Pub Date : 2018-07-26DOI: 10.17323/1609-4514-2019-19-1-89-106
O. Manita, A. Veretennikov
Rate of convergence is studied for a diffusion process on the half line with a non-sticky reflection to a heavy-tailed 1D invariant distribution which density on the half line has a polynomial decay at infinity. Starting from a standard receipt which guarantees some polynomial convergence, it is shown how to construct a new non-degenerate diffusion process on the half line which converges to the same invariant measure exponentially fast uniformly with respect to the initial data.
{"title":"On Convergence of 1D Markov Diffusions to Heavy-Tailed Invariant Density","authors":"O. Manita, A. Veretennikov","doi":"10.17323/1609-4514-2019-19-1-89-106","DOIUrl":"https://doi.org/10.17323/1609-4514-2019-19-1-89-106","url":null,"abstract":"Rate of convergence is studied for a diffusion process on the half line with a non-sticky reflection to a heavy-tailed 1D invariant distribution which density on the half line has a polynomial decay at infinity. Starting from a standard receipt which guarantees some polynomial convergence, it is shown how to construct a new non-degenerate diffusion process on the half line which converges to the same invariant measure exponentially fast uniformly with respect to the initial data.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45054101","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 : 2018-07-25DOI: 10.17323/1609-4514-2021-21-3-507-565
I. Entova-Aizenbud, V. Serganova
We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras $mathfrak{p}(n)$ as $n to infty$. The paper gives a construction of the tensor category $Rep(underline{P})$, possessing nice universal properties among tensor categories over the category $mathtt{sVect}$ of finite-dimensional complex vector superspaces. First, it is the "abelian envelope" of the Deligne category corresponding to the periplectic Lie superalgebra, in the sense of arXiv:1511.07699. Secondly, given a tensor category $mathcal{C}$ over $mathtt{sVect}$, exact tensor functors $Rep(underline{P})longrightarrow mathcal{C}$ classify pairs $(X, omega)$ in $mathcal{C}$ where $omega: X otimes X to Pi mathbf{1}$ is a non-degenerate symmetric form and $X$ not annihilated by any Schur functor. The category $Rep(underline{P})$ is constructed in two ways. The first construction is through an explicit limit of the tensor categories $Rep(mathfrak{p}(n))$ ($ngeq 1$) under Duflo-Serganova functors. The second construction (inspired by P. Etingof) describes $Rep(underline{P})$ as the category of representations of a periplectic Lie supergroup in the Deligne category $mathtt{sVect} boxtimes Rep(underline{GL}_t)$. An upcoming paper by the authors will give results on the abelian and tensor structure of $Rep(underline{P})$.
{"title":"Deligne Categories and the Periplectic Lie Superalgebra","authors":"I. Entova-Aizenbud, V. Serganova","doi":"10.17323/1609-4514-2021-21-3-507-565","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-3-507-565","url":null,"abstract":"We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras $mathfrak{p}(n)$ as $n to infty$. \u0000The paper gives a construction of the tensor category $Rep(underline{P})$, possessing nice universal properties among tensor categories over the category $mathtt{sVect}$ of finite-dimensional complex vector superspaces. \u0000First, it is the \"abelian envelope\" of the Deligne category corresponding to the periplectic Lie superalgebra, in the sense of arXiv:1511.07699. \u0000Secondly, given a tensor category $mathcal{C}$ over $mathtt{sVect}$, exact tensor functors $Rep(underline{P})longrightarrow mathcal{C}$ classify pairs $(X, omega)$ in $mathcal{C}$ where $omega: X otimes X to Pi mathbf{1}$ is a non-degenerate symmetric form and $X$ not annihilated by any Schur functor. \u0000The category $Rep(underline{P})$ is constructed in two ways. The first construction is through an explicit limit of the tensor categories $Rep(mathfrak{p}(n))$ ($ngeq 1$) under Duflo-Serganova functors. The second construction (inspired by P. Etingof) describes $Rep(underline{P})$ as the category of representations of a periplectic Lie supergroup in the Deligne category $mathtt{sVect} boxtimes Rep(underline{GL}_t)$. \u0000An upcoming paper by the authors will give results on the abelian and tensor structure of $Rep(underline{P})$.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48625365","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 : 2018-07-24DOI: 10.17323/1609-4514-2020-20-1-93-126
Robert Laugwitz, V. Retakh
We introduce a ring of noncommutative shifted symmetric functions based on an integer-indexed sequence of shift parameters. Using generating series and quasideterminants, this multiparameter approach produces deformations of the ring of noncommutative symmetric functions. Shifted versions of ribbon Schur functions are defined and form a basis for the ring. Further, we produce analogues of Jacobi-Trudi and N"agelsbach-Kostka formulas, a duality anti-algebra isomorphism, shifted quasi-Schur functions, and Giambelli's formula in this setup. In addition, an analogue of power sums is provided, satisfying versions of Wronski and Newton formulas. Finally, a realization of these noncommutative shifted symmetric functions as rational functions in noncommuting variables is given. These realizations have a shifted symmetry under exchange of the variables and are well-behaved under extension of the list of variables.
