The Betti numbers of forests

This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs

The Betti numbers of forests

This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs

Microbiota, Oral Microbiome, and Pancreatic Cancer

Only 30% of patients diagnosed with pancreatic cancer survive one year post-diagnosis. Progress in understanding the causes of pancreatic cancer has been made, including solidifying the associations with obesity and diabetes, and a proportion of cases should be preventable through lifestyle modifications. Unfortunately, identifying reliable biomarkers of early pancreatic cancer has been extremely challenging, and no effective screening modality is currently available for this devastating form of cancer. Recent data suggest the microbiota may play a role in the disease process, but many questions remain. Future studies focusing on the human microbiome, both etiologically and as a marker of disease susceptibility, should shed light on how to better tackle prevention, early detection, and treatment of this highly fatal disease

Information capacity of the Hopfield model

The information capacity of general forms of memory is formalized. The number of bits of information that can be stored in the Hopfield model of associative memory is estimated. It is found that the asymptotic information capacity of a Hopfield network of N neurons is of the order N^3b. The number of arbitrary state vectors that can be made stable in a Hopfield network of N neurons is proved to be bounded above by N

Universality of thermally assisted magnetic domain wall motion under spin torque

Thermally assisted motion of magnetic domain wall under spin torque is studied theoretically. It is shown that the wall velocity $v$ depends exponentially on the spin current, \Is, below the threshold value, in the same way as in a thermally activated motion driven by a force. A novel property of the spin torque driven case at low temperature is that the linear term in spin current is universal, i.e., \ln v \sim \frac{\pi\hbar}{2e}(\Is/\kB T). This behavior, which is independent of pinning and material constants, could be used to confirm experimentally the spin torque as the driving mechanism

Diffusion models for Jupiter's radiation belt

Solutions are given for the diffusion of trapped particles in a planetary magnetic field in which the first and second adiabatic invariants are preserved but the third is not, using as boundary conditions a fixed density at the outer boundary (the magnetopause) and a zero density at an inner boundary (the planetary surface). Losses to an orbiting natural satellite are included and an approximate evaluation is made of the effects of the synchrotron radiation on the energy of relativistic electrons. Choosing parameters appropriate to Jupiter, the electrons required to produce the observed synchrotron radiation are explained. If a speculative mechanism in which the diffusion is driven by ionospheric wind is the true explanation of the electrons producing the synchrotron emission it can be concluded that Jupiter's inner magnetosphere is occupied by an energetic proton flux that would be a serious hazard to spacecraft

Modelling resonances and orbital chaos in disk galaxies. Application to a Milky Way spiral model

Context: Resonances in the stellar orbital motion under perturbations from spiral arms structure play an important role in the evolution of the disks of spiral galaxies. The epicyclic approximation allows the determination of the corresponding resonant radii on the equatorial plane (for nearly circular orbits), but is not suitable in general. Aims: We expand the study of resonant orbits by analysing stellar motions perturbed by spiral arms with Gaussian-shaped profiles without any restriction on the stellar orbital configurations, and we expand the concept of Lindblad (epicyclic) resonances for orbits with large radial excursions. Methods: We define a representative plane of initial conditions, which covers the whole phase space of the system. Dynamical maps on representative planes are constructed numerically, in order to characterize the phase-space structure and identify the precise location of resonances. The study is complemented by the construction of dynamical power spectra, which provide the identification of fundamental oscillatory patterns in the stellar motion. Results: Our approach allows a precise description of the resonance chains in the whole phase space, giving a broader view of the dynamics of the system when compared to the classical epicyclic approach, even for objects in retrograde motion. The analysis of the solar neighbourhood shows that, depending on the current azimuthal phase of the Sun with respect to the spiral arms, a star with solar kinematic parameters may evolve either inside the stable co-rotation resonance or in a chaotic zone. Conclusions: Our approach contributes to quantifying the domains of resonant orbits and the degree of chaos in the whole Galactic phase-space structure. It may serve as a starting point to apply these techniques to the investigation of clumps in the distribution of stars in the Galaxy, such as kinematic moving groups.Comment: 17 pages, 15 figures. Matches accepted version in A&

Gesunder Boden - gesunde Pflanzen

Obwohl viele ältere Biobetriebe kleinere Nährstoffgehalte im Boden aufwiesen, waren diese noch nicht ertragsbegrenzend. Die abnehmenden Nährstoffgehalte im Boden und den Pflanzen zeigen, dass regelmässige Bodenanalysen auf dem Biobetrieb nötig sind. Die Suppressivität der Böden hing nicht direkt von den Biojahren ab. Durch die Gestaltung der Fruchtfolge und den Hofdüngereinsatz lässt sie sich aber steigern. Der Zusammenhang zwischen Bewirtschaftung Bodeneigenschaften und der Suppressivität muss genauer untersucht werden