An integrality theorem of Grosshans over arbitrary base ring

We revisit a theorem of Grosshans and show that it holds over arbitrary commutative base ring $k$. One considers a split reductive group scheme $G$ acting on a $k$-algebra $A$ and leaving invariant a subalgebra $R$. If $R^U=A^U$ then the conclusion is that $A$ is integral over $R$.Comment: 5 pages; final versio

Local generation of hydrogen for enhanced photothermal therapy.

By delivering the concept of clean hydrogen energy and green catalysis to the biomedical field, engineering of hydrogen-generating nanomaterials for treatment of major diseases holds great promise. Leveraging virtue of versatile abilities of Pd hydride nanomaterials in high/stable hydrogen storage, self-catalytic hydrogenation, near-infrared (NIR) light absorption and photothermal conversion, here we utilize the cubic PdH0.2 nanocrystals for tumour-targeted and photoacoustic imaging (PAI)-guided hydrogenothermal therapy of cancer. The synthesized PdH0.2 nanocrystals have exhibited high intratumoural accumulation capability, clear NIR-controlled hydrogen release behaviours, NIR-enhanced self-catalysis bio-reductivity, high NIR-photothermal effect and PAI performance. With these unique properties of PdH0.2 nanocrystals, synergetic hydrogenothermal therapy with limited systematic toxicity has been achieved by tumour-targeted delivery and PAI-guided NIR-controlled release of bio-reductive hydrogen as well as generation of heat. This hydrogenothermal approach has presented a cancer-selective strategy for synergistic cancer treatment

The delta invariant and the various GIT-stability notions of toric Fano varieties

In this article, we give combinatorial proofs of the following two theorems: (1) If a Gorenstein toric Fano variety is asymptotically Chow semistable then it is Ding polystable. (2) For a smooth toric Fano manifold $X$, the delta invariant $\delta(X)$ defined by Fujita and Odaka coincides with the greatest Ricci lower curvature $R(X)$. In the proof, neither toric test configuration nor toric Minimal Model Program (MMP) we use. We also verify the reductivity of automorphism group of toric Fano $3$-folds by computing Demazure's roots for each. All the results are listed in Table $1$ with the value of $\delta(X)$ and $R(X)$.Comment: 19 pages, 2 figures, 1 table. Fixed an error in Proposition 4.3. Section 5 in the previous version removed. The appendix added. The title changed from the first versio