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 .

     

Spatiotemporal stochastic resonance and its consequences in neural model systems

The realization of spatiotemporal stochastic resonance is studied in a two-dimensional FitzHugh-Nagumo system, and in a one-dimensional system of integrate-and-fire neurons. We show that spatiotemporal stochastic resonance occurs in these neural model systems, independent of the method of modeling. Moreover, the ways of realization are analogous in the two model systems. The biological implications and open questions are discussed. (c) 2001 American Institute of Physics.     

Neutron dosimetry in low-earth orbit using passive detectors

This paper summarizes neutron dosimetry measurements made by the USF Physics Research Laboratory aboard US and Russian LEO spacecraft over the past 20 years using two types of passive detector. Thermal/resonance neutron detectors exploiting the 6Li(n,T) alpha reaction were used to measure neutrons of energies <1 MeV. Fission foil neutron detectors were used to measure neutrons of energies above 1 MeV. While originally analysed in terms of dose equivalent using the NCRP-38 definition of quality factor, for the purposes of this paper the measured neutron data have been reanalyzed and are presented in terms of ambient dose equivalent. Dose equivalent rate for neutrons <1 MeV ranged from 0.80 microSv/d on the low altitude, low inclination STS-41B mission to 22.0 microSv/d measured in the Shuttle's cargo bay on the highly inclined STS-51F Spacelab-2 mission. In one particular instance a detector embedded within a large hydrogenous mass on STS-61 (in the ECT experiment) measured 34.6 microSv/d. Dose equivalent rate measurements of neutrons >1 MeV ranged from 4.5 microSv/d on the low altitude STS-3 mission to 172 microSv/d on the ~6 year LDEF mission. Thermal neutrons (<0.3 eV) were observed to make a negligible contribution to neutron dose equivalent in all cases. The major fraction of neutron dose equivalent was found to be from neutrons >1 MeV and, on LDEF, neutrons >1 MeV are responsible for over 98% of the total neutron dose equivalent. Estimates of the neutron contribution to the total dose equivalent are somewhat lower than model estimates, ranging from 5.7% at a location under low shielding on LDEF to 18.4% on the highly inclined (82.3 degrees) Biocosmos-2044 mission.     

Real rational curves in Grassmannians

Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some given general figures. For the problem of plane conics tangent to five general (real) conics, the surprising answer is that all 3264 may be real. Similarly, given any problem of enumerating -planes incident on some given general subspaces, there are general real subspaces such that each of the (finitely many) incident -planes is real. We show that the problem of enumerating parameterized rational curves in a Grassmannian satisfying simple (codimension 1) conditions may have all of its solutions real.

     

Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology

The mod 2 Steenrod algebra and Dyer-Lashof algebra have both striking similarities and differences arising from their common origins in ``lower-indexed' algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra , whose module actions are equivalent to, but quite different from, those of and . The exact relationships emerge as ``sheared algebra bijections', which also illuminate the role of the cohomology of . As a bialgebra, has a particularly attractive and potentially useful structure, providing a bridge between those of and , and suggesting possible applications to the Miller spectral sequence and the structure of Dickson algebras.

     

Homoclinic bifurcation in a Hodgkin-Huxley model of thermally sensitive neurons

We study global bifurcations of the chaotic attractor in a modified Hodgkin-Huxley model of thermally sensitive neurons. The control parameter for this model is the temperature. The chaotic behavior is realized over a wide range of temperatures and is visualized using interspike intervals. We observe an abrupt increase of the interspike intervals in a certain temperature region. We identify this as a homoclinic bifurcation of a saddle-focus fixed point which is embedded in the chaotic attractors. The transition is accompanied by intermittency, which obeys a universal scaling law for the average length of trajectory segments exhibiting only short interspike intervals with the distance from the onset of intermittency. We also present experimental results of interspike interval measurements taken from the crayfish caudal photoreceptor, which qualitatively demonstrate the same bifurcation structure. (c) 2000 American Institute of Physics.     

A data-analysis method for identifying differential effects of time-delayed feedback forces and periodic driving forces in stochastic systems

In this work a method is developed for analyzing time series of periodically driven stochastic systems involving time-delayed feedback. The proposed data-analysis method yields dynamical models in terms of stochastic delay differential equations. On the basis of these dynamical models differential effects of driving forces and time-delayed feedback forces can be identified.