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Bulletin of the Korean Mathematical Society, ISSN: 10158634, eISSN: 22343016, Pages: 597-605, Published: 2020
Japan Journal of Industrial and Applied Mathematics, ISSN: 09167005, eISSN: 1868937X, Pages: 115-130, Published: 1 January 2020 Springer Science and Business Media LLC
In this paper, we give a theoretical justification of the idea by G. N. Hounsfield for the practicalization of the computerized tomography (CT). He developed a device for medical CT, for which he was awarded Nobel Prize for Physiology or Medicine. There being a number of researches concerning Hounsfield’s idea, we give a new stochastic approach to prove that it is theoretically right. We also discuss its applications.
Applied Sciences (Switzerland), eISSN: 20763417, Published: 2019 MDPI AG
X-ray tomography and magnetic resonance imaging (MRI) are excellent techniques for non-destructive or non-invasive inspections, however, they have shotcomings including the expensive cost in both the devices themselves and their protection facilities, the harmful side effects of the X-rays to human bodies and to the environment. In view of this argument, it is necessary to develop new, inexpensive, safe and reliable tomographic techniques, especially in medical imaging and non-destructive inspections. There are new tomographic techniques under development such as optical tomography, photo-acoustic tomography, ultrasonic tomography and so on, from which we take ultrasonic tomography as the topic in this paper. We introduce a review of the known ultrasonic tomographic techniques and discuss their future development.
Journal of Mathematical Fluid Mechanics, ISSN: 14226928, eISSN: 14226952, Pages: 2161-2175, Published: 1 December 2018 Springer Science and Business Media LLC
We discuss representation of the vortex sheets in the perfect fluid. In the preface of his book, I. Imai claimed “hyperfunction = vortex layer”, the proof of which had not been given. In 2009, K. Uchikoshi-Y. Noro gave a hyperfunctional representation of the vortex layers in the 2-dimensional fluid. Their idea being highly dependent on identifying the two dimensional Euclidean plane with the complex plane, it is difficult to interpret their representation in the real fluid phenomena. Its 3-dimensional extension was also difficult. To solve these problems, we give a new representation of the vortex sheets (or layers) by the real flow velocity vectors, whose other merits are also discussed.
Complex Variables and Elliptic Equations, ISSN: 17476933, eISSN: 17476941, Pages: 1507-1519, Published: November 2014 Informa UK Limited
We study a unique continuation property of microlocally analytic functions. This property depends on the regularity of the functions. In this article, we mainly discuss the unique continuation property of the microlocally analytic quasi-analytic ultradistributions and hyperfunctions, whose relation with their structure is also discussed.
Tsutomu Sakurai and Takashi Takiguchi
Complex Variables and Elliptic Equations, ISSN: 17476933, eISSN: 17476941, Pages: 1501-1506, Published: November 2014 Informa UK Limited
In this article, we review the achievements by Professor Akira Kaneko, who mainly contributed to the continuation of regular solutions to linear partial differential equations and to the theory of hyperfunctions.
Acta Mathematica Scientia, ISSN: 02529602, Pages: 1637-1646, Published: July 2012 Elsevier BV
Complex Variables and Elliptic Equations, ISSN: 17476933, eISSN: 17476941, Pages: 745-756, Published: August 2009 Informa UK Limited
We study the structure of generalized functions. We introduce the structure theorem for the quasi-analytic ultradistributions. We also discuss the relation between the structure theorem and the generalized unique continuation property. †Dedicated to Professor Luigi Rodino on his 60th birthday. ‡This article is devoted to the special issue ‘Växjö Conference 2008’.
Tadashi Ohsawa and Takashi Takiguchi
Inverse Problems in Science and Engineering, ISSN: 17415977, eISSN: 17415985, Pages: 461-471, Published: June 2009 Informa UK Limited
In this article, we first discuss inversion methods of the Abel-type integral transforms. It is well known that the support theorem of the Radon transform does not hold unless the function decreases rapidly at infinity. We prove that this support theorem holds for L1 functions if they are radial. The inversion methods of the Abel-type integrals play an important role to prove the support theorem for the radial functions. We also prove inversion formulae of the exterior problems for the Radon transform of the radial functions.
