In this paper,we consider the Hopf lemma of the following mixed local and nonlocal weighted semilinear elliptic equations{-div(|x|^(-2α)■u)+(-△)_(α)^(s)u=0,x∈U,u(x^(^))=-u(x),x∈H,u(x)=0,x∈R^(N)\U,where H belong...In this paper,we consider the Hopf lemma of the following mixed local and nonlocal weighted semilinear elliptic equations{-div(|x|^(-2α)■u)+(-△)_(α)^(s)u=0,x∈U,u(x^(^))=-u(x),x∈H,u(x)=0,x∈R^(N)\U,where H belong to R^(N)with 0∈H is an open and affine half space,U belong to H is an open and bounded set,s∈(0,1),α∈[0,N-2s/2),(-△)_(α)^(s)is weighted fractional Laplacian with a weighted function.展开更多
Although there is evident homology among reproductive organs when comparing Poaceae （grass） and eudicots, the identity of grass specific organs, such as lodicules, palea, lemma, and glumes has been the subject of a ...Although there is evident homology among reproductive organs when comparing Poaceae （grass） and eudicots, the identity of grass specific organs, such as lodicules, palea, lemma, and glumes has been the subject of a vast and largely inconclusive discussion. Here we provide some direct evidence to support the idea that the empty glumes of rice （Oryza sativa） are counterparts of lemmas. We show that the development of empty glumes is regulated by ELE （elongated empty glume）, which belongs to a plant specific novel gene family. Mutations at the ELE locus cause elongated empty glumes, which mimic the lemmas and have the epidermal morphology of lemmas with four or five vascular bundles. As a nuclear-localized gene, ELE is specifically expressed at the empty glumes of immature spikelets, and its ectopic expression causes many floral development defects, including lemma-like palea, extra palea-like structures, elongated lodicules, extra stamens and stigmas. Our result suggests that empty glumes are lemmas of the sterile florets located at the lateral side of the rice spikelet, and ELE acts as a regulator restraining its growth to maintain its small size in wild-type plants.展开更多
In this note, we consider a holomorphic mapping f from the unit disk C in C to p-ball B^p = {z∈C^n;i=1∑n｜zi｜p〈1,1〈p〈＋∞. It is proved that for such f,｜ ｜ ｜f｜｜（z）｜≤1-｜｜f（z）｜｜^2/1-｜z｜^2,z∈D. Th...In this note, we consider a holomorphic mapping f from the unit disk C in C to p-ball B^p = {z∈C^n;i=1∑n｜zi｜p〈1,1〈p〈＋∞. It is proved that for such f,｜ ｜ ｜f｜｜（z）｜≤1-｜｜f（z）｜｜^2/1-｜z｜^2,z∈D. The extremal problem is also discussed when p is an even number. This result extends some related results on Schwarz lemma.展开更多
This note presents a set of new versions of Barbalat＇s lemma combining with positive （negative） definite functions.Based on these results,a set of new formulations of Lyapunov-like lemma are established.A simple ex...This note presents a set of new versions of Barbalat＇s lemma combining with positive （negative） definite functions.Based on these results,a set of new formulations of Lyapunov-like lemma are established.A simple example shows the usefulness of our results.展开更多
On the basis of Hill＇s lemma for classical Cauchy continuum, a version of Hill＇s lemma for micro-macro homogenization modeling of heterogeneous Cosserat continuum is presented in the flame of average-field theory. T...On the basis of Hill＇s lemma for classical Cauchy continuum, a version of Hill＇s lemma for micro-macro homogenization modeling of heterogeneous Cosserat continuum is presented in the flame of average-field theory. The admissible boundary conditions required to prescribe on the representative volume element for the modeling are extracted and discussed to ensure the satisfaction of Hill-Mandel energy condition and the first-order average field theory.展开更多
Let B2,p:= {z ∈ C2: |z1|2+ |z2|p< 1}（0 < p < 1）. Then, B2,p（0 < p < 1） is a non-convex complex ellipsoid in C2 without smooth boundary. In this article, we establish a boundary Schwarz lemma at z0 ...Let B2,p:= {z ∈ C2: |z1|2+ |z2|p< 1}（0 < p < 1）. Then, B2,p（0 < p < 1） is a non-convex complex ellipsoid in C2 without smooth boundary. In this article, we establish a boundary Schwarz lemma at z0 ∈ ?B2,p for holomorphic self-mappings of the non-convex complex ellipsoid B2,p, where z0 is any smooth boundary point of B2,p.展开更多
