Controllability and Qualitative properties of the solutions to SPDEs driven by boundary L\'evy noise

Let $u$ be the solution to the following stochastic evolution equation (1) du(t,x)& = &A u(t,x) dt + B \sigma(u(t,x)) dL(t),\quad t>0; u(0,x) = x taking values in an Hilbert space \HH, where $L$ is a \RR valued L\'evy process, $A:H\to H$ an infinitesimal generator of a strongly continuous semigroup, \sigma:H\to \RR bounded from below and Lipschitz continuous, and B:\RR\to H a possible unbounded operator. A typical example of such an equation is a stochastic Partial differential equation with boundary L\'evy noise. Let \CP=(\CP_t)_{t\ge 0} %{\CP_t:0\le t<\infty}$the corresponding Markovian semigroup. We show that, if the system (2) du(t) = A u(t)\: dt + B v(t),\quad t>0 u(0) = x is approximate controllable in time$T>0$, then under some additional conditions on$B$and$A$, for any$x\in H$the probability measure$\CP_T^\star \delta_x$is positive on open sets of$H$. Secondly, as an application, we investigate under which condition on$\HH$and on the L\'evy process$L$and on the operator$A$and$B$ the solution of Equation [1] is asymptotically strong Feller, respective, has a unique invariant measure. We apply these results to the damped wave equation driven by L\'evy boundary noise

Selective Synthesis of a Salt and a Cocrystal of the Ethionamide-Salicylic Acid System

Herein is presented a rare example of salt/cocrystal polymorphism involving the adduct between ethionamide (ETH) and salicylic acid (SAL). Both the salt and cocrystal forms have the same stoichiometry and composition and are both stable at room temperature. The synthetic procedure was successfully optimized in order to selectively obtain both polymorphs. The two adducts' structures were thoroughly investigated by means of single-crystal X-ray diffraction, solid-state NMR spectroscopy, and density functional theory (DFT) calculations. From the solid-state NMR point of view, the combination of mono- and multinuclear experiments (1H MAS, 13C and 15N CPMAS, 1H-{14N} D-HMQC, 1H-14N PM-S-RESPDOR) provided undoubted spectroscopic evidence about the different positions of the hydrogen atom along the main N\ub7\ub7\ub7H\ub7\ub7\ub7O interaction. In particular, the 1H-14N PM-S-RESPDOR allowed N-H distance measurements through the 1H detected signal at a very high spinning speed (70 kHz), which remarkably agree with those derived by DFT optimized X-ray diffraction, even on a natural abundance real system. The thermodynamic relationship between the salt and the cocrystal was inquired from the experimental and computational points of view, enabling the characterization of the two polymorphs as enantiotropically related. The performances of the two forms in terms of dissolution rate are comparable to each other but significantly higher with respect to the pure ETH

Linear Operator Inequality and Null Controllability with Vanishing Energy for unbounded control systems

We consider linear systems on a separable Hilbert space $H$, which are null controllable at some time $T_0>0$ under the action of a point or boundary control. Parabolic and hyperbolic control systems usually studied in applications are special cases. To every initial state $y_0 \in H$ we associate the minimal "energy" needed to transfer $y_0$ to $0$ in a time $T \ge T_0$ ("energy" of a control being the square of its $L^2$ norm). We give both necessary and sufficient conditions under which the minimal energy converges to $0$ for $T\to+\infty$. This extends to boundary control systems the concept of null controllability with vanishing energy introduced by Priola and Zabczyk (Siam J. Control Optim. 42 (2003)) for distributed systems. The proofs in Priola-Zabczyk paper depend on properties of the associated Riccati equation, which are not available in the present, general setting. Here we base our results on new properties of the quadratic regulator problem with stability and the Linear Operator Inequality.Comment: In this version we have also added a section on examples and applications of our main results. This version is similar to the one which will be published on "SIAM Journal on Control and Optimization" (SIAM

Anti-prion drug mPPIg5 inhibits PrP(C) conversion to PrP(Sc).

Prion diseases, also known as transmissible spongiform encephalopathies, are a group of fatal neurodegenerative diseases that include scrapie in sheep, bovine spongiform encephalopathy (BSE) in cattle and Creutzfeldt-Jakob disease (CJD) in humans. The 'protein only hypothesis' advocates that PrP(Sc), an abnormal isoform of the cellular protein PrP(C), is the main and possibly sole component of prion infectious agents. Currently, no effective therapy exists for these diseases at the symptomatic phase for either humans or animals, though a number of compounds have demonstrated the ability to eliminate PrPSc in cell culture models. Of particular interest are synthetic polymers known as dendrimers which possess the unique ability to eliminate PrP(Sc) in both an intracellular and in vitro setting. The efficacy and mode of action of the novel anti-prion dendrimer mPPIg5 was investigated through the creation of a number of innovative bio-assays based upon the scrapie cell assay. These assays were used to demonstrate that mPPIg5 is a highly effective anti-prion drug which acts, at least in part, through the inhibition of PrP(C) to PrP(Sc) conversion. Understanding how a drug works is a vital component in maximising its performance. By establishing the efficacy and method of action of mPPIg5, this study will help determine which drugs are most likely to enhance this effect and also aid the design of dendrimers with anti-prion capabilities for the future

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup T on a Banach space X we define its adjoint on an appropriate closed subspace X^o of the norm dual X'. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology (X^o,X). An application is the following: For K a Polish space we consider operator semigroups on the space C(K) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(K) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(K) are precisely those that are adjoints of a bi-continuous semigroups on C(K). We also prove that the class of bi-continuous semigroups on C(K) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if K is not Polish space this is not the case

Exponential Ergodicity of stochastic Burgers equations driven by α\alpha-stable processes

In this work, we prove the strong Feller property and the exponential ergodicity of stochastic Burgers equations driven by $\alpha/2$-subordinated cylindrical Brownian motions with $\alpha\in(1,2)$. To prove the results, we truncate the nonlinearity and use the derivative formula for SDEs driven by $\alpha$-stable noises established in Zhang (arXiv:1204.2630v2).Comment: 17p