In 1993, Tsai proved that a proper holomorphic mapping f: Ω → Ω′ from an irreducible bounded symmetric domain Ω of rank ? 2 into a bounded symmetric domain Ω′ is necessarily totally geodesic provided that r′:=...In 1993, Tsai proved that a proper holomorphic mapping f: Ω → Ω′ from an irreducible bounded symmetric domain Ω of rank ? 2 into a bounded symmetric domain Ω′ is necessarily totally geodesic provided that r′:= rank(Ωg’) ? rank(Ω):= r, proving a conjecture of the author’s motivated by Hermitian metric rigidity. As a first step in the proof, Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1. Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding, this means that the germ of f at a general point preserves the varieties of minimal rational tangents (VMRTs).In another completely different direction Hwang-Mok established with very few exceptions the Cartan-Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard number 1, showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs. We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1, especially in the case of classical manifolds such as rational homogeneous spaces of Picard number 1, by a non-equidimensional analogue of the Cartan-Fubini extension principle. As an illustration we show along this line that standard embeddings between complex Grassmann manifolds of rank ? 2 can be characterized by the VMRT-preserving property and a non-degeneracy condition, giving a new proof of a result of Neretin’s which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1, on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures.展开更多
In this paper we present the most important definitions and results of the theory of parabolic-like mappings, and we will give an example. The proofs of the results can be found in [2,4] and [3].
This paper proposes three fractional discrete chaotic systems based on the Rulkov,Chang,and Zeraoulia–Sprott rational maps.The dynamics of the proposed maps are investigated by means of phase plots and bifurcations d...This paper proposes three fractional discrete chaotic systems based on the Rulkov,Chang,and Zeraoulia–Sprott rational maps.The dynamics of the proposed maps are investigated by means of phase plots and bifurcations diagrams.Adaptive stabilization schemes are proposed for each of the three maps and the convergence of the states is established by using the Lyapunov method.Furthermore,a combination synchronization scheme is proposed whereby a combination of the fractional Rulkov and Chang maps is synchronized to the fractional Zeraoulia-Sprott map.Numerical results are used to confirm the findings of the paper.展开更多
A fundamental result in the theory of minimal rational curves on projective manifolds is Cartan- Fubini extension theorem proved by Hwang and Mok, which describes the extensibility of biholomorphisms between connected...A fundamental result in the theory of minimal rational curves on projective manifolds is Cartan- Fubini extension theorem proved by Hwang and Mok, which describes the extensibility of biholomorphisms between connected open subsets of two Fano manifolds of Picard number 1 which preserve varieties of minimal rational tangents （VMRT）, under a mild geometric assumption on the second fundamental forms of VMRT＇s. Hong and Mok have developed Cartan-Fubini extension for non-equidimensional holomorphic immersions from a connected open subset of a Pano manifold of Picard number 1 into a uniruled projective manifold, under the assumptions that the map sends VMRT＇s onto linear sections of VMRT＇s and it satisfies a mild geometric condition formulated in terms of second fundamental forms on VMRT＇s. In the current paper, we give a generalization of Hong and Mok＇s result, under the same condition on second fundamental forms, assuming only that the holomorphic immersions send VMRT＇s to VMRT＇s. Our argument is different from Hong and Mok＇s and is based on the study of natural foliations on the total family of VMRT＇s. This gives a substantially simpler proof than Hong and Mok＇s argument.展开更多
We examine when a meromorphic quadratic differential φ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of φ around each pole ...We examine when a meromorphic quadratic differential φ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of φ around each pole c, the most singular term should take the form(1- d2)/(2(z- c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles(i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko(2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by φ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative.Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball.展开更多
Rational proper holomorphic maps from the unit ball in C2 into the unit ball CN with degree 2 are studied. Any such map must be equivalent to one of the four types of maps.
The first part of the article provides an overview of the theoretical evidence, the main provisions, and the implementation strategy of information support for bioresource and ecosystem research in the north-west Paci...The first part of the article provides an overview of the theoretical evidence, the main provisions, and the implementation strategy of information support for bioresource and ecosystem research in the north-west Pacific, which has been conducted over the past 20 years in the Russian Far East Research Institute TINRO-Center. In short, the concept consists of a combination of the following four assertions: 1) For the steady and sustainable development of the Russian Far East, the entire Russian Federation and the Asia-Pacific Region in general, environmental, food, economic, and other security is required, which cannot be achieved without the rational use of bioresources based on the ecosystem approach to the management of aquatic bioresources. 2) For the inventory, appraisal, monitoring, forecasting of the state of and management the natural water resources when applying this approach, statistically relevant quantitative information is required on the greatest possible number of constituents of marine biocenosis of the north-western Pacific for the longest possible period of time, which is only available at the TINRO-Center. 3) This valuable data should be organized into databases, based on which geo-information and other electronic information systems are prepared, and based on these map atlases and reference books on natural water resources, using automated workplaces created especially for this. 4) The resulting unique information support will be of great value not only for practical purposes, but also for science, both applied and fundamental. Next comes a summary of the many years of work on the practical implementation of this concept and the key achievements in this field obtained by the TINRO-Center by the end of 2015 are reviewed. At the end, some plans for the near future are outlined.展开更多
基金This research is partially supported by a Competitive Earmarked Research Grant of the Research Grants Council of Hong Kong,China
文摘In 1993, Tsai proved that a proper holomorphic mapping f: Ω → Ω′ from an irreducible bounded symmetric domain Ω of rank ? 2 into a bounded symmetric domain Ω′ is necessarily totally geodesic provided that r′:= rank(Ωg’) ? rank(Ω):= r, proving a conjecture of the author’s motivated by Hermitian metric rigidity. As a first step in the proof, Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1. Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding, this means that the germ of f at a general point preserves the varieties of minimal rational tangents (VMRTs).In another completely different direction Hwang-Mok established with very few exceptions the Cartan-Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard number 1, showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs. We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1, especially in the case of classical manifolds such as rational homogeneous spaces of Picard number 1, by a non-equidimensional analogue of the Cartan-Fubini extension principle. As an illustration we show along this line that standard embeddings between complex Grassmann manifolds of rank ? 2 can be characterized by the VMRT-preserving property and a non-degeneracy condition, giving a new proof of a result of Neretin’s which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1, on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures.
