Journal de Mathematiques Pures et Appliquees最新文献

{"title":"On the Almgren minimality of the product of a paired calibrated set with a calibrated set of codimension 1 with singularities, and new Almgren minimal cones","authors":"Xiangyu Liang","doi":"10.1016/j.matpur.2024.01.006","DOIUrl":"10.1016/j.matpur.2024.01.006","url":null,"abstract":"<div><p>In this paper, we prove that the product of a paired calibrated set and a set of codimension 1 calibrated by a coflat calibration with small singularity set is Almgren minimal. This is motivated by the attempt to classify all possible singularities for Almgren minimal sets–Plateau's problem in the setting of sets. In particular, a direct application of the above result leads to various types of new singularities for Almgren minimal sets, e.g. the product of any paired calibrated cone (such as the cone over the <span><math><mi>d</mi><mo>−</mo><mn>2</mn></math></span> skeleton of the unit cube in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>,</mo><mi>d</mi><mo>≥</mo><mn>4</mn></math></span>) with homogeneous area minimizing hypercones (such as the Simons cone).</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"183 ","pages":"Pages 137-169"},"PeriodicalIF":2.3,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139658525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nonlocal critical growth elliptic problems with jumping nonlinearities","authors":"Giovanni Molica Bisci ,&nbsp;Kanishka Perera ,&nbsp;Raffaella Servadei ,&nbsp;Caterina Sportelli","doi":"10.1016/j.matpur.2024.01.005","DOIUrl":"10.1016/j.matpur.2024.01.005","url":null,"abstract":"<div><p>In this paper we study a nonlocal critical growth elliptic problem driven by the fractional Laplacian in the presence of a jumping nonlinearity. By using variational and topological methods and applying some new linking theorems recently proved by Perera and Sportelli in <span>[19]</span>, we prove the existence of a nontrivial solution for the problem under consideration.</p><p>The results we obtain here are the nonlocal counterparts of the ones obtained in <span>[19]</span> in the context of a local equation. Due to the nonlocal nature of our problem, some additional difficulties arise, and the arguments employed in the local setting need to be improved or reconceived. In fact, the proofs of our main theorems require some refined techniques and new regularity results for weak solutions of nonlocal problems that are of independent interest.</p><p>We would like to point out that our results are specifically for a nonlocal problem with the fractional operator in integral form. However, we do not exclude the possibility that our results may have a counterpart for the spectral operator studied in <span>[27]</span>. Since nonlocal operators in integral form are being widely investigated in the current literature, especially in connection with geometric problems, we have restricted ourselves to elliptic equations driven by a fractional operator in integral form here.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"183 ","pages":"Pages 170-196"},"PeriodicalIF":2.3,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000114/pdfft?md5=f51fc8015e6f845f984b0ffcc0c1be63&pid=1-s2.0-S0021782424000114-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Borel (α,β)-multitransforms and quantum Leray–Hirsch: Integral representations of solutions of quantum differential equations for P1-bundles","authors":"Giordano Cotti","doi":"10.1016/j.matpur.2024.01.003","DOIUrl":"10.1016/j.matpur.2024.01.003","url":null,"abstract":"<div><p>In this paper, we address the integration problem of the isomonodromic system of quantum differential equations (<em>qDE</em>s) associated with the quantum cohomology of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-bundles on Fano varieties. It is shown that bases of solutions of the <em>qDE</em> associated with the total space of the <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-bundle can be reconstructed from the datum of bases of solutions of the <em>qDE</em> associated with the base space. This represents a quantum analog of the classical Leray–Hirsch theorem in the context of the isomonodromic approach to quantum cohomology. The reconstruction procedure of the solutions can be performed in terms of some integral transforms, introduced in <span>[17]</span>, called <em>Borel</em> <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span><em>-multitransforms</em>. We emphasize the emergence, in the explicit integral formulas, of an interesting sequence of special functions (closely related to iterated partial derivatives of the Böhmer–Tricomi incomplete Gamma function) as integral kernels. Remarkably, these integral kernels have a <em>universal</em> feature, being independent of the specifically chosen <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-bundle. When applied to projective bundles on products of projective spaces, our results give Mellin–Barnes integral representations of solutions of <em>qDE</em>s. As an example, we show how to integrate the <em>qDE</em> of blow-up of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> at one point via Borel multitransforms of solutions of the <em>qDE</em> of <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"183 ","pages":"Pages 102-136"},"PeriodicalIF":2.3,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000096/pdfft?md5=454aefc44b3bd17a37fbb4d16963075b&pid=1-s2.0-S0021782424000096-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139658186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the weak solutions for the MHD systems with controllable total energy and cross helicity","authors":"Changxing Miao ,&nbsp;Weikui Ye","doi":"10.1016/j.matpur.2023.12.010","DOIUrl":"10.1016/j.matpur.2023.12.010","url":null,"abstract":"<div><p>In