Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string

We consider a one-dimensional wave equation, which governs the vibrations of a damped string with spatially nonhomogeneous density and damping coefficients. We introduce a family of boundary conditions depending on a complex parameter . Corresponding to different values of , the problem describes either vibrations of a finite string or propagation of elastic waves on an infinite string. Our main object of interest is the family of non-selfadjoint operators in the energy space of two-component initial data. These operators are the generators of the dynamical semigroups corresponding to the above boundary-value problems. We show that the operators are dissipative, simple, maximal operators, which differ from each other by rank-one perturbations. We also prove that the operator coincides with the generator of the Lax-Phillips semigroup, which plays an important role in the aforementioned scattering problem. The results of this work are applied in our two forthcoming papers both to the proof of the Riesz basis property of the eigenvectors and associated vectors of the operators and to establishing the exact and approximate controllability of the system governed by the damped wave equation.

     

Rotational spectrum and internal dynamics of tetrahydrofuran-krypton

The rotational spectrum of the tetrahydrofuran-krypton van der Waals complex has been investigated by pulsed-jet Fourier transform microwave spectroscopy. The spectra of the (84)Kr and (86)Kr isotopologues have been assigned and the krypton atom is located nearly over the oxygen atom, almost perpendicular to the COC plane. Each rotational transition is split into two component lines due to, according to the observed Coriolis coupling term between the tunneling states, the residual pseudorotational effects of the ring in the complex. The splitting between the two vibrational sublevels is 87.462(2) and 87.062(2) MHz for the (84)Kr and (86)Kr isotopologues, respectively. These splittings have been used to determine the barrier to inversion, B(2) = 67 cm(-1). The dissociation energy has been estimated to be 3.7 kJ mol(-1) from centrifugal distortion effects.     

Coordination properties of 3-functionalized beta-cyclodextrins. Thermodynamic stereoselectivity of copper(II) complexes of the A,B-diamino derivative and its exploitation in LECE

The bis-amino AB derivative of beta-cyclodextrin on the secondary rim was synthesised and spectroscopically characterised by different techniques. Its binary systems both with protons and copper(II) were thermodynamically characterised by pH-metric potentiometry. In addition the ternary systems with each of the enantiomers of tryptophan and alanine were investigated. A thermodynamic stereoselectivity was observed for the tryptophan enantiomers and this was exploited to separate them by capillary electrophoresis through a ligand exchange mechanism (LECE). LECE separation of racemates of phenylalanine and tyrosine was also obtained.     

Control problems for energy harvester model and interpolation in Hardy space

Three control problems for the system of two coupled differential equations governing the dynamics of an energy harvesting model are studied. The system consists of the equation of an Euler–Bernoulli beam model and the equation representing the Kirchhoff's electric circuit law. Both equations contain coupling terms representing the inverse and direct piezoelectric effects. The system is reformulated as a single evolution equation in the state space of 3-component functions. The control is introduced as a separable forcing term $\text{◂⋅▸}\mathbf{g}(x)f(t)$ on the right-hand side of the operator equation. The first control problem deals with an explicit construction of $f(t)$ that steers an initial state to zero on a time interval [0, T]. The second control problem deals with the construction of $f(t)$ such that the voltage output is equal to some given function $\mathbf{v}(t)$ (with $\mathbf{g}(x)$ being given as well). The third control problem deals with an explicit construction of both the force profile, $\mathbf{g}(x)$, and the control, $f(t)$, which generate the desired voltage output $\mathbf{v}(t)$. Interpolation theory in the Hardy space of analytic functions is used in the solution of the second and third problems.     

How To Make Davies' Theorem Visible

We prove that for an arbitrary measurable set A R² and a -finiteBorel measure µ on the plane, there is a Borel set oflines L such that for each point in A, the set of directionsof those lines from L containing the point is a residual set,and, moreover, We show how this result may be used to characterise the sets of the planefrom which an invisible set is visible. We also characterisethe rectifiable sets C₁, C₂ for which there is a set which isvisible from C₁ and invisible from C₂.     

Asymptotics of eigenfrequencies and eigenmodes of non‐homogeneous inextensible filament with an end load

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. We derive the spectral asymptotics for the aforementioned two‐parameter family of non‐selfadjoint operators. In the forthcoming papers, based on the asymptotical results of the present paper, we will prove the Riesz basis property of the eigenfunctions. The spectral results obtained in the aforementioned papers will allow us to solve boundary and/or distributed controllability problems for the filament using the spectral decomposition method. Copyright © 2001 John Wiley & Sons, Ltd.     

