On the Inverse of an Integral Operator

Wolfe, Peter

doi:10.2307/2037240

Cited by 3 publications

(3 citation statements)

References 3 publications

(4 reference statements)

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“…Regarding this question we want to refer here to the papers [40] and [6]. Namely, in [40] it is shown that the operator M I is a homeomorphism from a weighted L 2 -space onto a weighted space of smoother functions. In [6], Theorem 3.1, mapping properties for M I as an operator between Sobolev spaces, which are based on L pspaces, are presented, as we have already mentioned in the introduction.…”

Section: Concluding Remarks and Outlookmentioning

confidence: 99%

A new and self‐contained presentation of the theory of boundary operators for slit diffraction and their logarithmic approximations

Gorenflo

Kunik

2011

Mathematische Nachrichten

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We present a new and self-contained theory for mapping properties of the boundary operators for slit diffraction occurring in Sommerfeld's diffraction theory, covering two different cases of the polarisation of the light. This theory is entirely developed in the context of the boundary operators with a Hankel kernel and not based on the corresponding mixed boundary value problem for the Helmholtz equation. For a logarithmic approximation of the Hankel kernel we also study the corresponding mapping properties and derive explicit solutions together with certain regularity results.

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Section: Concluding Remarks and Outlookmentioning

confidence: 99%

A new and self‐contained presentation of the theory of boundary operators for slit diffraction and their logarithmic approximations

Gorenflo

Kunik

2011

Mathematische Nachrichten

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“…a function. Regarding this question we want to refer here to the papers 40 and 6. Namely, in 40 it is shown that the operator M I is a homeomorphism from a weighted L 2 ‐space onto a weighted space of smoother functions.…”

Section: Concluding Remarks and Outlookmentioning

confidence: 99%

“…Regarding this question we want to refer here to the papers 40 and 6. Namely, in 40 it is shown that the operator M I is a homeomorphism from a weighted L 2 ‐space onto a weighted space of smoother functions. In 6, Theorem 3.1, mapping properties for M I as an operator between Sobolev spaces, which are based on L p ‐spaces, are presented, as we have already mentioned in the introduction.…”

Section: Concluding Remarks and Outlookmentioning

confidence: 99%

A Comparison of Different Methods To Determine the End of Exponential Growth in CHO Cell Cultures for Optimization of Scale-Up

Schoenherr¹,

Stapp²,

Ryll³

2000

Biotechnol. Prog.

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Maximizing cell growth rate and cell yield are among the most important features of a successful mammalian cell culture production process. To minimize time and resources needed to scale up cell mass it is important to maintain the cultures in exponential growth at every scale. Here we report results comparing viable cell counts, packed cell volume, intracellular nucleotide ratios, cell cycle analysis, and on-line oxygen uptake rates (OUR) and optical density for the determination of the end of exponential growth to optimize transfer times during scale-up of CHO cell cultures. Viable cell concentration, packed cell volume, and relative abundance of cells in S-phase were not very reliable at determining the end of exponential growth during the process. In contrast, on-line determination of OUR and off-line determination of intracellular nucleotide ratios (U-ratio) were very sensitive to changes in growth rate, enabling clear determination of the end of exponential growth within a short time. Although on-line OUR was found to be the most convenient and fastest method, it is restricted to instrumented and continuously monitored cultures. In contrast the nucleotide method can be applied with any culture scale and condition but needs the availability of an operator running an HPLC system and takes about an hour from sampling to result. Optical density showed an inflection along with OUR and U-ratio but was less sensitive in determining the end of exponential growth.

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The decay of solutions of the two dimensional wave equation in the exterior of a straight strip

Wolfe¹

1977

Trans. Amer. Math. Soc.

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Abstract.We study an initial boundary value problem for the wave equation in the exterior of a straight strip. We assume the initial data has compact support and that the solution vanishes on the strip. We then show that at any point in space the solution is 0(1/1) as t -» oo. This is the same rate of decay as obtains for the solution of the initial boundary value problem posed in the exterior of a smooth star shaped region. Our method is to use a Laplace transform. This reduces the problem to a consideration of a boundary value problem for the Helmholtz equation. We derive estimates for the solution of the Helmholtz equation for both high and low frequencies which enable us to obtain our results by estimating the Laplace inversion integral asymptotically. 0. Introduction. We consider an initial boundary value problem for the two dimensional wave equation in the exterior of a straight strip. We assume that the initial data have compact support and that the solution vanishes on the strip. Our object is to obtain the rate of decay of the solution to this problem as / -* oo. In a recent paper [10] we considered the problem of diffraction of a plane pulse by a strip. Here we shall proceed much as in [10] and will use some results from that paper. We use the method of Laplace transformation. Our results are obtained by evaluating the inversion integral asymptotically. We show that the contour of the inversion integral can be deformed into the left half plane. Then by considering separately the integrals over the portions of the contour near the origin and far away from it we show that the solution decays at any point in space like 1//.Most of the work of the paper goes into proving estimates for solutions of the reduced wave equation. To obtain these estimates we study the integral equation

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On the Inverse of an Integral Operator

Abstract: We wish to consider the integral equation * f1 (i) i i(1) /(*) = -I Ho (k\x-t\ ) Show more

Cited by 3 publications

References 3 publications

A new and self‐contained presentation of the theory of boundary operators for slit diffraction and their logarithmic approximations

A new and self‐contained presentation of the theory of boundary operators for slit diffraction and their logarithmic approximations

A Comparison of Different Methods To Determine the End of Exponential Growth in CHO Cell Cultures for Optimization of Scale-Up

The decay of solutions of the two dimensional wave equation in the exterior of a straight strip

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