Abstract We consider a generalization of the inverse problem of the electrocardiography in the framework of the theory of elliptic and parabolic differential operators. More precisely, starting with the standard bidomain mathematical model related to the problem of the reconstruction of the transmembrane potential in the myocardium from known body surface potentials, we formulate a more general transmission problem for elliptic and parabolic equations in the Sobolev type spaces and describe conditions, providing uniqueness theorems for its solutions. Next, the new transmission problem is interpreted in the framework of the elasticity theory applied to composite media. Finally, we prove a uniqueness theorem for an evolutionary transmission problem that can be easily adopted to many models involving the diffusion type equations.
{"title":"On the uniqueness theorems for transmission problems related to models of elasticity, diffusion and electrocardiography","authors":"Alexander Shlapunov, Yulia Shefer","doi":"10.1515/jiip-2021-0071","DOIUrl":"https://doi.org/10.1515/jiip-2021-0071","url":null,"abstract":"Abstract We consider a generalization of the inverse problem of the electrocardiography in the framework of the theory of elliptic and parabolic differential operators. More precisely, starting with the standard bidomain mathematical model related to the problem of the reconstruction of the transmembrane potential in the myocardium from known body surface potentials, we formulate a more general transmission problem for elliptic and parabolic equations in the Sobolev type spaces and describe conditions, providing uniqueness theorems for its solutions. Next, the new transmission problem is interpreted in the framework of the elasticity theory applied to composite media. Finally, we prove a uniqueness theorem for an evolutionary transmission problem that can be easily adopted to many models involving the diffusion type equations.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"58 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234219","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The paper is devoted to the regularization of ill-posed stochastic Cauchy problems in Hilbert spaces: (0.1) du(t)=Au(t)dt+BdW(t),t>0,u(0)=ξ. du(t)=Au(t)dt+BdW(t),quad t>0,qquad u(0)=xi. The need for regularization is connected with the fact that in the general case the operator A is not supposed to generate a strongly continuous semigroup and with the divergence of the series defining the infinite-dimensional Wiener process {W(t):t≥0} {{W(t):tgeq 0}} . The construction of regularizing operators uses the technique of Dunford–Schwartz operators, regularized semigroups, generalized Fourier transform and infinite-dimensional Q -Wiener processes.
{"title":"Correctness and regularization of stochastic problems","authors":"Irina V. Melnikova, Vadim A. Bovkun","doi":"10.1515/jiip-2023-0011","DOIUrl":"https://doi.org/10.1515/jiip-2023-0011","url":null,"abstract":"Abstract The paper is devoted to the regularization of ill-posed stochastic Cauchy problems in Hilbert spaces: (0.1) <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mi>d</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi>A</m:mi> <m:mo></m:mo> <m:mi>u</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo></m:mo> <m:mi>d</m:mi> <m:mo></m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>B</m:mi> <m:mo></m:mo> <m:mi>d</m:mi> <m:mo></m:mo> <m:mi>W</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo rspace=\"12.5pt\">,</m:mo> <m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo rspace=\"22.5pt\">,</m:mo> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mi>ξ</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> du(t)=Au(t)dt+BdW(t),quad t>0,qquad u(0)=xi. The need for regularization is connected with the fact that in the general case the operator A is not supposed to generate a strongly continuous semigroup and with the divergence of the series defining the infinite-dimensional Wiener process <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mrow> <m:mi>W</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>:</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:math> {{W(t):tgeq 0}} . The construction of regularizing operators uses the technique of Dunford–Schwartz operators, regularized semigroups, generalized Fourier transform and infinite-dimensional Q -Wiener processes.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135549144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dmitrii Sergeevich Anikonov, Sergey G. Kazantsev, Dina S. Konovalova
Abstract We study the problem of the integral geometry, in which the functions are integrated over hyperplanes in the n -dimensional Euclidean space, n=2m+1 {n=2m+1} . The integrand is the product of a function of n variables called the density and weight function depending on 2n {2n} variables. Such an integration is called here the weighted Radon transform, which coincides with the classical one if the weight function is equal to one. It is proved the uniqueness for the problem of determination of the surface on which the integrand is discontinuous.
{"title":"A uniqueness result for the inverse problem of identifying boundaries from weighted Radon transform","authors":"Dmitrii Sergeevich Anikonov, Sergey G. Kazantsev, Dina S. Konovalova","doi":"10.1515/jiip-2023-0038","DOIUrl":"https://doi.org/10.1515/jiip-2023-0038","url":null,"abstract":"Abstract We study the problem of the integral geometry, in which the functions are integrated over hyperplanes in the n -dimensional Euclidean space, <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mi>m</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> {n=2m+1} . The integrand is the product of a function of n variables called the density and weight function depending on <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mi>n</m:mi> </m:mrow> </m:math> {2n} variables. Such an integration is called here the weighted Radon transform, which coincides with the classical one if the weight function is equal to one. It is proved the uniqueness for the problem of determination of the surface on which the integrand is discontinuous.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135547568","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article we characterize the range of the attenuated and non-attenuated X -ray transform of compactly supported symmetric tensor fields in the Euclidean plane. The characterization is in terms of a Hilbert-transform associated with A -analytic maps in the sense of Bukhgeim.
