Simple Systems with Anomalous Dissipation and Energy Cascade

We analyze a class of dynamical systems of the type where f _n (t) is a forcing term with only for and the coupling coefficients c _n satisfy a condition ensuring the formal conservation of energy . Despite being formally conservative, we show that these dynamical systems support dissipative solutions (suitably defined) and, as a result, may admit unique (statistical) steady states when the forcing term f _n (t) is nonzero. This claim is demonstrated via the complete characterization of the solutions of the system above for specific choices of the coupling coefficients c _n . The mechanism of anomalous dissipations is shown to arise via a cascade of the energy towards the modes with higher n; this is responsible for solutions with interesting energy spectra, namely scales as as n→∞. Here the exponents α depend on the coupling coefficients c _n and denotes expectation with respect to the equilibrium measure. This is reminiscent of the conjectured properties of the solutions of the Navier-Stokes equations in the inviscid limit and their accepted relationship with fully developed turbulence. Hence, these simple models illustrate some of the heuristic ideas that have been advanced to characterize turbulence, similar in that respect to the random passive scalar or random Burgers equation, but even simpler and fully solvable.     

Axial field shaping under high-numerical-aperture focusing

Kant reported [J. Mod. Optics 47, 905 (2000)] a formulation for solving the inverse problem of vector diffraction, which accurately models high-NA focusing. Here, Kant's formulation is adapted to the method of generalized projections to obtain an algorithm for designing diffractive optical elements (DOEs) that reshape the axial point-spread function (PSF). The algorithm is applied to design a binary phase-only DOE that superresolves the axial PSF with controlled increase in axial sidelobes. An 11-zone DOE is identified that axially narrows the PSF central lobe by 29% while maintaining the sidelobe intensity at or below 52% of the peak intensity. This DOE could improve the resolution achievable in several applications without significantly complicating the optical system.     

Diversity of new structural types in polynuclear iron chemistry with a tridentate N,N,O ligand

The syntheses, crystal structures, and magnetochemical characterization of four new iron clusters [Fe7O4(O2CPh)11(dmem)2] (1), [Fe7O4(O2CMe)11(dmem)2] (2), [Fe6O2(OH)4(O2CBut)8(dmem)2] (3), and [Fe3O(O2CBut)2(N3)3(dmem)2] (4) (dmemH=Me2NCH2CH2N(Me)CH2CH2OH)=2-{[2-(dimethylamino)ethyl]methylamino}ethanol) are reported. The reaction of dmemH with [Fe3O(O2CR)6(H2O)3](NO3) (R=Ph (1), Me (2), and But (3)) gave 1, 2, and 3, respectively, whereas 4 was obtained from the reaction of 3 with sodium azide. The complexes all possess rare or novel core topologies. The core of 1 comprises two [Fe4(mu3-O)2]8+ butterfly units sharing a common body Fe atom. The core of 2 consists of a [Fe3O3] ring with each doubly bridging O2- ion becoming mu3 by also bridging to a third, external Fe atom; a seventh Fe atom is attached on the outside of this core via an additional mu3-O2- ion. The core of 3 consists of a [Fe4(mu3-O)2]8+ butterfly unit with an Fe atom attached above and below this by bridging O atoms. Finally, the core of 4 is an isosceles triangle bridged by a mu3-O2- ion with a rare T-shaped geometry and with the azide groups all bound terminally. Variable-temperature, solid-state dc, and ac magnetization studies were carried out on complexes 1-4 in the 5.0-300 K range. Fitting of the obtained magnetization (M) vs field (H) and temperature (T) data by matrix diagonalization and including only axial anisotropy (zero-field splitting) established that 1, 2, and 4 each possess an S=5/2 ground state spin, whereas 3 has an S=5 ground state. As is usually the case, good fits of the magnetization data could be obtained with both positive and negative D values. To obtain more accurate values and to determine the sign of D, high-frequency EPR studies were carried out on single crystals of representative complexes 1.4MeCN and 3.2MeCN, and these gave D=+0.62 cm-1 and |E|>or=0.067 cm-1 for 1.4MeCN and D=-0.25 cm-1 for 3.2MeCN. The magnetic susceptibility data for 4 were fit to the theoretical chiM vs T expression derived by the use of an isotropic Heisenberg spin Hamiltonian and the Van Vleck equation, and this revealed the pairwise exchange parameters to be antiferromagnetic with values of Ja=-3.6 cm-1 and Jb=-45.9 cm-1. The combined results demonstrate the ligating flexibility of dmem and its usefulness in the synthesis of a variety of Fex molecular species.     

The highest nuclearity metal oxime clusters: Ni14 and Ni12Na2 complexes from the use of 2-pyridinealdoximate and azide ligands

The reactions of 2-pyridinealdoxime [(py)CHNOH] with Ni(ClO(4))(2).6H(2)O in the presence of NaOH and NaN(3) have led to Ni(II)(14) and Ni(II)(12)Na(2) clusters; the Ni(14) compound is the biggest metal oxime cluster discovered to date, as well as the largest Ni azide cluster.     

A discrete Fe18 'molecular chain'

The use of a new O,N,N,O chelate has led to two new Fe6 and Fe18 molecular compounds, the latter with an unusual double-headed serpentine chain structure.     