The Q gene in common wheat encodes an APETALA2(AP2) transcription factor that causes the free threshing attribute. Wheat spikelets bearing several florets are subtended by a pair of soft glumes that allow free liberat...The Q gene in common wheat encodes an APETALA2(AP2) transcription factor that causes the free threshing attribute. Wheat spikelets bearing several florets are subtended by a pair of soft glumes that allow free liberation of seeds. In wild species, the glumes are tough and rigid,making threshing difficult. However, the nature of these "soft glumes", caused by the domestication allele Q is not clear. Here, we found that over expression of Q in common wheat leads to homeotic florets at glume positions. We provide phenotypic, microscopy, and marker genes evidence to demonstrate that the soft glumes of common wheat are in fact lemma-like organs, or so-called sterile-lemmas. By comparing the structures subtending spikelets in wheat and other crops such as rice and maize, we found that AP2 genes may play conserved functions in grasses by manipulating vestigial structures, such as floret-derived soft glumes in wheat and empty glumes in rice. Conversion of these seemingly vegetative organs to reproductive organs may be useful in yield improvement of crop species.展开更多
The strict bounded real lemma for linear system with finite discrete jumps was considered. Especially, the case where D matrices in the system are not assumed to be zero was dealt. Several versions of the bounded real...The strict bounded real lemma for linear system with finite discrete jumps was considered. Especially, the case where D matrices in the system are not assumed to be zero was dealt. Several versions of the bounded real lemma are presented in terms of solution to Riccati differential equations or inequalities with finite discrete jumps. Both the finite and infinite horizon cases are considered. These results generalize the existed bounded real lemma for linear systems.展开更多
In this paper we prove a Schwarz-Pick lemma for the modulus of holomorphic mappings from the polydisk into the unit ball. This result extends some related results.
Rice florets are subtended by two sterile lemmas,whose origin and biological functions have not been studied extensively.Here we demonstrate that two putative transcription factors,LAX PANICLE1(LAX1)and FRIZZY PANICLE...Rice florets are subtended by two sterile lemmas,whose origin and biological functions have not been studied extensively.Here we demonstrate that two putative transcription factors,LAX PANICLE1(LAX1)and FRIZZY PANICLE(FZP),synergistically control the development of sterile lemmas.Both LAX1 and FZP are previously known for their roles in panicle and floret development.Disruption of either LAX1 or FZP greatly reduces the number of floret development.We generated new lax1 mutants(lax1-c)using CRISPR/Cas9 gene editing technology.In addition to the expected lax panicle phenotypes,we noticed that a significant number of spikelets of lax1-c developed elongated sterile lemmas.Moreover,our characterization of lax1-RNAi plants also revealed sterile lemma phenotypes similar to lax1-c mutants.We isolated a weak allele of fzp(fzp-14)in a genetic screen for lax1–1 enhancers.The fzp-14 lax1–1 double mutants completely eliminated flower development.Interestingly,the isolated fzp-14 produced spikelets with elongated sterile lemmas.Furthermore,fzp-14 was haploid-insufficient in the lax1–1 background whereas fzp-14 heterozygous plants were indistinguishable from wild type plants.The lax1–1 fzp-14+/−also developed elongated sterile lemma as observed in lax1-c,lax1-RNAi,and fzp-14,suggesting that LAX1 and FZP synergistically control sterile lemma development.展开更多
This paper derives the bounded real lemmas corresponding to L∞norm and H∞norm(L-BR and H-BR) of fractional order systems. The lemmas reduce the original computations of norms into linear matrix inequality(LMI) probl...This paper derives the bounded real lemmas corresponding to L∞norm and H∞norm(L-BR and H-BR) of fractional order systems. The lemmas reduce the original computations of norms into linear matrix inequality(LMI) problems, which can be performed in a computationally efficient fashion. This convex relaxation is enlightened from the generalized Kalman-YakubovichPopov(KYP) lemma and brings no conservatism to the L-BR. Meanwhile, an H-BR is developed similarly but with some conservatism.However, it can test the system stability automatically in addition to the norm computation, which is of fundamental importance for system analysis. From this advantage, we further address the synthesis problem of H∞control for fractional order systems in the form of LMI. Three illustrative examples are given to show the effectiveness of our methods.展开更多