{"title":"On the <i>X</i>-ray transform of planar symmetric tensors","authors":"Kamran Sadiq, Otmar Scherzer, Alexandru Tamasan","doi":"10.1515/jiip-2022-0055","DOIUrl":"https://doi.org/10.1515/jiip-2022-0055","url":null,"abstract":"Abstract In this article we characterize the range of the attenuated and non-attenuated X -ray transform of compactly supported symmetric tensor fields in the Euclidean plane. The characterization is in terms of a Hilbert-transform associated with A -analytic maps in the sense of Bukhgeim.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135547693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Numerical differentiation of a function over the unit interval of the real axis, which is contaminated with noise, by inverting the simple integration operator J mapping in L2 {L^{2}} is discussed extensively in the literature. The complete singular system of the compact operator J is explicitly given with singular values σn(J) {sigma_{n}(J)} asymptotically proportional to 1n {frac{1}{n}} . This indicates a degree one of ill-posedness for the associated inverse problem of differentiation. We recall the concept of the degree of ill-posedness for linear operator equations with compact forward operators in Hilbert spaces. In contrast to the one-dimensional case, there is little specific material available about the inverse problem of mixed differentiation, where the d-dimensional analog Jd {J_{d}} to J , defined over unit d -cube, is to be inverted. In this note, we show for that problem that the degree of ill-posedness stays at one for all dimensions d∈ℕ {din{mathbb{N}}} . Some more discussion refers to the two-dimensional case in order to characterize the range of the operator J2 {J_{2}} .
{"title":"A note on the degree of ill-posedness for mixed differentiation on the d-dimensional unit cube","authors":"Bernd Hofmann, Hans-Jürgen Fischer","doi":"10.1515/jiip-2023-0025","DOIUrl":"https://doi.org/10.1515/jiip-2023-0025","url":null,"abstract":"Abstract Numerical differentiation of a function over the unit interval of the real axis, which is contaminated with noise, by inverting the simple integration operator J mapping in <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L^{2}} is discussed extensively in the literature. The complete singular system of the compact operator J is explicitly given with singular values <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>σ</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>J</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> {sigma_{n}(J)} asymptotically proportional to <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mfrac> <m:mn>1</m:mn> <m:mi>n</m:mi> </m:mfrac> </m:math> {frac{1}{n}} . This indicates a degree one of ill-posedness for the associated inverse problem of differentiation. We recall the concept of the degree of ill-posedness for linear operator equations with compact forward operators in Hilbert spaces. In contrast to the one-dimensional case, there is little specific material available about the inverse problem of mixed differentiation, where the d-dimensional analog <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>J</m:mi> <m:mi>d</m:mi> </m:msub> </m:math> {J_{d}} to J , defined over unit d -cube, is to be inverted. In this note, we show for that problem that the degree of ill-posedness stays at one for all dimensions <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>d</m:mi> <m:mo>∈</m:mo> <m:mi>ℕ</m:mi> </m:mrow> </m:math> {din{mathbb{N}}} . Some more discussion refers to the two-dimensional case in order to characterize the range of the operator <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>J</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> {J_{2}} .","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"221 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135548380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-01DOI: 10.1515/jiip-2023-frontmatter5
{"title":"Frontmatter","authors":"","doi":"10.1515/jiip-2023-frontmatter5","DOIUrl":"https://doi.org/10.1515/jiip-2023-frontmatter5","url":null,"abstract":"","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134934693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article, we consider an inverse problem for determining simultaneously a fractional order and a time-dependent source term in a multi-dimensional time-fractional diffusion-wave equation by a nonlocal condition. Based on a uniformly bounded estimate of the Mittag-Leffler function given in this paper, we prove the uniqueness of the inverse problem and the Lipschitz continuity properties for the direct problem. Then we employ the Levenberg–Marquardt method to recover simultaneously the fractional order and the time source term, and establish a finite-dimensional approximation algorithm to find a regularized numerical solution. Moreover, a fast tensor method for solving the direct problem in the three-dimensional case is provided. Some numerical results in one and multidimensional spaces are presented for showing the robustness of the proposed algorithm.
