Applications of pseudocanonical covers to tight closure problems

It is well known that nice conditions on the canonical module of a ring have a strong impact in the study of strong F-regularity and F-purity. For example, in the Gorenstein (or even -Gorenstein) case strong F-regularity is equivalent to weak F-regularity, and it is conjectured that this is true in general. In this note, we will show how to use the double cover of a ring to obtain sufficient conditions for strong F-regularity and F-purity. Our results involve a closure operation for a pair of ideals that could be of relevance for the conjecture mentioned above.     

On the diophantine equation

In this note, we find all positive integer solutions of the diophantine equation from the title with a prime power.

     

-injective rings and -stable primes

The notion of stability of the highest local cohomology module with respect to the Frobenius functor originates in the work of R. Hartshorne and R. Speiser. R. Fedder and K.-i. Watanabe examined this concept for isolated singularities by relating it to -rationality. The purpose of this note is to study what happens in the case of non-isolated singularities and to show how this stability concept encapsulates a few of the subtleties of tight closure theory. Our study can be seen as a generalization of the work by Fedder and Watanabe. We introduce two new ring invariants, the -stability number and the set of -stable primes. We associate to every ideal generated by a system of parameters and an ideal of multipliers denoted and obtain a family of ideals . The set is independent of and consists of finitely many prime ideals. It also equals prime ideal such that is -stable. The maximal height of such primes defines the -stability number.

     

On a conjecture of Erdos and Stewart

For any , let be the th prime number. In this paper, we confirm a conjecture of Erdos and Stewart concerning all the solutions of the diophantine equation , when .

     

Crystal structure of ethene-/α-olefin copolymers with various long comonomers (C8–C26)

Ethene-/α-olefin copolymers having a wide range of comonomers between 8 and 26 carbon atoms in length were characterized by wide angle X-ray diffraction (WAXD). It was found that the crystallinities for the shorter comonomers (C₈, C₁₂) matched fairly well with the crystallinities from DSC and volumetric measurements, while for the longer comonomers (C₁₈, C₂₆) distinct differences between the three methods were found. This was explained by the presence of an additional crystalline phase of the side chains slightly deforming the amorphous peak.     

A Microgel Construction Kit for Bioorthogonal Encapsulation and pH‐Controlled Release of Living Cells

pH‐Cleavable cell‐laden microgels with excellent long‐term viabilities were fabricated by combining bioorthogonal strain‐promoted azide–alkyne cycloaddition (SPAAC) and droplet‐based microfluidics. Poly(ethylene glycol)dicyclooctyne and dendritic poly(glycerol azide) served as bioinert hydrogel precursors. Azide conjugation was performed using different substituted acid‐labile benzacetal linkers that allowed precise control of the microgel degradation kinetics in the interesting pH range between 4.5 and 7.4. By this means, a pH‐controlled release of the encapsulated cells was achieved upon demand with no effect on cell viability and spreading. As a result, the microgel particles can be used for temporary cell encapsulation, allowing the cells to be studied and manipulated during the encapsulation and then be isolated and harvested by decomposition of the microgel scaffolds.     

Shallow lake economics run deep: nonlinear aspects of an economic-ecological interest conflict

Outcomes of the shallow lake long-term interest conflict in a number of different settings are presented, in particular in the contexts of quasi-static and dynamic social planning and of quasi-static non-cooperative play. Also the effect of trigger strategies in repeated quasi-static play is analysed. A characteristic feature of these interest conflicts, and of pollution problems in general, is the qualitative dichotomy in possible outcomes: the lake (or the ecosystem, or the climate) ends up in either a clean or in a polluted state, both of which, if attained, is stabilised by some kind of feedback mechanism. The social choice therefore always incorporates a qualitative aspect: the decision maker has to decide for or against production, for or against conserving the ecosystem.