{"title":"Noncommutative Shifted Symmetric Functions","authors":"Robert Laugwitz, V. Retakh","doi":"10.17323/1609-4514-2020-20-1-93-126","DOIUrl":"https://doi.org/10.17323/1609-4514-2020-20-1-93-126","url":null,"abstract":"We introduce a ring of noncommutative shifted symmetric functions based on an integer-indexed sequence of shift parameters. Using generating series and quasideterminants, this multiparameter approach produces deformations of the ring of noncommutative symmetric functions. Shifted versions of ribbon Schur functions are defined and form a basis for the ring. Further, we produce analogues of Jacobi-Trudi and N\"agelsbach-Kostka formulas, a duality anti-algebra isomorphism, shifted quasi-Schur functions, and Giambelli's formula in this setup. In addition, an analogue of power sums is provided, satisfying versions of Wronski and Newton formulas. Finally, a realization of these noncommutative shifted symmetric functions as rational functions in noncommuting variables is given. These realizations have a shifted symmetry under exchange of the variables and are well-behaved under extension of the list of variables.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46393252","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 : 2018-07-18DOI: 10.17323/1609-4514-2020-20-1-67-91
A. Kolesnikov
We study the transportation problem on the unit sphere $S^{n-1}$ for symmetric probability measures and the cost function $c(x,y) = log frac{1}{langle x, y rangle}$. We calculate the variation of the corresponding Kantorovich functional $K$ and study a naturally associated metric-measure space on $S^{n-1}$ endowed with a Riemannian metric generated by the corresponding transportational potential. We introduce a new transportational functional which minimizers are solutions to the symmetric log-Minkowski problem and prove that $K$ satisfies the following analog of the Gaussian transportation inequality for the uniform probability measure ${sigma}$ on $S^{n-1}$: $frac{1}{n} Ent(nu) ge K({sigma}, nu)$. It is shown that there exists a remarkable similarity between our results and the theory of the K{"a}hler-Einstein equation on Euclidean space. As a by-product we obtain a new proof of uniqueness of solution to the log-Minkowski problem for the uniform measure.
{"title":"Mass Transportation Functionals on the Sphere with Applications to the Logarithmic Minkowski Problem","authors":"A. Kolesnikov","doi":"10.17323/1609-4514-2020-20-1-67-91","DOIUrl":"https://doi.org/10.17323/1609-4514-2020-20-1-67-91","url":null,"abstract":"We study the transportation problem on the unit sphere $S^{n-1}$ for symmetric probability measures and the cost function $c(x,y) = log frac{1}{langle x, y rangle}$. \u0000We calculate the variation of the corresponding Kantorovich functional $K$ and study a naturally associated metric-measure space on $S^{n-1}$ endowed with a Riemannian \u0000metric generated by the corresponding transportational potential. We introduce a new transportational functional which minimizers are \u0000solutions to the symmetric log-Minkowski problem and prove that $K$ satisfies the following analog of the Gaussian transportation inequality for the uniform probability measure ${sigma}$ on $S^{n-1}$: \u0000$frac{1}{n} Ent(nu) ge K({sigma}, nu)$. It is shown that there exists a remarkable similarity between our results and the theory of the K{\"a}hler-Einstein equation on Euclidean space. \u0000As a by-product we obtain a new proof of uniqueness of solution to the log-Minkowski problem for the uniform measure.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46226818","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 : 2018-07-11DOI: 10.17323/1609-4514-2020-2-423-436
A. Soldatenkov
This note is inspired by the work of Deligne on the local behavior of Hodge structures at infinity. We study limit mixed Hodge structures of degenerating families of compact hyperk"ahler manifolds. We show that when the monodromy action on $H^2$ has maximal index of unipotency, the limit mixed Hodge structures on all cohomology groups are of Hodge-Tate type.
