Probability densities that are not uniquely determined by their moments are said to be “moment-indeterminate,” or “M-indeterminate.” Determining whether or not a density is M-indeterminate, or how to generate an M-indeterminate density, is a challenging problem with a long history. Quantum mechanics is inherently probabilistic, yet the way in which probability densities are obtained is dramatically different in comparison with standard probability theory, involving complex wave functions and operators, among other aspects. Nevertheless, the end results are standard probabilistic quantities, such as expectation values, moments and probability density functions. We show that the quantum mechanics procedure to obtain densities leads to a simple method to generate an infinite number of M-indeterminate densities. Different self-adjoint operators can lead to new classes of M-indeterminate densities. Depending on the operator, the method can produce densities that are of the Stieltjes class or new formulations that are not of the Stieltjes class. As such, the method complements and extends existing approaches and opens up new avenues for further development. The method applies to continuous and discrete probability densities. A number of examples are given.
Summary.
A coupled semilinear elliptic problem modelling an
irreversible, isothermal chemical reaction is introduced, and
discretised using the usual piecewise linear Galerkin finite element
approximation. An interesting feature of the problem is that a reaction order of
less than one gives rise to a "dead core" region. Initially,
one
reactant is assumed to be acting as a catalyst and is kept constant. It
is shown that error bounds previously obtained for a scheme involving
numerical integration can be improved upon by considering a quadratic regularisation
of the nonlinear term.
This technique is then applied to the full coupled problem, and optimal
and error bounds
are proved in the absence of
quadrature. For a scheme involving numerical integration,
bounds similar to those
obtained for the catalyst problem are shown to hold.
Received May 25, 1993 / Revised version received July 5, 1994 相似文献
A new inisurf (acting as surfactant and initiator) molecule for ring-opening metathesis polymerization (ROMP) was synthesized and used in aqueous solution in order to control the size and shape of polymer nodules grown from liposomes. Nodules were observed to grow in size with conversion of monomer, and depending on the monomer used, they adopted either a spherical or comet-like shape. Here, we investigate polymer production from a liposome surface. We use a hydrophobic derivative of the Grubbs catalyst positioned at the liposome surface to allow for ROMP of monomers dissolved in the aqueous outer phase. We obtain nodules of polymer that can grow up to tens of micrometers, unveiling new efficient possibilities of polymerization from a membrane in an aqueous solution. 相似文献
Neutral pi-conjugated molecules and their radical cations co-exist in [(EDT-TTF-CONHMe+*)4(EDT-TTF-CONHMe0)2] [Re6Se8(CN)6]4- (CH3CN)2(CH2Cl2)2 whose crystal structure reveals that, upon one-electron oxidation, an activation of the N-H and C-H hydrogen bond donor ability is coupled to a deactivation of the hydrogen bond acceptor character of the carbonyl oxygen atom: this is expressed in the supramolecular hydrogen bond pattern and, ultimately, into charge localisation and partition in the solid state. 相似文献
The oxidation of thiophene (1) with peracids in a strongly acidic environment yielded thiophen-2-one (4) as the product of an apparent direct hydroxylation of the thiophene aromatic ring together with the anticipated thiophene-S-oxide dimers, 2a,b, as the main products. Formation of the latter dimers can be rationalized in a straightforward manner by initial oxidation at the sulfur atom of thiophene (1) to yield thiophene-S-oxide followed by subsequent dimerization in a Diels-Alder type reaction. Trapping experiments in the presence of a competing dienophile indicated that thiophen-2-one (4) did not originate from the monomeric thiophene-S-oxide but was the product of an independent reaction pathway. The extent of thiophen-2-one (4) formation correlated with the acidity of the reaction medium and was suppressed in the presence of water, the latter presumably acting as a competing base. As evidenced by the use of 2,5-dideuterated thiophene (1-D), its mechanism of formation involved a 1,2-hydride shift, a feature commonly described in the peracid-mediated epoxidation of aromatic hydrocarbons and indicative for the occurrence of cationic intermediates. In agreement with all these observations we propose a mechanism involving initial protonation of thiophene followed by nucleophilic attack of the peracid in position 2 of the thiophene ring. Intramolecular epoxidation may lead to the formation of thiophene 2,3-epoxide as a highly reactive intermediate that then undergoes heterolytic ring opening and a 1,2-hydride shift to yield thiophen-2-one (4) after a final, acid-catalyzed, isomerization of the double bond. 相似文献
