Gaussian quadrature 4D-VAR

A new incremental four-dimensional variational (4D-Var) data assimilation algorithm is introduced. The algorithm does not require the computationally expensive integrations with the nonlinear model in the outer loops. Nonlinearity is accounted for by modifying the linearization trajectory of the observation operator based on integrations with the tangent linear (TL) model. This allows us to update the linearization trajectory of the observation operator in the inner loops at negligible computational cost. As a result the distinction between inner and outer loops is no longer necessary. The key idea on which the proposed 4D-Var method is based is that by using Gaussian quadrature it is possible to get an exact correspondence between the nonlinear time evolution of perturbations and the time evolution in the TL model. It is shown that J-point Gaussian quadrature can be used to derive the exact adjoint-based observation impact equations and furthermore that it is straightforward to account for the effect of multiple outer loops in these equations if the proposed 4D-Var method is used. The method is illustrated using a three-level quasi-geostrophic model and the Lorenz (1996) model.

Gál-type GCD sums beyond the critical line

We prove that ∑k,ℓ=1N(nk,nℓ)2α(nknℓ)α≪N2−2α(logN)b(α) holds for arbitrary integers 1≤n1<⋯sums when 0<α<1/20<α<1/2.

Synthesis, reactivity, and X-ray crystal structure of some mixed-ligand oxovanadium(V) complexes: first report of binuclear oxovanadium(V) complexes containing 4,4 '-Bipyridine type bridge

Reaction of the tridentate ONO Schiff-base ligand 2-hydroxybenzoylhydrazone of 2-hydroxybenzoylhydrazine (H2L) with VO(acac)(2) in ethanol medium produces the oxoethoxovanadium(V) complex [VO(OEt)L] (A), which reacts with pyridine to form [VO(OEt)L center dot(py)] (1). Complex 1 is structurally characterized. It has a distorted octahedral O4N2 coordination environment around the V(V) acceptor center. Both complexes A and 1 in ethanol medium react with neutral monodentate Lewis bases 2-picoline, 3-picoline, 4-picoline, 4-amino pyridine, imidazole, and 4-methyl imidazole, all of which are stronger bases than pyridine, to produce dioxovanadium(V) complexes of general formula BH[VO2L]. Most of these dioxo complexes are structurally characterized, and the complex anion [VO2L](-) is found to possess a distorted square pyramidal structure. When a solution/suspension of a BH[VO2L] complex in an alcohol (ROH) is treated with HCl in the same alcohol, it is converted into the corresponding monooxoalkoxo complex [ O(OR)L], where R comes from the alcohol used as the reaction medium. Both complexes A and 1 produce the 4,4'-bipyridine-bridged binuclear complex [VO(OEt)L](2)(mu-4,4'-bipy) (2), which, to the best of our knowledge, represents the first report of a structurally characterized 4,4'-bipyridine-bridged oxovanadium(V) binuclear complex. Two similar binuclear oxovanadium(V) complexes 3 and 4 are also synthesized and characterized. All these binuclear complexes (2-4), on treatment with base B, produce the corresponding mononuclear dioxovanadium(V) complexes (5-10).

A Wiener-Hopf type factorization for the exponential functional of Levy processes

For a Lévy process ξ=(ξt)t≥0 drifting to −∞, we define the so-called exponential functional as follows: Formula Under mild conditions on ξ, we show that the following factorization of exponential functionals: Formula holds, where × stands for the product of independent random variables, H− is the descending ladder height process of ξ and Y is a spectrally positive Lévy process with a negative mean constructed from its ascending ladder height process. As a by-product, we generate an integral or power series representation for the law of Iξ for a large class of Lévy processes with two-sided jumps and also derive some new distributional properties. The proof of our main result relies on a fine Markovian study of a class of generalized Ornstein–Uhlenbeck processes, which is itself of independent interest. We use and refine an alternative approach of studying the stationary measure of a Markov process which avoids some technicalities and difficulties that appear in the classical method of employing the generator of the dual Markov process.