Asymptotic analysis of nonselfadjoint operators generated by coupled Euler‐Bernoulli and Timoshenko beam model

In the current paper, we present a series of results on the asymptotic and spectral analysis of coupled Euler‐Bernoulli and Timoshenko beam model. The model is well‐known in the different branches of the engineering sciences, such as in mechanical and civil engineering (in modelling of responses of the suspended bridges to a strong wind), in aeronautical engineering (in predicting and suppressing flutter in aircraft wings, tails, and control surfaces), in engineering and practical aspects of the computer science (in suppressing bending‐torsional flutter of a new generation of hard disk drives, which is expected to pack high track densities (20,000+TPI) and rotate at very high speeds (25,000+RPM)), in medical science (in bio mechanical modelling of bloodcarrying vessels in the body, which are elastic and collapsible). The aforementioned mathematical model is governed by a system of two coupled differential equations and a two parameter family of boundary conditions representing the action of the self‐straining actuators. This linear hyperbolic system is equivalent to a single operator evolution equation in the energy space. That equation defines a semigroup of bounded operators and a dynamics generator of the semigroup is our main object of interest. We formulate and proof the following results: (a) the dynamics generator is a nonselfadjoint operator with compact resolvent from the class ??p with p > 1; (b) precise spectral asymptotics for the two‐branch discrete spectrum; (c) a nonselfadjoint operator, which is the inverse of the dynamics generator, is a finite‐rank perturbation of a selfadjoint operator. The latter fact is crucial for the proof that the root vectors of the dynamics generator form a complete and minimal set. In our forthcoming paper, we will use the spectral results to prove that the dynamics generator is Riesz spectral, which will allow us to solve several boundary and distributed controllability problems via the spectral decomposition method. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Austrodoral and austrodoric acid: nor-sesquiterpenes with a new carbon skeleton from the Antarctic nudibranch Austrodoris kerguelenensis

Two unprecedented nor-sesquiterpenes, austrodoral (4) and its oxidised derivative austrodoric acid (5), have been isolated from the skin of the marine dorid Austrodoris kerguelenensis, collected in Antarctica. The structures and the relative stereochemistry were elucidated on the basis of spectroscopic data. A role of stress-metabolites could be suggested for these compounds.     

Bis(arylimino)pyridine derivatives of Group 4 metals: preparation, characterization and activity in ethylene polymerization

Titanium tetrachloride reacts with 2,6-bis[(1-phenylimino)ethyl]pyridine, 1, and 2,6-bis[1-(2,6-diisopropylphenylimino)ethyl]pyridine, 2, giving the adducts of general formulae [Ti1Cl3]Cl, 3, and [Ti2Cl3]Cl, 6, the latter through the intermediacy of the covalently bonded [Ti2Cl4], 4. Heating 6 leads to reduction to the titanium(III) derivative [Ti2Cl3], 12, the latter characterized by X-ray diffraction methods. The reaction of [Ti1Cl3]Cl with a toluene solution of MAO proceeds with methylation at the ortho-position of the pyridine ring to give the titanium(iv) derivative [Ti(C22H21N3)Cl3], 8. The reaction of [Ti2Cl3]Cl with MAO gives a mixture of products containing [Ti2Cl2(OAlCl3)], 9. Compound 9, which has been prepared independently by reacting 6 with AlOCl, is a rare case of a compound containing the -OAlCl3 moiety, as shown by a single-crystal X-ray diffraction study. From the tetrachlorides of zirconium and hafnium with 1 or 2, the corresponding adducts [M(L)Cl4] have been obtained in high yields. These derivatives of Group 4 metals act as ethylene polymerization catalytic precursors: the substitution of the phenyl ring of the imino fragment strongly influences the catalytic activity which is 5,544 kg(PE) mol(Ti)(-1) h(-1) in the case of 3 and 267 kg(PE) mol(Ti)(-1) h(-1) with 6. Catalytic activity has been observed for zirconium and hafnium too, the activity decreasing from zirconium to hafnium, under comparable conditions.     

Monitoring the Evolution of Asynchrony between Mean Arterial Pressure and Mean Cerebral Blood Flow via Cross-Entropy Methods

Cerebrovascular control is carried out by multiple nonlinear mechanisms imposing a certain degree of coupling between mean arterial pressure (MAP) and mean cerebral blood flow (MCBF). We explored the ability of two nonlinear tools in the information domain, namely cross-approximate entropy (CApEn) and cross-sample entropy (CSampEn), to assess the degree of asynchrony between the spontaneous fluctuations of MAP and MCBF. CApEn and CSampEn were computed as a function of the translation time. The analysis was carried out in 23 subjects undergoing recordings at rest in supine position (REST) and during active standing (STAND), before and after surgical aortic valve replacement (SAVR). We found that at REST the degree of asynchrony raised, and the rate of increase in asynchrony with the translation time decreased after SAVR. These results are likely the consequence of the limited variability of MAP observed after surgery at REST, more than the consequence of a modified cerebrovascular control, given that the observed differences disappeared during STAND. CApEn and CSampEn can be utilized fruitfully in the context of the evaluation of cerebrovascular control via the noninvasive acquisition of the spontaneous MAP and MCBF variability.