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

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

(15)N NMR spectroscopy unambiguously establishes the coordination mode of the diimine linker 2-(2'-pyridyl)pyrimidine-4-carboxylic acid (cppH) in Ru(II) complexes

We investigated the reactivity of three Ru(II) precursors - trans,cis,cis-[RuCl2(CO)2(dmso-O)2], cis,fac-[RuCl2(dmso-O)(dmso-S)3], and trans-[RuCl2(dmso-S)4] - towards the diimine linker 2-(2'-pyridyl)pyrimidine-4-carboxylic acid (cppH) or its parent compound 4-methyl-2-(2'-pyridyl)pyrimidine ligand (mpp), in which a methyl group replaces the carboxylic group on the pyrimidine ring. In principle, both cppH and mpp can originate linkage isomers, depending on how the pyrimidine ring binds to ruthenium through the nitrogen atom ortho (N(o)) or para (N(p)) to the group in position 4. The principal aim of this work was to establish a spectroscopic fingerprint for distinguishing the coordination mode of cppH/mpp also in the absence of an X-ray structural characterization. By virtue of the new complexes described here, together with the others previously reported by us, we successfully recorded (1)H,(15)N-HMBC NMR spectra at natural abundance of the (15)N isotope on a consistent number of fully characterized Ru(ii)-cppH/mpp compounds, most of them being stereoisomers and/or linkage isomers. Thus, we found that (15)N NMR chemical shifts unambiguously establish the binding mode of cppH and mpp - either through N(o) or N(p) - and can be conveniently applied also in the absence of the X-ray structure. In fact, coordination of cppH to Ru(ii) induces a marked upfield shift for the resonance of the N atoms directly bound to the metal, with coordination induced shifts (CIS) ranging from ca. -45 to -75 ppm, depending on the complex, whereas the unbound N atom resonates at a frequency similar to that of the free ligand. Similar results were found for the complexes of mpp. This work confirmed our previous finding that cppH has no binding preference, whereas mpp binds exclusively through N(p). Interestingly, the two cppH linkage isomers trans,cis-[RuCl2(CO)2(cppH-\u3baN(p))] () and trans,cis-[RuCl2(CO)2(cppH-\u3baN(o))] () were easily obtained in pure form by exploiting their different solubility properties

Neutral 1,3,5-Triaza-7-phosphaadamantane-Ruthenium(II) Complexes as Precursors for the Preparation of Highly Water-Soluble Derivatives

The monodentate phosphane ligand 1,3,5-triaza-7-phosphaadamantane (PTA) imparts excellent water solubility to its complexes. We aimed to prepare precursors with one or more PTA coligands for solubility and one or more labile ligands for facile replacement by a linker. For this purpose, we investigated the reactivity of the neutral isomers trans- and cis-RuCl2(PTA)4 (1 and 2) towards 2,2\u2032-bipyridine (bpy), as a model chelating diimine linker. The new derivatives mer-[Ru(bpy)Cl(PTA)3]Cl (9) and fac-[Ru(bpy)Cl(PTA)3]Cl (10) were prepared and characterized. We also found that PTA reacts rapidly with cis,fac-RuCl2(dmso-O)(dmso-S)3 (11) and trans-RuCl2(dmso-S)4 (13) under mild conditions through the replacement of pairs of mutually trans dmso ligands with high selectivity, even when in stoichiometric defect. Thus, 11 affords cis,cis,trans-RuCl2(dmso-S)2(PTA)2 (12), whereas 13 gives 1. The two dmso ligands of 12 can be replaced selectively by chelating diimines such as bpy to afford the less symmetrical all-cis product cis,cis-Ru(bpy)Cl2(PTA)2 (16)

The face of Ebola: changing frequency of haemorrhage in the West African compared with Eastern-Central African outbreaks

BACKGROUND: The West-African (WA) Zaire Ebolavirus disease (EVD) outbreak was characterized by an exceptionally high number of cases and deaths as compared with the Eastern-Central African (ECA) outbreaks. Despite the Zaire Ebolavirus being the most lethal for humans, case-fatality rate, close to 80 % in ECA outbreaks, almost halved to 47 % in Guinea-Liberia-Sierra Leone (WA). Such an improvement was due to the remarkable implementation of international humanitarian aids. Some studies also suggested that the long human-to-human transmission cycle occurred in WA, gave rise to human adaptation and consequent immune escape. Haemorrhage, the main feature in seriously infected EVD patients, is due to the immune system that triggers the infected endothelial cells which expose the spike-like glycoprotein (GP) of the virion on their surface. If the human adaptation hypothesis holds true, the proportion of EVD patients with haemorrhage in the WA outbreak should be lower than in the ECA outbreaks due to immune escape. Therefore, the aim of this meta-analysis was to compare the relative frequencies of three typical haemorrhagic symptoms (conjunctival -CB, nasal -NB, gingival -GB- bleedings) in the ECA and WA outbreaks. METHODS: Literature searches were performed through PubMed and Scopus using generic keywords; surveys including at least ten patients reporting CB, NB, GB relative frequencies were extracted and split into ECA and WA. The meta-analytical methods chosen were based on the levels of between-study heterogeneity and publication bias. Pooled CB, NB, GB relative frequencies in ECA and WA were estimated and compared. Subgroup analysis including only studies on Zaire Ebolavirus also was performed. RESULTS: Fifteen studies (10 ECA, 5 WA) were located with 4,867 (CB), 3,859 (NB), 4,278 (GB) EVD patients overall. GB pooled relative frequency was 45.3 % (95 % confidence interval -95 CI, 34.7-56.1 %) and 18.0 % (95 CI, 6.0-34.5 %), in ECA and WA; NB was 10.6 % (95 CI, 5.7-16.8 %) and 1.3 % (1.0-1.8 %); GB was 24.2 % (95 CI, 11.9-39.2 %) and 1.9 % (95 CI, 1.4-2.4 %). Subgroup analysis confirmed these results. CONCLUSIONS: During the WA outbreak the relative frequency of GB decreased by two thirds, while NB and GB almost disappeared, suggesting that the Zaire Ebolavirus human adaptation hypothesis is plausible

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

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