{"title":"Limit Mixed Hodge Structures of Hyperkähler Manifolds","authors":"A. Soldatenkov","doi":"10.17323/1609-4514-2020-2-423-436","DOIUrl":"https://doi.org/10.17323/1609-4514-2020-2-423-436","url":null,"abstract":"This note is inspired by the work of Deligne on the local behavior of Hodge structures at infinity. We study limit mixed Hodge structures of degenerating families of compact hyperk\"ahler manifolds. We show that when the monodromy action on $H^2$ has maximal index of unipotency, the limit mixed Hodge structures on all cohomology groups are of Hodge-Tate type.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46763447","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 : 2018-07-10DOI: 10.17323/1609-4514-2021-21-1-99-127
Deniz Kus, R. Venkatesh
We develop a Borel-de Siebenthal theory for affine reflection systems by classifying their maximal closed subroot systems. Affine reflection systems (introduced by Loos and Neher) provide a unifying framework for root systems of finite-dimensional semi-simple Lie algebras, affine and toroidal Lie algebras, and extended affine Lie algebras. In the special case of nullity $k$ toroidal Lie algebras, we obtain a one-to-one correspondence between maximal closed subroot systems with full gradient and triples $(q,(b_i),H)$, where $q$ is a prime number, $(b_i)$ is a $n$-tuple of integers in the interval $[0,q-1]$ and $H$ is a $(ktimes k)$ Hermite normal form matrix with determinant $q$. This generalizes the $k=1$ result of Dyer and Lehrer in the setting of affine Lie algebras.
{"title":"Borel–de Siebenthal Theory for Affine Reflection Systems","authors":"Deniz Kus, R. Venkatesh","doi":"10.17323/1609-4514-2021-21-1-99-127","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-1-99-127","url":null,"abstract":"We develop a Borel-de Siebenthal theory for affine reflection systems by classifying their maximal closed subroot systems. Affine reflection systems (introduced by Loos and Neher) provide a unifying framework for root systems of finite-dimensional semi-simple Lie algebras, affine and toroidal Lie algebras, and extended affine Lie algebras. In the special case of nullity $k$ toroidal Lie algebras, we obtain a one-to-one correspondence between maximal closed subroot systems with full gradient and triples $(q,(b_i),H)$, where $q$ is a prime number, $(b_i)$ is a $n$-tuple of integers in the interval $[0,q-1]$ and $H$ is a $(ktimes k)$ Hermite normal form matrix with determinant $q$. This generalizes the $k=1$ result of Dyer and Lehrer in the setting of affine Lie algebras.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45318907","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 : 2018-07-04DOI: 10.17323/1609-4514-2022-22-4-565-593
Peter Beelen, M. Datta, S. Ghorpade
We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible ${mathbb{F}}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in $m+1$ variables with coefficients in the finite field ${mathbb{F}}q$ with $q$ elements, when $d
{"title":"A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields","authors":"Peter Beelen, M. Datta, S. Ghorpade","doi":"10.17323/1609-4514-2022-22-4-565-593","DOIUrl":"https://doi.org/10.17323/1609-4514-2022-22-4-565-593","url":null,"abstract":"We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible ${mathbb{F}}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in $m+1$ variables with coefficients in the finite field ${mathbb{F}}q$ with $q$ elements, when $d<q$. It is shown that this formula holds in the affirmative for several values of $r$. In the general case, we give explicit lower and upper bounds for $e_r(d,m)$ and show that they are sometimes attained. Our approach uses a relatively recent result, called the projective footprint bound, together with results from extremal combinatorics such as the Clements-Lindstr\"om Theorem and its variants. Applications to the problem of determining the generalized Hamming weights of projective Reed-Muller codes are also included.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48938692","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 : 2018-06-09DOI: 10.17323/1609-4514-2019-19-4-739-760
V. Grines, E. Gurevich, O. Pochinka
J.~Palis found necessary conditions for a Morse-Smale diffeomorphism on a closed $n$-dimensional manifold $M^n$ to embed into a topological flow and proved that these conditions are also sufficient for $n=2$. For the case $n=3$ a possibility of wild embedding of closures of separatrices of saddles is an additional obstacle for Morse-Smale cascades to embed into topological flows. In this paper we show that there are no such obstructions for Morse-Smale diffeomorphisms without heteroclinic intersection given on the sphere $S^n, ,ngeq 4$, and Palis's conditions again are sufficient for such diffeomorphisms.