Computational (solute–solvent cluster + PCM) study of medium effects on the experimental 13C and 1H NMR chemical shifts of lactones and lactams

Solvent effects on the ¹H and ¹³C NMR chemical shifts of some lactones: β‐propiolactone, γ‐butyrolactone, δ‐valerolactone and ε‐caprolactone, as well as lactams: azetidin‐2‐one, pyrrolidin‐2‐one, δ‐valerolactam and ε‐caprolactam have been investigated and discussed in a wide range of solvents. The experimental results were compared with density functional calculations using a large basis set. Solvent effects were computed by means of an integrated approach including the polarizable continuum model and an optimum number of explicit solvent molecules surrounding the solute. The agreement between computed and experimental chemical shifts fully validates our integrated approach. In order to quantify and elucidate the origin of the solvent effects on the ¹H and ¹³C chemical shifts of the selected compounds, a multi‐linear regression analysis has been carried out using the empirical Kamlet–Abboud–Taft solvatochromic parameters. It has been found that there is a good correlation between the solvent‐induced chemical shifts of ¹³C and the π* scale of solvent dipolarity polarizability. ¹H chemical shifts are affected mainly by the dipolarity–polarizability and the basicity of the solvent. An excellent agreement has been obtained between the calculated and the experimental data. Copyright © 2011 John Wiley & Sons, Ltd.     

New mixed-valent Mn clusters from the use of N,N,N',N'-tetrakis(2-hydroxyethyl)ethylenediamine (edteH4): Mn3, Mn4, Mn6, and Mn10

The syntheses, crystal structures, and magnetochemical characterization are reported for the new mixed-valent Mn clusters [Mn(2)(II)Mn(III)(O(2)CMe)(2)(edteH(2))(2)](ClO(4)) (1), [Mn(II)(2)Mn(III)(2)(edteH(2))(2)(hmp)(2)Cl(2)](Mn(II)Cl(4)) (2), [Mn(III)(6)O(2)(O(2)CBu(t))(6)(edteH)(2)(N(3))(2)] (3), [Na(2)Mn(III)(8)Mn(II)(2)O(4)(OMe)(2)(O(2)CEt)(6)(edte)(2)(N(3))(6)] (4), and (NEt(4))(2)[Mn(8)(III)Mn(2)(II)O(4)(OH)(2)-(O(2)CEt)(6)(edte)(2)(N(3))(6)](5), where edteH(4) is N,N,N',N'-tetrakis-(2-hydroxyethyl)ethylenediamine and hmpH is 2-(hydroxymethyl)pyridine. 1-5 resulted from a systematic exploration of the effect of different Mn sources, carboxylates, the presence of azide, and other conditions, on the Mn/edteH(4) reaction system. The core of 1 consists of a linear Mn(II)Mn(III)Mn(II) unit, whereas that of 2 is a planar Mn(4) rhombus within a [Mn(II)(2)Mn(III)(2)(μ(3)-OR)(2)] incomplete-dicubane unit. The core of 3 comprises a central [Mn(III)(4)(OR)(2)] incomplete-dicubane on either side of which is edge-fused a triangular [Mn(III)(3)(μ(3)-O)] unit. The cores of 4 and 5 are similar and consist of a central [Mn(II)(2)Mn(III)(2)(μ(3)-OR)(2)] incomplete-dicubane on either side of which is edge-fused a distorted [Mn(II)Mn(III)(3)(μ(3)-O)(2)(μ(3)-OR)(2)] cubane unit. Variable-temperature, solid-state direct current (dc) and alternating current (ac) magnetization studies were carried out on 1-5 in the 5.0-300 K range, and they established the complexes to have ground state spin values of S = 3 for 1, S = 9 for 2, and S = 4 for 3. The study of 3 provided an interesting caveat of potential pitfalls from particularly low-lying excited states. For 4 and 5, the ground state is in the S = 0-4 range, but its identification is precluded by a high density of low-lying excited states.     

Coupling discontinuous Galerkin methods and retarded potentials for transient wave propagation on unbounded domains

This work deals with the numerical simulation of wave propagation on unbounded domains with localized heterogeneities. To do so, we propose to combine a discretization based on a discontinuous Galerkin method in space and explicit finite differences in time on the regions containing heterogeneities with the retarded potential method to account the unbounded nature of the computational domain. The coupling formula enforces a discrete energy identity ensuring the stability under the usual CFL condition in the interior. Moreover, the scheme allows to use a smaller time step in the interior domain yielding to quasi-optimal discretization parameters for both methods. The aliasing phenomena introduced by the local time stepping are reduced by a post-processing by averaging in time obtaining a stable and second order consistent (in time) coupling algorithm. We compute the numerical rate of convergence of the method for an academic problem. The numerical results show the feasibility of the whole discretization procedure.     

Dirichlet-to-Neumann semigroup acts as a magnifying glass

The first aim of this paper is to illustrate numerically that the Dirichlet-to-Neumann semigroup represented by P. Lax acts as a magnifying glass. In this perspective, we used the finite element method for discretizing of the correspondent boundary dynamical system using the implicit and explicit Euler schemes. We prove by using the Chernoff’s Theorem that the implicit and explicit Euler methods converge to the exact solution and we use the (P1)-finite elements to illustrate this convergence through a FreeFem++ implementation which provides a movie available online. In the Dirichlet-to-Neumann semigroup represented by P. Lax the conductivity \(\gamma \) is the identity matrix \(I_n\) , but for a different conductivity \(\gamma \) , the authors of Cornean et al. (J Inverse Ill-posed Prob 12:111–134, 2006) supplied an estimation of the operator norm of the difference between the Dirichlet-to-Neumann operator \(\Lambda _\gamma \) and \(\Lambda _1\) , when \(\gamma =\beta I_n\) and \(\beta =1\) near the boundary \(\partial \Omega \) (see Lemma 2.1). We will use this result to estimate the accuracy between the correspondent Dirichlet-to-Neumann semigroup and the Lax semigroup, for \(f\in H^{1/2}(\partial \Omega )\) .