this paper, we prove the non-uniqueness of three-dimensional magneto-hydrodynamic (MHD) system in <span><math><mi>C</mi><mo>(</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo><mo>;</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo><mo>)</mo></math></span><span> for any initial data in </span><span><math><msup><mrow><mi>H</mi></mrow><mrow><mover><mrow><mi>β</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msup><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math></span> (<span><math><mover><mrow><mi>β</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>&gt;</mo><mn>0</mn></math></span><span>), by exhibiting that the total energy and the cross helicity<span><span> can be controlled in a given positive time interval. Our results extend the non-uniqueness results of the ideal MHD system to the viscous and resistive MHD system. Different from the ideal MHD system, the dissipative effect in the viscous and resistive MHD system prevents the </span>nonlinear term from balancing the stress error </span></span><span><math><mo>(</mo><msub><mrow><mover><mrow><mi>R</mi></mrow><mrow><mo>˚</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><msub><mrow><mover><mrow><mi>M</mi></mrow><mrow><mo>˚</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> as doing in <span>[4]</span>. We introduce the box flows and construct the perturbation consisting in seven different kinds of flows in convex integral scheme, which ensures that the iteration works and yields the non-uniqueness.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"181 ","pages":"Pages 190-227"},"PeriodicalIF":2.3,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138493208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Khovanskii bases for semimixed systems of polynomial equations – Approximating stationary nonlinear Newtonian dynamics","authors":"Viktoriia Borovik ,&nbsp;Paul Breiding ,&nbsp;Javier del Pino ,&nbsp;Mateusz Michałek ,&nbsp;Oded Zilberberg","doi":"10.1016/j.matpur.2023.12.005","DOIUrl":"10.1016/j.matpur.2023.12.005","url":null,"abstract":"<div><p>We provide an approach to counting roots of polynomial systems, where each polynomial is a general linear combination of prescribed, fixed polynomials. Our tools rely on the theory of Khovanskii bases, combined with toric geometry, the Bernstein–Khovanskii–Kushnirenko (BKK) Theorem, and fiber products.</p><p>As a direct application of this theory, we solve the problem of counting the number of approximate stationary states for coupled driven nonlinear resonators. We set up a system of polynomial equations that depends on three numbers <span><math><mi>N</mi><mo>,</mo><mi>n</mi></math></span> and <em>M</em> and whose solutions model the stationary states. The parameter <em>N</em> is the number of coupled resonators, <span><math><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></math></span> is the degree of nonlinearity of the underlying differential equation, and <em>M</em> is the number of frequencies used in the approximation. We use our main theorems, that is, the generalized BKK <span>Theorem 2.5</span> and the Decoupling <span>Theorem 3.8</span>, to count the number of (complex) solutions of the polynomial system for an arbitrary degree of nonlinearity <span><math><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo>⩾</mo><mn>3</mn></math></span>, any number of resonators <span><math><mi>N</mi><mo>⩾</mo><mn>1</mn></math></span>, and <span><math><mi>M</mi><mo>=</mo><mn>1</mn></math></span> harmonic. We also solve the case <span><math><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>=</mo><mn>2</mn></math></span> and <span><math><mi>M</mi><mo>=</mo><mn>2</mn></math></span> and give a computational way to check the number of solutions for <span><math><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>=</mo><mn>2</mn></math></span> and <span><math><mi>M</mi><mo>&gt;</mo><mn>2</mn></math></span>. This extends the results of <span>[1]</span>.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"182 ","pages":"Pages 195-222"},"PeriodicalIF":2.3,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782423001563/pdfft?md5=41eba82621f640de61de196186037d89&pid=1-s2.0-S0021782423001563-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138493962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"C1,α-regularity for solutions of degenerate/singular fully nonlinear parabolic equations","authors":"Ki-Ahm Lee ,&nbsp;Se-Chan Lee ,&nbsp;Hyungsung Yun","doi":"10.1016/j.matpur.2023.12.002","DOIUrl":"10.1016/j.matpur.2023.12.002","url":null,"abstract":"<div><p>We establish the interior <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span><span>-estimate for viscosity solutions<span> of degenerate/singular fully nonlinear parabolic equations</span></span><span><span><span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>|</mo><mi>D</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>γ</mi></mrow></msup><mi>F</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo><mo>+</mo><mi>f</mi><mspace></mspace><mtext>in </mtext><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo></math></span></span></span> where <span><math><mi>γ</mi><mo>&gt;</mo><mo>−</mo><mn>1</mn></math></span> and <span><math><mi>f</mi><mo>∈</mo><mi>C</mi><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>∩</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span>. For this purpose, we prove the well-posedness of the regularized Cauchy-Dirichlet problem<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd><mtd><mo>=</mo><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mo>|</mo><mi>D</mi><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mi>γ</mi><mo>/</mo><mn>2</mn></mrow></msup><mi>F</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>)</mo></mtd><mtd><mspace></mspace></mtd><mtd><mrow><mtext>in </mtext><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mtd></mtr><mtr><mtd><mi>u</mi></mtd><mtd><mo>=</mo><mi>φ</mi></mtd><mtd><mspace></mspace></mtd><mtd><mrow><mtext>on </mtext><msub><mrow><mo>∂</mo></mrow><mrow><mi>p</mi></mrow></msub><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>γ</mi><mo>&gt;</mo><mo>−</mo><mn>2</mn></math></span>. Our approach utilizes the Bernstein method with approximations in view of the difference quotient.