Non parametric estimation of the diffusion coefficients of a diffusion with jumps

In this article, we consider a jump diffusion process ${\left(X_t\right)}_{t\ge 0}$, with drift function $b$, diffusion coefficient $\sigma$ and jump coefficient ${\xi }^2$. This process is observed at discrete times $t=0,\Delta ,\dots ,n\Delta$. The sampling interval $\Delta$ tends to 0 and the time interval $n\Delta$ tends to infinity. We assume that ${\left(X_t\right)}_{t\ge 0}$ is ergodic, strictly stationary and exponentially $\beta$-mixing. We use a penalized least-square approach to compute adaptive estimators of the functions ${\sigma }^2+{\xi }^2$ and ${\sigma }^2$. We provide bounds for the risks of the two estimators.     

Asymptotic expansion for vector-valued sequences of random variables with focus on Wiener chaos

We develop the asymptotic expansion theory for vector-valued sequences ${\left(F_N\right)}_{N\ge 1}$ of random variables in terms of the convergence of the Stein–Malliavin matrix associated with the sequence $F_N$. Our approach combines the classical Fourier approach and the recent Stein–Malliavin theory. We find the second order term of the asymptotic expansion of the density of $F_N$ and we illustrate our results by several examples.     

Powder‐XRD and 14N magic angle‐spinning solid‐state NMR spectroscopy of some metal nitrides

Some metal nitrides (TiN, ZrN, InN, GaN, Ca₃N₂, Mg₃N₂, and Ge₃N₄) have been studied by powder X‐ray diffraction (XRD) and ¹⁴N magic angle‐spinning (MAS) solid‐state NMR spectroscopy. For Ca₃N₂, Mg₃N₂, and Ge₃N₄, no ¹⁴N NMR signal was observed. Low speed (ν_r = 2 kHz for TiN, ZrN, and GaN; ν_r = 1 kHz for InN) and ‘high speed’ (ν_r = 15 kHz for TiN; ν_r = 5 kHz for ZrN; ν_r = 10 kHz for InN and GaN) MAS NMR experiments were performed. For TiN, ZrN, InN, and GaN, powder‐XRD was used to identify the phases present in each sample. The number of peaks observed for each sample in their ¹⁴N MAS solid‐state NMR spectrum matches perfectly well with the number of nitrogen‐containing phases identified by powder‐XRD. The ¹⁴N MAS solid‐state NMR spectra are symmetric and dominated by the quadrupolar interaction. The envelopes of the spinning sidebands manifold are Lorentzian, and it is concluded that there is a distribution of the quadrupolar coupling constants Q_cc's arising from structural defects in the compounds studied. Copyright © 2015 John Wiley & Sons, Ltd.     

Eisenstein type series for Calabi–Yau varieties

In this article we introduce an ordinary differential equation associated to the one parameter family of Calabi–Yau varieties which is mirror dual to the universal family of smooth quintic three folds. It is satisfied by seven functions written in the q-expansion form and the Yukawa coupling turns out to be rational in these functions. We prove that these functions are algebraically independent over the field of complex numbers, and hence, the algebra generated by such functions can be interpreted as the theory of (quasi) modular forms attached to the one parameter family of Calabi–Yau varieties. Our result is a reformulation and realization of a problem of Griffiths around seventies on the existence of automorphic functions for the moduli of polarized Hodge structures. It is a generalization of the Ramanujan differential equation satisfied by three Eisenstein series.     

Reactivity of a Cyclopentadienyl Containing Heterobimetallic Alkoxide

The synthesis of the new heteroleptic heterotrimetallic cluster, [(C₅H₅)Sn(μ-OBu^t)₂Ge(OBu^t)Mo(CO)₅] (1) has been achieved by a thermally induced CO substitution of the transition metal derivative, Mo(CO)₆, by the basic germanium atom of the cyclopentadienyl heterobimetallic alkoxide, [(C₅H₅)Sn(μ-OBu^t)₂Ge(OBu^t)]. The microanalysis, molecular weight (monomer in benzene), IR and multinuclear NMR data and X-ray diffraction study is consistent with the formulation of 1. The Sn atom has a trigonal pyramidal coordination environment formed by a aysmmterically π-bonded terminal C₅H₅ ring and two symmetrically bridged tert-butoxy groups. The four-coordinate Ge atom lies at the centre of a distorted tetrahedron and is terminally attached to a tert-butoxy group and a Mo(CO)₅5^? fragment.     