Jordan's lemma can be used for a wider range than the original one. The extended Jordan's lemma can be described as follows. Let f(z) be analytic in the upper half of the z plane (Imz≥0), with the exception o...Jordan's lemma can be used for a wider range than the original one. The extended Jordan's lemma can be described as follows. Let f(z) be analytic in the upper half of the z plane (Imz≥0), with the exception of a finite number of isolated singularities, and for P>o, if then where z=Rei and CR is the open semicircle in the upper half of the z plane.With the extended Jordan's lemma one can find that Laplace transform and Fourier transform are a pair of integral transforms which relate to each other.展开更多
Some preliminary results on strict bounded real lemma for time-varying continuous linear systems are proposed, where uncertainty in initial conditions, terminal cost and extreme of the cost function are dealt with exp...Some preliminary results on strict bounded real lemma for time-varying continuous linear systems are proposed, where uncertainty in initial conditions, terminal cost and extreme of the cost function are dealt with explicitly. Based on these results, a new recursive approach is proposed in the necessity proof of strict bounded real lemma for generalized linear system with finite discrete jumps.展开更多
A. Robinson＇s sequential lemma is extended to nets in general topological space, and obviously the case of nets in ^＊R is its corollary. As its application, the paper proves a property about topology of uniform conv...A. Robinson＇s sequential lemma is extended to nets in general topological space, and obviously the case of nets in ^＊R is its corollary. As its application, the paper proves a property about topology of uniform convergence.展开更多
Matrices and integral operators with off-diagonal decay appear in numerous areas of mathematics including numerical analysis and harmonic analysis, and they also play impor-tant roles in engineering science including ...Matrices and integral operators with off-diagonal decay appear in numerous areas of mathematics including numerical analysis and harmonic analysis, and they also play impor-tant roles in engineering science including signal processing and communication engineering. Wiener’s lemma states that the localization of matrices and integral operators are preserved un-der inversion. In this introductory note, we re-examine several approaches to Wiener’s lemma for matrices. We also review briefly some recent advances on localization preservation of operations including nonlinear inversion, matrix factorization and optimization.展开更多
In this article, we present a Schwarz lemma at the boundary for pluriharmonic mappings from the unit polydisk to the unit ball, which generalizes classical Schwarz lemma for bounded harmonic functions to higher dimens...In this article, we present a Schwarz lemma at the boundary for pluriharmonic mappings from the unit polydisk to the unit ball, which generalizes classical Schwarz lemma for bounded harmonic functions to higher dimensions. It is proved that if the pluriharmonic mapping f ∈ P（Dn, BN） is C1＋α at z0 ∈ ErDn with f（0） = 0 and f（z0） = ω0∈BN for any n,N ≥ 1, then there exist a nonnegative vector λf =（λ1,0,…,λr,0,…,0）T∈R2 nsatisfying λi≥1/（22 n-1） for 1 ≤ i ≤ r such that where z’0 and w’0 are real versions of z0 and w0, respectively.展开更多
To distinguish the taxonomy of Yihuang wild rice found in Yihuang County, Jiangxi Province, China in 1999, the botanical characteristics of Yihuang wild rice were retrieved by genus, and species; at the same time, the...To distinguish the taxonomy of Yihuang wild rice found in Yihuang County, Jiangxi Province, China in 1999, the botanical characteristics of Yihuang wild rice were retrieved by genus, and species; at the same time, the tubercle characters on lemma in Yihuang wild rice, O. rufipogon, 0. officinalis, 0. meyeriana, and O. granulata, were also observed by scanning electron microscopy. The retrieval result by genus showed that Yihuang wild rice is Leersia Soland. Ex Swartz., not Oryza L., Hygroryza Nees and Chikusichloa Koidz. The retrieval result by species showed that Yihuang wild rice is L. sayanuka Ohwi, not L. hexandra Swartz., L. japonica （Makino） Honda and L. oryzoides （L.） Swartz. The results of the SEM observation on tubercles on lemma showed that the tubercle characters on lemma of Yihuang wild rice belongs to ＂inlaid- silica-peak＂ type, which is different from the three types of wild rice existing in China, O. rufipogon, O. officinalis, O. Meyeriana, and also O. granulata, but is similar to L. tisseranti, thereby gives a proof of the above retrieval results. It could be concluded that Yihuang wild rice does not belong to Oryza L., but to Leersia sayanuka Ohwi of Leersia Soland. Ex Swartz.展开更多
