Recursive kernel density estimators under a weak dependence condition

Let X _t, t= ..., \s-1,0,1,... be a strietly stationary sequence of random variables (r.v.'s) defined on a probability space (,P) and taking values in R ^d.Let X ₁,...,X _nbe n consecutive observations of X _t.Let f be the density of X ₁.As an estimator of f(x), we shall consider % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaaiaacaqabeaadaqaaqGaaO% qaaiqadAgagaqcamaaBaaaleaacaWGUbaabeaakiaacIcacaWG4bGa% aiykaiabg2da9iaad6gadaahaaWcbeqaaiabgkHiTiaaigdaaaGcda% aeWbqaaiaadkgadaWgaaWcbaGaamOAaaqabaGcdaahaaWcbeqaaiab% gkHiTiaadsgaaaGccaWGlbGaaiikaiaacIcacaWG4bGaeyOeI0Iaam% iwamaaBaaaleaacaWGQbaabeaakiaacMcacaGGVaGaamOyamaaBaaa% leaacaWGQbaabeaakiaacMcaaSqaaiaadQgacqGH9aqpcaaIXaaaba% GaamOBaaqdcqGHris5aaaa!58A9!\[\hat f_n (x) = n^{ - 1} \sum\limits_{j = 1}^n {b_j ^{ - d} K((x - X_j )/b_j )} \]. Here K is a kernel function and b _nis a esquence of bandwidths tending to zero as n . The asymptotic distribution and uniform convergence of f n are obtained under general conditions. Appropriate bandwidths are given explicitly. The process X _tis assumed to satisfy a weak dependence condition defined in terms of joint densities. The results are applicable to a large class of time series models.     

Synthesis of Eupolauridine and Its Benzo-Annulated Derivative

The unique diazafluoranthene alkaloid eupolauridine (1) was synthesized by a three-step route utilizing as the initial step the thermal rearrangement of the oxime O-crotyl ether 12, which afforded onychine (2). Bracher pyridine synthesis procedure subsequently converted 2 into 1. On the other hand, Friedländer reaction was employed for the synthesis of 3, which is a benzo-annulated derivative of eupolauridine (1).     

Experimental study of developing turbulent flow and heat transfer in ribbed convergent/divergent square ducts

The local heat transfer and pressure drop characteristics of developing turbulent flows of air in three stationary ribbed square ducts have been investigated experimentally. These are: ribbed square duct with constant cross-section (straight duct), ribbed divergent square duct and ribbed convergent square duct. The convergent/divergent duct has an inclination angle of 1°. The measurement was conducted within the range of Reynolds numbers from 10 000 to 77 000. The heat transfer performance of the divergent/convergent ducts is compared with the ribbed straight duct under three constraints: identical mass flow rate, identical pumping power and identical pressure drop. Because of the streamwise flow acceleration or deceleration, the local heat transfer characteristics of the divergent and convergent ducts are quite different from those of the straight duct. In the straight duct, the fluid flow and heat transfer become fully developed after 2–3 ribs, while in the divergent and convergent ducts there is no such trend. The comparison shows that among the three ducts, the divergent duct has the highest heat transfer performance, the convergent duct has the lowest, while the straight duct locates somewhere in between.     

C-H arylation of unactivated arenes with aryl halides catalyzed by cobalt porphyrin

A general procedure for cobalt-catalyzed direct C-H arylation of unactivated arenes has been discovered. This method employs aryl halides as the direct coupling partners with arenes without using any Grignard-type reagents. This catalysis opens a new methodology for the preparation of symmetrical as well as unsymmetrical biaryls in a user-friendly approach.     

Catalytic carbon-carbon σ-bond hydrogenation with water catalyzed by rhodium porphyrins

The catalytic carbon-carbon σ-bond activation and hydrogenation of [2.2]paracyclophane with water in a neutral reaction medium is demonstrated. The hydrogen from water is transferred to the hydrocarbon to furnish hydrogen enrichment in good yields.     

On a functional equation arising from comparison of utility representations

We solve the functional equation F₁(t)−F₁(t+s)=F₂[F₃(t)+F₄(s)] for real functions defined on intervals, assuming that F₂ is positive valued and strictly monotonic and that F₃ is continuous. The equation arose from the equivalence problem of utility representations under assumptions of separability, homogeneity and segregation (e-distributivity).     

Microwave assisted dissolution of aluminium oxide samples

The dependence on time of the efficiency of microwave assisted digestion of Al₂O₃ samples with various physico-chemical properties in H₂SO₄ solution (1 + 1) at a temperature of 170 °C and a pressure of 207 kPa has been investigated. Therefore, the concentrations of Al and that of the impurities in the solutions obtained, as well as the masses of residues after different times have been measured. The efficiency of the dissolution was shown to depend on the specific surface area and on the phase-state of the Al₂O₃. The time demand for total dissolution of the samples ranged from 0.1 to 7 hours.