Hermite series estimates of a probability density and its derivatives

The following estimate of the pth derivative of a probability density function is examined: $\text{Σ}{}_{\mathrm{k\; =\; 0}}{}^{\mathrm{N}}\text{a}\text{?}{}_{\mathrm{k}}\text{h}{}_{\mathrm{k}}\text{(x)}$, where h_k is the kth Hermite function and $\text{a}\text{?}{}_{\mathrm{k}}\text{= (}\text{(?1)}{}^{\mathrm{p}}\text{n}\text{)}$Σ_{i = 1}ⁿh_k^(p)(X_i) is calculated from a sequence X₁,…, X_n of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n^?α) and O(n^?α log n), respectively, where $\text{α =}\text{2(r ? p)}\text{(2r + 1)}$. Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n^?β) and $\text{O(n}{}^{\mathrm{<mtext>?\beta </mtext><mtext>2</mtext>}}\text{log}\text{n)}$, respectively, where $\text{β =}\text{(2(r ? p) ? 1)}\text{(2r + 1)}$.     

X-ray structure and NMR assignment of 3,4-dimethylthioquinoline

The title compound (C₁₁H₁₁NS₂) is monoclinic:P2₁/c,a=15.200(4),b=14.644(4),c=10.098(3),Z=8. The structure was solved by direct methods, and refined to anR value of 0.047 with 2886 independent reflections. There are two nonequivalent molecules in the unit cell. BothS-methyl groups have different spatial orientation: the-methyl group side-chain is approximately coplanar with the pyridine ring and turned to the ortho-position, but -methyl group side-chain is turned over this ring. Both¹H and¹³C NMR spectra were assigned using 1D and 2D experiments. The NOE measurements are consistent with inter-proton distances from X-ray data.Part XV in the series of Azinyl sulfides.     

X-ray structure of 4-chloro-3′-methylthio-3,4′-diquinolinyl sulfide

The title compound crystallizes in a monoclinic space groupP2₁/n witha=9.524(3),b=6.782(2),c=26.217(6) Å, =92.52(4)°. The values of the Cl...S and S...S distances are well below the sum of the van der Waals radii of the heteroatoms. Also, the Cl–C–C(S) and (Cl)C–C–S bond angles are below the standard 120°. This indicates a attractive interaction between both pairs ofortho-situated heteroatoms.Part XXXVIII in the Series of Azinyl Sulfides.     

Liquid‐Crystalline Properties of trans‐A2B2‐Porphyrins with Extended π‐Electron Systems

Incorporation of four trialkoxyphenyl substituents combined with extending the π‐conjugated system has allowed porphyrins to display liquid‐crystalline columnar phases at room temperature. 2D and 3D columnar structures were observed as well as a biaxial smectic phase.     

Conformational flexibility and pseudosymmetric aggregation in a betainium salt hydrate

Structural Chemistry - The cation of 3-(trimethylammonium)-benzoic acid exhibits a considerable conformational flexibility connected to the orientation of carboxyl group, coupled to the proton...     

Synthesis of New Primary-Tertiary Alkyl Bisperoxides by Reaction of Tertiary Hydroperoxides with Bromoalkanesin a PTC System

Summary.  Syntheses of novel primary-tertiary bisperoxides were carried out under phase transfer catalysis conditions. The bisperoxides were obtained in good yields, the reaction times not exceeding four hours. Received January 15, 2001. Accepted (revised) February 9, 2001