A pilot-scale sewage treatment system consisting of two upflow anaerobic sludge bed (UASB) reactors followed by five waste stabilization ponds (WSPs) in series was studied under subtropical conditions. The first UASB reactor started up in only 1 mo (stable operation, high chemical oxygen demand [COD] removal efficiency, low volatile fatty acids concentration in the effluent, alkalinity ratio above 0.7, biogas production above 0.1 Nm3/kg of CODremoved). Removal efficiencies up to 90% were obtained in the anaerobic steps at a hydraulic retention time of 6 + 4 h (80% removal in the first step). Fecal coliform removal in the whole system was 99.9999% (99.94% in anaerobic steps and 99.98% in WSPs). COD balances over UASB reactors are provided. A minimum set of data necessary to build COD balances is proposed. Intermittent sludge washout was detected in the reactors with the COD balances. Sludge washout from single-step UASB reactors should be monitored and minimized in order to ensure constant compliance with discharge standards, especially when no posttreatment is provided. The system combined high COD and fecal coliform removal efficiency with an extremely low effluent concentration, complying with discharge standards, and making it an attractive option for sewage treatment in subtropical regions. 相似文献
This paper presents a method for programming the flow rate of liquids inside open microfluidic networks (MFNs). A MFN comprises a number of independent flow paths, each of which starts with an open filling port, has a sealed microchannel in which assays can be performed, and an open capillary pump (CP). The MFN is placed over Peltier elements and its flow paths initially fill owing to capillary forces when liquids are added to the filling ports. A cooling Peltier element underneath the filling ports dynamically prevents evaporation in all filling ports using the ambient temperature and relative humidity as inputs. Another Peltier element underneath the CPs heats the pumps thereby inducing evaporation in the CPs and setting the flow rate in the microchannels. This method achieves flow rates in the microchannels ranging from approximately 1.2 nL s(-1) to approximately 30 pL s(-1), and is able to keep 90% of a 0.6 microL solution placed in an open filling port for 60 min. This simple and efficient method should be applicable to numerous assays or chemical reactions that require small and precise flow of liquids and reagents inside microfluidics. 相似文献
The scope and utility of the metal-catalyzed rearrangement of propargyl acetates first reported by Rautenstrauch were expanded. Treatment of a series of appropriate acetate substrates with Pd(II)- and Hg(II)-catalysts afforded synthetically useful fused 5,6-bicyclic-1,4-cyclopentadienyl acetates and 2-cyclopentenones. It was found that the substituents at the terminal alkynyl and alkenyl positions of the acetate substrate had a significant impact on the outcome of the reaction. 相似文献
The novel ternary reduced molybdenum oxide Mn(approximately)(2.4)Mo(6)O(9) has been synthesized by solid-state reaction at 1400 degrees C for 96 h in sealed molybdenum crucibles. Electron diffraction studies showed that Mn(approximately)(2.4)Mo(6)O(9) presents a complex crystal structure with a 3d incommensurate modulation. The average crystal structure was determined on a single-crystal by X-ray diffraction in the orthorhombic space group Pnma with the following lattice parameters: a = 16.4824(2) A, b = 2.8273(2) A, c = 17.3283(2) A, Z = 4. The Mo network consists of empty twin chains of trans-edge-sharing octahedra that occur for the first time in a solid-state compound. The Mo-Mo distances within the chains range from 2.62 to 2.92 A, and the Mo-O distances from 1.99 to 2.17 A as usually observed in the reduced molybdenum oxides. Single-crystal resistivity measurements show that Mn(approximately)(2.4)Mo(6)O(9) is metallic between 4.2 and 300 K. The magnetic susceptibility data indicate paramagnetic behavior due to the Mn(2+) moment at high temperatures with a weak ferromagnetic behavior below 80 K. 相似文献