{"title":"Simultaneous inversion for a fractional order and a time source term in a time-fractional diffusion-wave equation","authors":"Kaifang Liao, Lei Zhang, Ting Wei","doi":"10.1515/jiip-2020-0057","DOIUrl":"https://doi.org/10.1515/jiip-2020-0057","url":null,"abstract":"Abstract In this article, we consider an inverse problem for determining simultaneously a fractional order and a time-dependent source term in a multi-dimensional time-fractional diffusion-wave equation by a nonlocal condition. Based on a uniformly bounded estimate of the Mittag-Leffler function given in this paper, we prove the uniqueness of the inverse problem and the Lipschitz continuity properties for the direct problem. Then we employ the Levenberg–Marquardt method to recover simultaneously the fractional order and the time source term, and establish a finite-dimensional approximation algorithm to find a regularized numerical solution. Moreover, a fast tensor method for solving the direct problem in the three-dimensional case is provided. Some numerical results in one and multidimensional spaces are presented for showing the robustness of the proposed algorithm.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136129014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper investigates the inverse problem for the non-stationary radiation transfer equation, which involves finding the attenuation coefficient using the data of serial irradiation of the medium with pulses of various durations. In the framework of single and double scattering approximations, we obtain asymptotic estimates of the scattered radiation flux density for a short duration of the probing pulse. We propose extrapolation procedures for the ballistic component of the radiation transfer equation solution using the data of multiple irradiations of the medium by pulsed radiation sources, which allows us to obtain approximate formulas for finding the attenuation coefficient. The results of numerical experiments with a well-known digital phantom confirm the effectiveness of the extrapolation algorithm for improving the quality of tomographic images of scattering media.
{"title":"An extrapolation method for improving the quality of tomographic images using multiple short-pulse irradiations","authors":"Ivan P. Yarovenko, Igor V. Prokhorov","doi":"10.1515/jiip-2023-0022","DOIUrl":"https://doi.org/10.1515/jiip-2023-0022","url":null,"abstract":"Abstract This paper investigates the inverse problem for the non-stationary radiation transfer equation, which involves finding the attenuation coefficient using the data of serial irradiation of the medium with pulses of various durations. In the framework of single and double scattering approximations, we obtain asymptotic estimates of the scattered radiation flux density for a short duration of the probing pulse. We propose extrapolation procedures for the ballistic component of the radiation transfer equation solution using the data of multiple irradiations of the medium by pulsed radiation sources, which allows us to obtain approximate formulas for finding the attenuation coefficient. The results of numerical experiments with a well-known digital phantom confirm the effectiveness of the extrapolation algorithm for improving the quality of tomographic images of scattering media.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136129013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This work presents an application of tensor field tomography for non-destructive reconstructions of axially symmetric residual stresses in a graded-index YAG single crystal for the case of beam deflection. The axis of the cylinder coincides with the crystallographic axis [001] of the single crystal and it has an axially symmetric refractive index distribution. The transformation of the polarization of light is measured in a plane orthogonal to the axis of the cylinder. Stresses are determined within the framework of the Maxwell piezo-optic law (linear dependence of the permittivity tensor on stresses) and small rotation of quasi principal stress axes. This paper generalizes the method of integrated photoelasticity for the case of ray deflection.
{"title":"Tensor tomography of the residual stress field in graded-index YAG’s single crystals","authors":"A. Puro, Egor Marin","doi":"10.1515/jiip-2021-0047","DOIUrl":"https://doi.org/10.1515/jiip-2021-0047","url":null,"abstract":"Abstract This work presents an application of tensor field tomography for non-destructive reconstructions of axially symmetric residual stresses in a graded-index YAG single crystal for the case of beam deflection. The axis of the cylinder coincides with the crystallographic axis [001] of the single crystal and it has an axially symmetric refractive index distribution. The transformation of the polarization of light is measured in a plane orthogonal to the axis of the cylinder. Stresses are determined within the framework of the Maxwell piezo-optic law (linear dependence of the permittivity tensor on stresses) and small rotation of quasi principal stress axes. This paper generalizes the method of integrated photoelasticity for the case of ray deflection.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46171170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study inverse spectral problems for Dirac-type functional-differential operators with two constant delays greater than two fifths the length of the interval, under Dirichlet boundary conditions. The inverse problem of recovering operators from four spectra has been studied. We consider cases when delays are greater or less than half the length of the interval. The main result of the paper refers to the proof that in both cases operators can be recovered uniquely from four spectra.
{"title":"Inverse problem for Dirac operators with two constant delays","authors":"B. Vojvodić, V. Vladičić, Nebojša Djurić","doi":"10.1515/jiip-2023-0047","DOIUrl":"https://doi.org/10.1515/jiip-2023-0047","url":null,"abstract":"Abstract We study inverse spectral problems for Dirac-type functional-differential operators with two constant delays greater than two fifths the length of the interval, under Dirichlet boundary conditions. The inverse problem of recovering operators from four spectra has been studied. We consider cases when delays are greater or less than half the length of the interval. The main result of the paper refers to the proof that in both cases operators can be recovered uniquely from four spectra.","PeriodicalId":50171,"journal":{"name":"Journal of Inverse and Ill-Posed Problems","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42476992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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