Mild Solutions for a Class of Fractional SPDEs and Their Sample Paths

In this article we introduce and analyze a notion of mild solution for a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset\mathbb{R}^{d}$ and driven by an infinite-dimensional fractional noise. The noise is derived from an $L^{2}(D)$-valued fractional Wiener process $W^{H}$ whose covariance operator satisfies appropriate restrictions; moreover, the Hurst parameter $H$ is subjected to constraints formulated in terms of $d$ and the H\"{o}lder exponent of the derivative $h^\prime$ of the noise nonlinearity in the equations. We prove the existence of such solution, establish its relation with the variational solution introduced in \cite{nuavu} and also prove the H\"{o}lder continuity of its sample paths when we consider it as an $L^{2}(D)$--valued stochastic processes. When $h$ is an affine function, we also prove uniqueness. The proofs are based on a relation between the notions of mild and variational solution established in Sanz-Sol\'e and Vuillermot 2003, and adapted to our problem, and on a fine analysis of the singularities of Green's function associated with the class of parabolic problems we investigate. An immediate consequence of our results is the indistinguishability of mild and variational solutions in the case of uniqueness.Comment: 37 page

Alteración de las biotitas y biotitas clorizadas de la granodiorita de Collserola (Barcelona)

[ES] Se estudia la alteración natural experimentada por las biotitas y biotitas cloritizadas de la granodiorita de la Serra de Collcerola (Barcelona), bajo clima mediterráneo. Se muestrearon nueve perfiles, separándose magnéticamente las biotitas de la fracción arena, analizándose químicamente (elementos totales), por difractometría de R.X. y microscopia electrónica de barrido. También se estudiaron con el microscopio petrográfico láminas delgadas de muestras no perturbadas. Las biotitas evolucionan progresivamente primero a interestratificados biotita-vermiculita y después a vermiculita, apreciándose una tendencia a la esmectización, sugiriendo que las condiciones medioambitentales en las que se ha producido la alteración son muy similares a las actuales. La fase clorita procedente de las biotitas cloritizadas permanece inalterada en dicho medio.[EN] The weathering of biotites and cloritized biotites from a granodiorite from Serra de Collcerola (Barcelona, Spain), under mediterranean climate, has been studied. The sand fraction from the horizons of nine pedological profiles were sampled and the biotites magnetically separated and analysed by XRD, SEM and chemically. Thin sections of undisturbed samples were also studied. The weathering sequence from non-cloritized biotites was Biotite - Interlayer B-V - Vermiculite with a trend towards smectitization. The chlorite mineral from chloritized biotites remains unchanged in this environment. These results suggest tliat the weathering environment has been quite similar to the present one.Peer reviewe

Inclusion of chemical effect in a fully coupled THM finite element code

Bentonite-rich clays can be used as a buffer / backfill material in deep geological repositories for nuclear waste. The prediction of the long-term performance of a buffer / backfill in such a complex environment, where the temperature, humidity and chemistry of water change, requires a fully thermo-hydro-mechanical-chemical (THMC) coupled numerical code. This paper presents a simple extension of a THM coupled finite element code to include chemical effects. After deriving the governing salt mass balance equation and discussing its implementation into the code, the paper verifies the extended framework against an analytical solution for 1D salt transport. In addition, the article presents a validation example in which the code replicates experimental data. The numerical results obtained from the extended THMC coupled finite element code encourage further investigation of the chemical effects on the mechanical and thermal behaviour of the material, which would serve the ultimate goal of achieving a safer design of the nuclear waste storage facility.Postprint (published version

Analysis of strain localization with a nonlocal plasticity model

In the present paper a nonlocal plasticity model is described, intended to reproduce the mechanical behaviour of stiff fine-grained soils, including the objective simulation of strain localization; the phenomenon of accumulation of deformations in narrow zones in the form of shear bands or fractures. A number of analyses have been performed to assess the developed formulation. Relevant aspects have been addressed such as the thickness of the shear band, its orientation, and the onset of localization in a boundary value problem (BVP). Results provide useful insigths into relevant aspects of the numerical simulation of strain localization

A linear construction for certain Kerdock and Preparata codes

The Nordstrom-Robinson, Kerdock, and (slightly modified) Pre\- parata codes are shown to be linear over \ZZ_4, the integers $\bmod~4$. The Kerdock and Preparata codes are duals over \ZZ_4, and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over \ZZ_4. This provides a simple definition for these codes and explains why their Hamming weight distributions are dual to each other. First- and second-order Reed-Muller codes are also linear codes over \ZZ_4, but Hamming codes in general are not, nor is the Golay code.Comment: 5 page