Computational Study of Cage Like (ZnO)12 Cluster Using Hybrid and Hybrid Meta Functionals

Density Functional Theory employing hybrid and M06 functionals in combination with three different basis sets is used to calculate the ground state of a cage like (ZnO)₁₂ nanocluster which has been consistently reported as the more stable cluster for its particular size. B3LYP and B3PW91 hybrid functionals combined with 6‐31+G*, Lanl2dz and SDD basis sets are employed to treat the ZnO molecular system. Alternatively, three M06 functionals in combination with three basis sets are employed in the nanostructure calculations. Results obtained by treating ZnO sodalite cage nanocluster with M06 functionals demonstrated comparable quality to results obtained with hybrid functionals. Within this study, efficient theoretical DFT methods with the widely known hybrid and the recently created M06 meta‐hybrid functionals are employed to study nanostructured ZnO. Our resulting parameters provide a fresh approach performance wise on the different theoretical methods to treat transition metal nanostructures, particularly, ZnO nanoclusters geometry and electronic structure.     

Primary Studies on Variation in Position of Trifluoromethyl Groups in Several Aromatic Group‐14 Derivatives by 19F‐NMR Spectroscopy

Variation in the position of CF₃ groups in several aromatic Group‐14 compounds was studied by ¹⁹F‐NMR spectroscopy. In these compounds R_nECl_4?n (n=1 or 2; E=Si, Ge, or Sn; R=2,4,6‐(CF₃)₃C₆H₂ (=Ar), 2,6‐(CF₃)₂C₆H₃ (=Ar′), or 2,4‐(CF₃)₂C₆H₃ (=Ar″)), Ar, Ar′, and Ar″ are all bulky, strongly electron‐withdrawing ligands. The ¹⁹F‐NMR studies of the variation in position of the CF₃ substituents in these compounds as revealed by chemical shifts could be correlated with the electronegativities of the central elements E, and with intramolecular E–F interactions derived from single‐crystal X‐ray diffraction data. These interactions are considered to play an important role in the stabilization of these compounds.     

Three New C19‐Diterpenoid Alkaloids from Aconitum forrestii

A further study of the alkaloid constituents of Aconitum forrestii led to the isolation of three new C₁₉‐diterpenoid alkaloids, named 14‐acetoxy‐8‐O‐methylsachaconitine ( 1 ), 14‐acetoxyscaconine ( 2 ), and 8‐O‐ethylcammaconine ( 3 ). Their structures were determined by UV, IR, and MS, 1D‐ and 2D‐NMR analyses.     

A New Taraxastane‐Type Triterpene from Vitex trifolia var. simplicifolia

3‐Oxotaraxer‐14‐en‐30‐al ( 1 ), a new taraxastane‐type triterpene, together with 14 known compounds, taraxerone ( 2 ), 3‐epiursolic acid ( 3 ), 2α,3β‐dihydroxyurs‐12‐en‐28‐oic acid ( 4 ), lupeol ( 5 ), betulinic acid ( 6 ), casticin ( 7 ), artemetin ( 8 ), luteolin ( 9 ), 4‐hydroxybenzoic acid ( 10 ), docosanoic acid ( 11 ), tetracosanoic acid ( 12 ), cerotic acid ( 13 ), β‐sitosterol ( 14 ), and β‐daucosterol ( 15 ), was isolated from the leaves and twigs of Vitex trifolia var. simplicifolia . Compounds 2 – 6 were found for the first time in this plant. Their structures were established by spectroscopic analysis, including 2D‐NMR techniques. Cytotoxic activities of compounds 3 , and 5 – 10 were tested on the three cancer cell lines, PANC‐1, K562, and BxPC‐3. Results revealed that 7 exhibited cytotoxicity against PANC‐1, K562, and BxPC‐3, with IC₅₀ values of 4.67, 0.72, and 4.01 μg/ml, respectively, whereas 8 was inactive against these cancer cell lines. The structure? activity relationship of compound 7 and 8 indicated that the 3′‐OH group in polymethoxyflavonoids is essential for antitumor activity.     

Varieties fibred over abelian varieties with fibres of log general type

Let (X,B)$(X,B)$ be a complex projective klt pair, and let f:X→Z$f:X\to Z$ be a surjective morphism onto a normal projective variety with maximal albanese dimension such that _KX+B$K_X+B$ is relatively big over Z. We show that such pairs have good log minimal models.