基金supported by National Natural Science Foundation of China (Grant Nos. 11401236 and 11471132)Self-Determined Research Funds of Central China Normal University (Grant No. CCNU17QN0009)
文摘The dynamical structure of the rational map ax+1/x on the projective line P^1(Q_2) over the field Q_2 of 2-adic numbers, is fully described.
文摘In this paper we present the most important definitions and results of the theory of parabolic-like mappings, and we will give an example. The proofs of the results can be found in [2,4] and [3].
基金supported by the Natural Science Foundation of China under Grant Nos.11726624,11726623,61473237the Natural Science Basic Research Plan in Shaxanxi Province of China under Grant No.2018GY-091the Natural Science Basic Research Plan in Shandong Province of China under Grant No.ZR2017PA008。
文摘This paper proposes three fractional discrete chaotic systems based on the Rulkov,Chang,and Zeraoulia–Sprott rational maps.The dynamics of the proposed maps are investigated by means of phase plots and bifurcations diagrams.Adaptive stabilization schemes are proposed for each of the three maps and the convergence of the states is established by using the Lyapunov method.Furthermore,a combination synchronization scheme is proposed whereby a combination of the fractional Rulkov and Chang maps is synchronized to the fractional Zeraoulia-Sprott map.Numerical results are used to confirm the findings of the paper.
基金supported by National Researcher Program of National Research Foundation of Korea(Grant No.2010-0020413)
文摘A fundamental result in the theory of minimal rational curves on projective manifolds is Cartan- Fubini extension theorem proved by Hwang and Mok, which describes the extensibility of biholomorphisms between connected open subsets of two Fano manifolds of Picard number 1 which preserve varieties of minimal rational tangents （VMRT）, under a mild geometric assumption on the second fundamental forms of VMRT＇s. Hong and Mok have developed Cartan-Fubini extension for non-equidimensional holomorphic immersions from a connected open subset of a Pano manifold of Picard number 1 into a uniruled projective manifold, under the assumptions that the map sends VMRT＇s onto linear sections of VMRT＇s and it satisfies a mild geometric condition formulated in terms of second fundamental forms on VMRT＇s. In the current paper, we give a generalization of Hong and Mok＇s result, under the same condition on second fundamental forms, assuming only that the holomorphic immersions send VMRT＇s to VMRT＇s. Our argument is different from Hong and Mok＇s and is based on the study of natural foliations on the total family of VMRT＇s. This gives a substantially simpler proof than Hong and Mok＇s argument.
基金supported by National Natural Science Foundation of China (Grant Nos. 11125106 and 11501383)Project LAMBDA (Grant No. ANR-13-BS01-0002)
文摘We examine when a meromorphic quadratic differential φ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of φ around each pole c, the most singular term should take the form(1- d2)/(2(z- c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles(i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko(2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by φ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative.Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball.
基金The first author in supported by National Natural Science Foundation of China (Grant No. 10571135).
文摘Rational proper holomorphic maps from the unit ball in C2 into the unit ball CN with degree 2 are studied. Any such map must be equivalent to one of the four types of maps.
文摘The first part of the article provides an overview of the theoretical evidence, the main provisions, and the implementation strategy of information support for bioresource and ecosystem research in the north-west Pacific, which has been conducted over the past 20 years in the Russian Far East Research Institute TINRO-Center. In short, the concept consists of a combination of the following four assertions: 1) For the steady and sustainable development of the Russian Far East, the entire Russian Federation and the Asia-Pacific Region in general, environmental, food, economic, and other security is required, which cannot be achieved without the rational use of bioresources based on the ecosystem approach to the management of aquatic bioresources. 2) For the inventory, appraisal, monitoring, forecasting of the state of and management the natural water resources when applying this approach, statistically relevant quantitative information is required on the greatest possible number of constituents of marine biocenosis of the north-western Pacific for the longest possible period of time, which is only available at the TINRO-Center. 3) This valuable data should be organized into databases, based on which geo-information and other electronic information systems are prepared, and based on these map atlases and reference books on natural water resources, using automated workplaces created especially for this. 4) The resulting unique information support will be of great value not only for practical purposes, but also for science, both applied and fundamental. Next comes a summary of the many years of work on the practical implementation of this concept and the key achievements in this field obtained by the TINRO-Center by the end of 2015 are reviewed. At the end, some plans for the near future are outlined.
