The Damped Wave Equation with Acoustic Boundary Conditions and Non-locally Reacting Surfaces

The aim of the paper is to study the problem $$u_{tt}+du_t-c^2\Delta u=0 \qquad \text{in $\mathbb{R}\times\Omega$,}$$ $$\mu v_{tt}- \text{div}_\Gamma (\sigma \nabla_\Gamma v)+\delta v_t+\kappa v+\rho u_t =0\qquad \text{on $\mathbb{R}\times \Gamma_1$,}$$ $$v_t =\partial_\nu u\qquad \text{on $\mathbb{R}\times \Gamma_1$,}$$ $$\partial_\nu u=0 \text{on $\mathbb{R}\times \Gamma_0$,}$$ $$u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x)\quad \text{in $\Omega$,}$$ $$v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x) \quad \text{on $\Gamma_1$,}$$ where $\Omega$ is a open domain of $\mathbb{R}^N$ with uniformly $C^r$ boundary ($N\ge 2$, $r\ge 1$), $\Gamma=\partial\Omega$, $(\Gamma_0,\Gamma_1)$ is a relatively open partition of $\Gamma$ with $\Gamma_0$ (but not $\Gamma_1$) possibly empty. Here $\text{div}_\Gamma$ and $\nabla_\Gamma$ denote the Riemannian divergence and gradient operators on $\Gamma$, $\nu$ is the outward normal to $\Omega$, the coefficients $\mu,\sigma,\delta, \kappa, \rho$ are suitably regular functions on $\Gamma_1$ with $\rho,\sigma$ and $\mu$ uniformly positive, $d$ is a suitably regular function in $\Omega$ and $c$ is a positive constant. In this paper we first study well-posedness in the natural energy space and give regularity results. Hence we study asymptotic stability for solutions when $\Omega$ is bounded, $\Gamma_1$ is connected, $r=2$, $\rho$ is constant and $\kappa,\delta,d\ge 0$.Comment: The paper also extends some results in arXiv:2105.0921

Power spectral density of a single Brownian trajectory: what one can and cannot learn from it

The power spectral density (PSD) of any time-dependent stochastic processXt is ameaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of Xt over an infinitely large observation timeT, that is, it is defined as an ensemble-averaged property taken in the limitT  ¥.Alegitimate question iswhat information on the PSDcan be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes thePSD of a single trajectory recorded for a finite observation timeT. In quest for this answer, for a d-dimensionalBrownian motion (BM) we calculate the probability density function of a single-trajectory PSDfor arbitrary frequency f, finite observation timeTand arbitrary number k of projections of the trajectory on different axes.We show analytically that the scaling exponent for the frequency-dependence of the PSDspecific to an ensemble ofBMtrajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is afluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail.Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncatedWiener representation ofBM, and the case of a discrete-time lattice randomwalk.Wehighlight some differences in the behavior of a single-trajectory PSDforBMand for the two latter situations.The framework developed herein will allow formeaningful physical analysis of experimental stochastic trajectories

The Interaction of the Antitumor Complexes Na[trans-RuCl4 (DMSO)(Im)] and Na[trans-RuCl4(DMSO)(Ind)] With Apotransferrin: a Spectroscopic Study

The interaction of two antitumor ruthenium(III) complexes,-Na[trans-RuCl4(DMSO)(Im)] and Na[trans-RuCl4(DMSO)(Ind)]- with human serum apotransferrin (apoTf) was investigated through a number of spectroscopic techniques such as UV-Vis absorption, CD and 1H NMR spectroscopy. Interestingly, the hydrolysis profiles of these complexes in a physiological buffer are markedly affected by the presence, in solution, of apoTf suggesting the occurrence of a specific interaction of their respective hydrolysis products with the protein. The formation of stable adducts with apotransferrin has been demonstrated by CD spectroscopy, and additional information obtained through 1H NMR of the hyperfine shifted signals. The bound ruthenium(III) species may be detached from these adducts by addition of excess citrate at low pH. The behavior of the investigated ruthenium(III) complexes with apoTf is compared with that of the recently described and strictly related ru-im and ru-ind antitumour complexes, and discussed in the frame of general strategies of drug targeting

Passive advection of fractional Brownian motion by random layered flows

We study statistical properties of the process $Y(t)$ of a passive advection by quenched random layered flows in situations when the inter-layer transfer is governed by a fractional Brownian motion $X(t)$ with the Hurst index $H \in (0,1)$. We show that the disorder-averaged mean-squared displacement of the passive advection grows in the large time $t$ limit in proportion to $t^{2 - H}$, which defines a family of anomalous super-diffusions. We evaluate the disorder-averaged Wigner-Ville spectrum of the advection process $Y(t)$ and demonstrate that it has a rather unusual power-law form $1/f^{3 - H}$ with a characteristic exponent which exceed the value $2$. Our results also suggest that sample-to-sample fluctuations of the spectrum can be very important.Comment: 18 pages, 4 figure

cis-Locked Ru(II)-DMSO Precursors for the Microwave-Assisted Synthesis of Bis-Heteroleptic Polypyridyl Compounds

