On Fredholm integral equations,Toeplitz equations and Kalman-Bucy filtering

In this paper we consider some new algorithms for computing the Kalman-Bucy gain in stationary systems requiring a number of equations of ordern (rather thann ² ) whenever the ordern of the system is much larger than the dimension of the output. These equations were independently obtained by Kailath and Lindquist in continuous and discrete time respectively. We briefly discuss the relations with some recent related results due to Casti, Kalaba & Murthy and Rissanen. Some of the reasons for these reductions are inherent in the properties of general stationary processes, and therefore a considerable portion of the paper is devoted to exploring the connections with some previous work by Levinson, Whittle and Wiggins & Robinson, and also with the Szegö theory of polynomials orthogonal on the unit circle and some continuous analogs due to Krein. We demonstrate that the Bellman-Krein formula is the fundamental relation in continuous time, the trick being to introduce a reversed time counterpart of the weighting function (Fredholm resolvent). This is suggested by the forward and backward innovation approach in a previous paper by the author, the essential relations of which we reformulate in terms of Fredholm integral equations (in continuous time) and Toeplitz equations (in discrete time). Therefore we also derive the discrete-time Bellman-Krein formulas of which there are actually two—one corresponding to the one-step predictor and one to the pure filter. In this way we shall be able to pin down the reasons for the striking discrepancies between the continuous-time and the discrete-time cases. Finally we clarify the relations between Levinson's equations and Chandrasekhar'sX- andY-functions.This research was carried out while the author held a visiting position at the Division of Applied Mathematics, Brown University, Providence, Rhode Island, and was supported by the National Science Foundation under grant NSF-46614.     

MAGNETIC FIELD EFFECT ON PHOTOPOLYMERIZATION OF STYRENE MICROEMULSION

External magnetic field increases the photo-induced polymerization rate of styrene microemulsion.The type of photoinitiator plays an important role. The photoinitiators, used are dimethoxyphenyl acetophenone(DMPA), 1-hydroxycyclohexyl phenylketone (Irgacure 184) and dimethylhydroxyacetophenone (Darocur 1173). No magnetic effect was observed by using dibenzylketone (DBK) as photoinitiator. The molecular weight of the polymer is slightly affected by magnetic field. The influence of temperature has also been investigated.     

Maximum sustainable yield harvesting in an age‐structured fishery population model

A generic type age‐structured fishery population model consisting of two harvestable age classes is formulated. Optimal harvest rates are determined with uniform fishing mortality and perfectly selective fishing, respectively. Selectivity allows for differentiating the fishing mortality among different age classes. Sustainable yield–biomass functions are developed, and the maximum sustainable yield (MSY) solutions are found under both exploitation schemes. The gain of perfectly selective fishing over uniform (or biomass) fishing is examined under various assumptions, and it is proved that the benefit of selective harvesting increases when the harvestable fish population becomes more heterogeneous in terms of weights, or values. In contrast to the surplus production model, or Clark model, the analysis also demonstrates that MSY with different age classes is not purely a biological concept.     

Autocatalytic water dissociation on Cu(110) at near ambient conditions

Autocatalytic dissociation of water on the Cu(110) metal surface is demonstrated on the basis of X-ray photoelectron spectroscopy studies carried out in situ under near ambient conditions of water vapor pressure (1 Torr) and temperature (275-520 K). The autocatalytic reaction is explained as the result of the strong hydrogen-bond in the H2O-OH complex of the dissociated final state, which lowers the water dissociation barrier according to the Br?nsted-Evans-Polanyi relations. A simple chemical bonding picture is presented which predicts autocatalytic water dissociation to be a general phenomenon on metal surfaces.     

Structure and composition of the nacre–prisms transition in the shell of <Emphasis Type="Italic">Pinctada margaritifera</Emphasis> (Mollusca,Bivalvia)

A microstructural, mineralogical, and chemical study of the nacre–prisms boundary in the shells of Pinctada margaritifera shows that this boundary is not an abrupt transition, but that there exists a distinct fibrous layer with clear topographic structures and evidence of growth lines. A three-step biomineralization process is proposed that involves changes in the chemical and biochemical composition of the last growth increments of the calcite prisms, formation of the fibrous layer, and development of regular tablets in the nacreous layer.     

Application of inductively coupled plasma–mass spectrometry (ICP–MS) and quality assurance to study the incorporation of strontium into bone,bone marrow,and teeth of dogs after one month of treatment with strontium malonate

The strontium content of serum, bone, marrow, and teeth was determined by inductively-coupled plasma mass spectrometry (ICP–MS). Significant correlations were obtained after the data were subjected to quality assurance (QA) performed according to validated procedures. After four weeks of treatment with strontium malonate, strontium levels increased from 76 ± 9 μg g⁻¹ in placebo-treated dogs to levels of 7.2 ± 1.7 mg g⁻¹, 9.5 ± 2.7 mg g⁻¹, and 9.8 ± 2.7 mg g⁻¹ in groups treated with 300, 1000, and 3000 mg kg⁻¹ day⁻¹, respectively. Strontium induced a highly significant increase in the bone formation marker, bone-specific alkaline phosphatase (BSAP), and an excellent correlation was found with the bone-strontium content. In females, the placebo-treated group showed a decrease in BSAP of 53%, whereas the three strontium malonate-treated groups showed an increase of 60, 276, and 278% for the groups treated with 300, 1000, and 3000 mg kg⁻¹ day⁻¹, respectively. For males the corresponding values were −44%, +142%, +194%, and +247% increases in BSAP in the placebo, 300, 1000, and 3000 mg kg⁻¹ day⁻¹ groups respectively.     

On the approximation of spectra of linear operators on Hilbert spaces

We present several new techniques for approximating spectra of linear operators (not necessarily bounded) on an infinite-dimensional, separable Hilbert space. Our approach is to take well-known techniques from finite-dimensional matrix analysis and show how they can be generalized to an infinite-dimensional setting to provide approximations of spectra of elements in a large class of operators. We conclude by proposing a solution to the general problem of approximating the spectrum of an arbitrary bounded operator by introducing the n-pseudospectrum and argue how that can be used as an approximation to the spectrum.     

A note on the Jacobian conjecture

In this paper we consider the Jacobian conjecture for a map of complex affine spaces of dimension . It is well known that if is proper, then the conjecture will hold. Using topological arguments, specifically Smith theory, we show that the conjecture holds if and only if is proper onto its image.

     

Detecting completeness from Ext-vanishing

Motivated by work of C. U. Jensen, R.-O. Buchweitz, and H. Flenner, we prove the following result. Let be a commutative noetherian ring and an ideal in the Jacobson radical of . Let be the -adic completion of . If is a finitely generated -module such that for all , then is -adically complete.