

We study the SL(2) transformation properties of spherically symmetric perturbations of the Bertotti-Robinson universe and identify an invariant μ that characterizes the backreaction of these linear solutions. The only backreaction allowed by Birkhoff’s theorem is one that destroys the AdS2× S2 boundary and builds the exterior of an asymptotically flat Reissner-Nordström black hole with $$ Q=M\sqrt{1-\mu /4} $$ Q = M 1 − μ / 4 . We call such backreaction with boundary condition change an anabasis. We show that the addition of linear anabasis perturbations to Bertotti-Robinson may be thought of as a boundary condition that defines a connected AdS2×S2. The connected AdS2 is a nearly-AdS2 with its SL(2) broken appropriately for it to maintain connection to the asymptotically flat region of Reissner-Nordström. We perform a backreaction calculation with matter in the connected AdS2× S2 and show that it correctly captures the dynamics of the asymptotically flat black hole.

A short note on extreme points of certain polytopes

We give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

Routine Psychological Testing of the Individual Is Not Valid

In this article, we present and argue our assertion that current routine psychological testing of individuals is not valid. To support our assertion, we review the concept of ergodicity, Birkhoff’s theorem, and Molenaar’s manifesto, which together support our contention that the direct transposition of population estimations for producing inferences about the individual is not valid. We argue that this practice of direct transposition is the root cause of why routine psychological testing of individual is not valid. We then provide an example of a common application of psychological testing of an individual, explaining why this practice is not valid. Finally, we discuss how the intraindividual (or within-person) approach provides some prospect for valid individual testing and also introduces new challenges. We hope that our questioning of current psychological testing practices motivates researchers to propose and study novel methodological propositions to address the issues raised by our assertion.