We describe a synthetic strategy for the preparation of bis-heteroleptic polypyridyl Ru(II) complexes of the type [Ru(L1)2(L2)]2+ (L1 and L2 = diimine ligands) from well-defined Ru(II) precursors. For this purpose, a series of six neutral, anionic, and cationic cis-locked Ru(II)-DMSO complexes (2-7) of the general formula [Y] fac-[RuX(DMSO-S)3(O-O)]n (where O-O is a symmetrical chelating anion: oxalate (ox), malonate (mal), acetylacetonate (acac); X = DMSO-O or Cl-; n = -1/0/+1 depending on the nature and charge of X and O-O; when present, Y = K+ or PF6-) were efficiently prepared from the well-known cis-[RuCl2(DMSO)4] (1). When treated with diimine chelating ligands (L1 = bpy, phen, dpphen), the compounds 2-7 afforded the target [Ru(L1)2(O-O)]0/+ complex together with the undesired (and unexpected) [Ru(L1)3]2+ species. Nevertheless, we found that the formation of [Ru(L1)3]2+can be minimized by carefully adjusting the reaction conditions: in particular, high selectivity toward [Ru(L1)2(O-O)]0/+ and almost complete conversion of the precursor was obtained within minutes, also on a 100-200 mg scale, when the reactions were performed in absolute ethanol at 150 \ub0C in a microwave reactor. Depending on the nature of L1 and concentration, with the oxalate and malonate precursors, the neutral product [Ru(L1)2(O-O)] can precipitate spontaneously from the final mixture, in pure form and acceptable-to-good yields. When spontaneous precipitation of the disubstituted product does not occur, purification from [Ru(L1)3]2+ can be rather easily accomplished by column chromatography or solvent extraction. By comparison, under the same conditions, compound 1 is much less selective, thus demonstrating that locking the geometry of the precursor through the introduction of O-O in the coordination sphere of Ru is a valid strategic approach. By virtue of its proton-sensitive nature, facile and quantitative replacement of O-O in [Ru(L1)2(O-O)]0/+ by L2, selectively affording [Ru(L1)2(L2)]2+, was accomplished in refluxing ethanol in the presence of a slight excess of trifluoroacetic acid or HPF6

New Insight into a Deceptively Simple Reaction: The Coordination of bpy to RuII-Carbonyl Precursors - The Central Role of thefac-[Ru(bpy)Cl(CO)3]+Intermediate and theChloride Rebound Mechanism

This work demonstrates how a careful reexamination of well-trodden fields can fill conceptual gaps that previously escaped full understanding. The coordination of 2,2'-bipyridine (bpy) to the known Ru(II)-chlorido-carbonyl precursors \u2013 the dinuclear [RuCl2(CO)3]2 (P1) and the polymeric [RuCl2(CO)2]n (P2) \u2013 has been investigated by several groups in the past, and a remarkably large number of ruthenium mono(bpy) carbonyls were identified and fully characterized. Many were investigated as catalysts or key intermediates for the photochemical, electrochemical, and photo-electrochemical reduction of CO2, and for the water\u2013gas shift reaction. Nevertheless, even though most \u2013 if not all \u2013 the reaction products are known already, a careful exam of the literature led us to believe that a convincing general scheme interconnecting them all was still missing and important questions remained unanswered. For this reason, we investigated the reactivity of two mononuclear Ru(II)-carbonyl-dmso precursors, trans,cis,cis-[RuCl2(CO)2(dmso-O)2] (P3) and fac-[RuCl2(CO)3(dmso-O)] (P4) \u2013 that can be considered as \u2018activated forms\u2019 of P2 and P1, respectively \u2013 towards the coordination of bpy. Compounds P3 and P4, allowed us to gain new mechanistic insight and a deeper level of understanding. In particular, we found that coordination of bpy to P4 (or P1) generates first the tricarbonyl cation fac-[Ru(bpy)Cl(CO)3]+.This key intermediate undergoes the facile and selective nucleophilic attack on the CO trans to Cl (by RO\u2013 in alcoholic solvents or OH\u2013 from adventitious water in other solvents), leading to all other species. We also demonstrated that Cl\u2013 \u2013 even when in large excess \u2013 is unable to replace a carbonyl on fac-[Ru(bpy)Cl(CO)3]+. However, the chloride set free from the precursor, competes efficiently with bpy for the coordination to Ru(II) (chloride rebound mechanism)

Approximate analytical description of the nonaffine response of amorphous solids

An approximation scheme for model disordered solids is proposed that leads to the fully analytical evaluation of the elastic constants under explicit account of the inhomogeneity (nonaffinity) of the atomic displacements. The theory is in quantitative agreement with simulations for central-force systems and predicts the vanishing of the shear modulus at the isostatic point with the linear law {\mu} ~ (z - 2d), where z is the coordination number. The vanishing of rigidity at the isostatic point is shown to be a consequence of the canceling out of positive affine and negative nonaffine terms

Frequency-frequency correlations of single-trajectory spectral densities of Gaussian processes

We investigate the stochastic behavior of the single-trajectory spectral density $S(\omega,\mathcal{T})$ of several Gaussian stochastic processes, i.e., Brownian motion, the Ornstein-Uhlenbeck process, the Brownian gyrator model and fractional Brownian motion, as a function of the frequency $\omega$ and the observation time $\mathcal{T}$. We evaluate in particular the variance and the frequency-frequency correlation of $S(\omega,\mathcal{T})$ for different values of $\omega$. We show that these properties exhibit different behaviors for different physical cases and can therefore be used as a sensitive probe discriminating between different kinds of random motion. These results may prove quite useful in the analysis of experimental data.Comment: 29 pages, 7 figure