Publications of the Research Institute for Mathematical Sciences, ISSN: 00345318, Pages: 425-442, Published: June 2007 European Mathematical Society - EMS - Publishing House GmbH
We study the structure of functions between distributions and hyperfunctions. The structure theorem is known for distributions, non-quasi-analytic ultradistributions and hyperfunctions. In this paper, we try to fill the gap among them. We prove the structure theorem for quasi-analytic ultradistributions. In this paper, we discuss the structure of generalized functions. It is wellknown that any distribution f is locally represented as f = P (D)g ,w hereP (D) is a finite order differential operator with constant coefficients and g is a continuous function, which is the structure theorem for distributions. The structure theorems for non-quasi-analytic ultradistributions ([1, 5]) and hyperfunctions () are also known. In this paper, we study the structure of functions between them, namely, the structure of quasi-analytic ultradistributions. We prove the structure theorem for non-analytic ultradistributions which includes both nonquasi-analytic and quasi-analytic ones. It is our main theorem to prove that any non-analytic ultradistribution f of the class ∗ is locally represented as f = P (D)g ,w hereP (D) is an ultradifferential operator of the class ∗ and g is an ultradifferentiable function of the class † > ∗. We also claim that this
Journal of Physics: Conference Series, ISSN: 17426588, eISSN: 17426596, Published: 1 June 2007 IOP Publishing
We discuss the reconstruction of the measurable plane sets from their two projections. Reconstruction of a measurable plane set F ⊂ 2with λ2(F) < ∞ from its orthogonal projections is well studied, where λi; is the Lebesgue measure on i, i= 1, 2. In this paper we first discuss generalization of the known results in the frame of general two projections. The main purpose is to study, for any measurable plane set F, whether there are suitable two angles such that Fis uniquely reconstructed from the pair of its projections in that two directions. We give an example to show that this problem is negatively solved.
Proceedings of the Japan Academy Series A: Mathematical Sciences, ISSN: 03862194, Pages: 87-91, Published: 2001 Project Euclid
Journal of Inverse and Ill-Posed Problems, ISSN: 09280219, Pages: 573-579, Published: 2000 Walter de Gruyter GmbH
Abstract - It is well known that for uniqueness of the exterior problem for the Radon transform it is necessary and sufficient that a function decays rapidly towards infinity. This is Helgason’s support theorem. Several years ago, J. Boman discussed a modification of this theorem by restricting the condition of rapid decay to an open cone. In this paper, we study this modification.
L. HUANG and T. TAKIGUCHI
Journal of Inverse and Ill-Posed Problems, ISSN: 09280219, Pages: 217-242, Published: 1998 Walter de Gruyter GmbH
The reconstruction problem of measurable plane sets from two projections means that: Given two non-negative, integrable functions fx and /y, find a measurable plane set F having projections equal to fx and fy almost everywhere. In this paper, we show that the reconstruction of uniquely determined plane sets is stable. Furthermore we give an algorithm to reconstruct uniquely determined plane set.
Takashi TAKIGUCHI and Akira KANEKO
Hokkaido Mathematical Journal, ISSN: 03854035, Pages: 63-103, Published: February 1995 Department of Mathematics, Hokkaido University
We define the Radon transform for a class of hyperfunctions which are not necessarily with bounded support. We give characterization of the image space for some basic spaces. Then we give a variant of support theorem by Helgason-Boman. In this article we define the Radon transform for a class of hyperfunctions and discuss its properties. Especially, we prove a variant of support theorem by Helgason-Boman. Although the significance of extending the Radon transform to hyperfunctions is not so clear from the viewpoint of applications to industrial tomography, it will be interesting from purely mathematical viewpoint. Here we only treat the codimension one case i.e. the case of hyperplane integrals. We should remark that in the theory of hyperfunctions there already exists another kind of Radon transformation theory (see e.g. [Kt]). Its viewpoint lies in the microlocalization of the classical Radon transformation and is different from ours laying stress on the global behavior of the transformation.