Generic transient memory formation in disordered systems with noise

Out-of-equilibrium disordered systems may form memories of external driving in a remarkable fashion. The system "remembers" multiple values from a series of training inputs yet "forgets" nearly all of them at long times despite the inputs being continually repeated. Here, learning and forgetting are inseparable aspects of a single process. The memory loss may be prevented by the addition of noise. We identify a class of systems with this behavior, giving as an example a model of non-brownian suspensions under cyclic shear.Comment: 4 pages, 3 figure

A finite-strain hyperviscoplastic model and undrained triaxial tests of peat

This paper presents a finite-strain hyperviscoplastic constitutive model within a thermodynamically consistent framework for peat which was categorised as a material with both rate-dependent and thermodynamic equilibrium hysteresis based on the data reported in the literature. The model was implemented numerically using implicit time integration and verified against analytical solutions under simplified conditions. Experimental studies on the undrained relaxation and loading-unloading-reloading behaviour of an undisturbed fibrous peat were carried out to define the thermodynamic equilibrium state during deviatoric loading as a prerequisite for further modelling, to fit particularly those model parameters related to solid matrix properties, and to validate the proposed model under undrained conditions. This validation performed by comparison to experimental results showed that the hyperviscoplastic model could simulate undrained triaxial compression tests carried out at five different strain rates with loading/unloading relaxation steps.Comment: 30 pages, 16 figures, 4 tables. This is a pre-peer reviewed version of manuscript submitted to the International Journal of Numerical and Analytical Methods in Geomechanic

Bounds and asymptotic minimal growth for Gorenstein Hilbert functions

We determine new bounds on the entries of Gorenstein Hilbert functions, both in any fixed codimension and asymptotically. Our first main theorem is a lower bound for the degree $i+1$ entry of a Gorenstein $h$-vector, in terms of its entry in degree $i$. This result carries interesting applications concerning unimodality: indeed, an important consequence is that, given $r$ and $i$, all Gorenstein $h$-vectors of codimension $r$ and socle degree $e\geq e_0=e_0(r,i)$ (this function being explicitly computed) are unimodal up to degree $i+1$. This immediately gives a new proof of a theorem of Stanley that all Gorenstein $h$-vectors in codimension three are unimodal. Our second main theorem is an asymptotic formula for the least value that the $i$-th entry of a Gorenstein $h$-vector may assume, in terms of codimension, $r$, and socle degree, $e$. This theorem broadly generalizes a recent result of ours, where we proved a conjecture of Stanley predicting that asymptotic value in the specific case $e=4$ and $i=2$, as well as a result of Kleinschmidt which concerned the logarithmic asymptotic behavior in degree $i= \lfloor \frac{e}{2} \rfloor$.Comment: Several minor changes; to appear in J. Algebr