The dimension representation has been a useful tool in studying the mysterious automorphism group of a shift of finite type, the classification of shifts of finite type, and surrounding problems. However, there has been little progress in understanding the image of the dimension representation. We discuss the importance of understanding the image of the dimension representation and discuss the ...

The class of Petrov type I curvature tensors is further divided into those for which the span set distinct principal null directions has dimension four (maximally spanning I) or three (nonmaximally I). Explicit examples are provided both vacuum and nonvacuum spacetimes.

Förster-type nonradiative energy transfer (NRET) provides us with the ability to transfer excitation energy between proximal nanostructures with high efficiency under certain conditions. Nevertheless, the well-known Förster theory was developed for the case of a single donor (e.g., a molecule, a dye) together with single acceptor. There is no complete understanding for the cases when the donors...

We prove the formula ed(X ) = cdim(X ) + ed(A) for any gerbe X banded by an algebraic group A which is the kernel of a homomorphism of algebraic tori Q → S with Q invertible and S split. This result is applied to prove new results on the essential dimension of algebraic groups.

Let $D$ be an integral domain and $star$ a semistar operation stable and of finite type on it. We define the semistar dimension (inequality) formula and discover their relations with $star$-universally catenarian domains and $star$-stably strong S-domains. As an application, we give new characterizations of $star$-quasi-Pr"{u}fer domains and UM$t$ domains in terms of dimension inequal...