{"title":"On Embedding of Multidimensional Morse–Smale Diffeomorphisms into Topological Flows","authors":"V. Grines, E. Gurevich, O. Pochinka","doi":"10.17323/1609-4514-2019-19-4-739-760","DOIUrl":"https://doi.org/10.17323/1609-4514-2019-19-4-739-760","url":null,"abstract":"J.~Palis found necessary conditions for a Morse-Smale diffeomorphism on a closed $n$-dimensional manifold $M^n$ to embed into a topological flow and proved that these conditions are also sufficient for $n=2$. For the case $n=3$ a possibility of wild embedding of closures of separatrices of saddles is an additional obstacle for Morse-Smale cascades to embed into topological flows. In this paper we show that there are no such obstructions for Morse-Smale diffeomorphisms without heteroclinic intersection given on the sphere $S^n, ,ngeq 4$, and Palis's conditions again are sufficient for such diffeomorphisms.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46174566","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 : 2018-05-29DOI: 10.17323/1609-4514-2022-22-1-133-168
S. Tanabé
We calculate the period integrals for a special class of affine hypersurfaces (deformed Delsarte hypersurfaces) in an algebraic torus by the aid of their Mellin transforms. A description of the relation between poles of Mellin transforms of period integrals and the mixed Hodge structure of the cohomology of the hypersurface is given. By interpreting the period integrals as solutions to Pochhammer hypergeometric differential equation, we calculate concretely the irreducible monodromy group of period integrals that correspond to the compactification of the affine hypersurface in a complete simplicial toric variety. As an application of the equivalence between oscillating integral for Delsarte polynomial and quantum cohomology of a weighted projective space $mathbb{P}_{bf B}$, we establish an equality between its Stokes matrix and the Gram matrix of the full exceptional collection on $mathbb{P}_{bf B}$.
{"title":"Period Integrals Associated to an Affine Delsarte Type Hypersurface","authors":"S. Tanabé","doi":"10.17323/1609-4514-2022-22-1-133-168","DOIUrl":"https://doi.org/10.17323/1609-4514-2022-22-1-133-168","url":null,"abstract":"We calculate the period integrals for a special class of affine hypersurfaces (deformed Delsarte hypersurfaces) in an algebraic torus by the aid of their Mellin transforms. A description of the relation between poles of Mellin transforms of period integrals and the mixed Hodge structure of the cohomology of the hypersurface is given. By interpreting the period integrals as solutions to Pochhammer hypergeometric differential equation, we calculate concretely the irreducible monodromy group of period integrals that correspond to the compactification of the affine hypersurface in a complete simplicial toric variety. As an application of the equivalence between oscillating integral for Delsarte polynomial and quantum cohomology of a weighted projective space $mathbb{P}_{bf B}$, we establish an equality between its Stokes matrix and the Gram matrix of the full exceptional collection on $mathbb{P}_{bf B}$.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45509205","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 : 2018-05-02DOI: 10.17323/1609-4514-2021-21-2-325-364
V. Gubarev
We state that all Rota---Baxter operators of nonzero weight on Grassmann algebra over a field of characteristic zero are projections on a subalgebra along another one. We show the one-to-one correspondence between the solutions of associative Yang---Baxter equation and Rota---Baxter operators of weight zero on the matrix algebra $M_n(F)$ (joint with P. Kolesnikov). We prove that all Rota---Baxter operators of weight zero on a unital associative (alternative, Jordan) algebraic algebra over a field of characteristic zero are nilpotent. For an algebra $A$, we introduce its new invariant the rb-index $mathrm{rb}(A)$ as the nilpotency index for Rota---Baxter operators of weight zero on $A$. We show that $2n-1leq mathrm{rb}(M_n(F))leq 2n$ provided that characteristic of $F$ is zero.
{"title":"Rota–Baxter Operators on Unital Algebras","authors":"V. Gubarev","doi":"10.17323/1609-4514-2021-21-2-325-364","DOIUrl":"https://doi.org/10.17323/1609-4514-2021-21-2-325-364","url":null,"abstract":"We state that all Rota---Baxter operators of nonzero weight on Grassmann algebra over a field of characteristic zero are projections on a subalgebra along another one. We show the one-to-one correspondence between the solutions of associative Yang---Baxter equation and Rota---Baxter operators of weight zero on the matrix algebra $M_n(F)$ (joint with P. Kolesnikov). \u0000We prove that all Rota---Baxter operators of weight zero on a unital associative (alternative, Jordan) algebraic algebra over a field of characteristic zero are nilpotent. For an algebra $A$, we introduce its new invariant the rb-index $mathrm{rb}(A)$ as the nilpotency index for Rota---Baxter operators of weight zero on $A$. We show that $2n-1leq mathrm{rb}(M_n(F))leq 2n$ provided that characteristic of $F$ is zero.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46391221","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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