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"181 ","pages":"Pages 152-189"},"PeriodicalIF":2.3,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138556531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Far field broadband approximate cloaking for the Helmholtz equation with a Drude-Lorentz refractive index","authors":"Fioralba Cakoni,&nbsp;Narek Hovsepyan,&nbsp;Michael S. Vogelius","doi":"10.1016/j.matpur.2023.12.001","DOIUrl":"10.1016/j.matpur.2023.12.001","url":null,"abstract":"<div><p><span>This paper concerns the analysis of a passive, broadband approximate cloaking scheme for the Helmholtz equation in </span><span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> for <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span> or <span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span>. Using ideas from transformation optics, we construct an approximate cloak by “blowing up” a small ball of radius <span><math><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></math></span> to one of radius 1. In the anisotropic cloaking layer resulting from the “blow-up” change of variables, we incorporate a Drude-Lorentz-type model for the index of refraction, and we assume that the cloaked object is a soft (perfectly conducting) obstacle. We first show that (for any fixed <em>ϵ</em>) there are no real transmission eigenvalues associated with the inhomogeneity representing the cloak, which implies that the cloaking devices we have created will not yield perfect cloaking at any frequency, even for a single incident time harmonic wave. Secondly, we establish estimates on the scattered field due to an arbitrary time harmonic incident wave. These estimates show that, as <em>ϵ</em> approaches 0, the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm of the scattered field outside the cloak, and its far field pattern, approach 0 uniformly over any bounded band of frequencies. In other words: our scheme leads to broadband approximate cloaking for arbitrary incident time harmonic waves.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"182 ","pages":"Pages 285-318"},"PeriodicalIF":2.3,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138556527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"High-order estimates for fully nonlinear equations under weak concavity assumptions","authors":"Alessandro Goffi","doi":"10.1016/j.matpur.2023.12.006","DOIUrl":"10.1016/j.matpur.2023.12.006","url":null,"abstract":"<div><p>This paper studies a priori and regularity estimates of Evans-Krylov type in Hölder spaces for fully nonlinear uniformly elliptic and parabolic equations of second order when the operator fails to be concave or convex in the space of symmetric matrices. In particular, it is assumed that either the level sets are convex or the operator is concave, convex or close to a linear function near infinity. As a byproduct, these results imply polynomial Liouville theorems for entire solutions of elliptic equations and for ancient solutions to parabolic problems.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"182 ","pages":"Pages 223-252"},"PeriodicalIF":2.3,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782423001575/pdfft?md5=ba1f0f1482cc16cb5e4884edc4186dea&pid=1-s2.0-S0021782423001575-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138493207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Asymptotic behavior of a plate with a non-planar top surface","authors":"G. Griso","doi":"10.1016/j.matpur.2023.12.004","DOIUrl":"10.1016/j.matpur.2023.12.004","url":null,"abstract":"<div><p>In this paper, we study the asymptotic behaviors<span> of a plate with non-planar top surface in the framework of linear elasticity. For this plate, we give a decomposition of the displacements. We show that every displacement of the plate is the sum of a Kirchhoff-Love displacement and a residual displacement that takes into account the deformations of the fibers of the plate and shear. We also prove Korn's type inequalities.</span></p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"182 ","pages":"Pages 1-30"},"PeriodicalIF":2.3,"publicationDate":"2023-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138493963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nonautonomous (p,q)-equations with unbalanced growth and competing nonlinearities","authors":"Zhenhai Liu ,&nbsp;Nikolaos S. Papageorgiou","doi":"10.1016/j.matpur.2023.12.008","DOIUrl":"10.1016/j.matpur.2023.12.008","url":null,"abstract":"<div><p>We consider a parametric<span><span> nonlinear Dirichlet problem driven by the double phase </span>differential operator<span> and a reaction that has the competing effects of parametric “concave” term and of a “convex” perturbation (concave-convex problem). Using variational tools together with truncation and comparison techniques and critical groups<span>, we show that for all small values of the parameter, the problem has at least three nontrivial bounded solutions and we provide sign information for all of them (positive, negative and nodal). Moreover, the solutions are ordered.</span></span></span></p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"182 ","pages":"Pages 164-194"},"PeriodicalIF":2.3,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138493206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}