The purpose of this paper is to verify the Smulyan lemma for the support function, and also the Gateaux differentiability of the support function is studied on its domain. Moreover, we provide a characterization of Fr...The purpose of this paper is to verify the Smulyan lemma for the support function, and also the Gateaux differentiability of the support function is studied on its domain. Moreover, we provide a characterization of Frechet differentiability of the support function on the extremal points.展开更多
A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in R^n＋1, and such that its mean curvature is constant, is a sphere. Here we study the p...A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in R^n＋1, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane Xn＋1 =constant in case M satisfies： for any two points （X^1, ^↑Xn＋1）, （X^1, Xn＋1） on M, with Xn＋1 〉^↑Xn＋1, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part I dealt with corresponding one dimensional problems.展开更多
This paper is to extend the Poincar’e Lemma for differential forms in a bounded, convex domain [1] in Rn to a more general domain that, we call, is deformable to every point in itself. Then we extend the homotopy ope...This paper is to extend the Poincar’e Lemma for differential forms in a bounded, convex domain [1] in Rn to a more general domain that, we call, is deformable to every point in itself. Then we extend the homotopy operator T in [1] to the domain defromed to every point of itself.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.12361026)Fundamental Research Funds for the Central Universities(Grant No.31920240069)Innovation Team Project of Northwest Minzu University.
文摘In this paper,we consider the Hopf lemma of the following mixed local and nonlocal weighted semilinear elliptic equations{-div(|x|^(-2α)■u)+(-△)_(α)^(s)u=0,x∈U,u(x^(^))=-u(x),x∈H,u(x)=0,x∈R^(N)\U,where H belong to R^(N)with 0∈H is an open and affine half space,U belong to H is an open and bounded set,s∈(0,1),α∈[0,N-2s/2),(-△)_(α)^(s)is weighted fractional Laplacian with a weighted function.
基金supported by grants from the Ministry of Sciences and Technology of China (No. 2008ZX08009-003 and 2009CB118500)the National Natural Sci-ence Foundation of China (No. 30621001 and 30900885)
文摘Although there is evident homology among reproductive organs when comparing Poaceae （grass） and eudicots, the identity of grass specific organs, such as lodicules, palea, lemma, and glumes has been the subject of a vast and largely inconclusive discussion. Here we provide some direct evidence to support the idea that the empty glumes of rice （Oryza sativa） are counterparts of lemmas. We show that the development of empty glumes is regulated by ELE （elongated empty glume）, which belongs to a plant specific novel gene family. Mutations at the ELE locus cause elongated empty glumes, which mimic the lemmas and have the epidermal morphology of lemmas with four or five vascular bundles. As a nuclear-localized gene, ELE is specifically expressed at the empty glumes of immature spikelets, and its ectopic expression causes many floral development defects, including lemma-like palea, extra palea-like structures, elongated lodicules, extra stamens and stigmas. Our result suggests that empty glumes are lemmas of the sterile florets located at the lateral side of the rice spikelet, and ELE acts as a regulator restraining its growth to maintain its small size in wild-type plants.
基金supported by National Natural Science Foundations of China(11011373,11201199,11271333)Zhejiang Provincial Natural Science Foundation of China(LY14A010008)
文摘In this note, we consider a holomorphic mapping f from the unit disk C in C to p-ball B^p = {z∈C^n;i=1∑n｜zi｜p〈1,1〈p〈＋∞. It is proved that for such f,｜ ｜ ｜f｜｜（z）｜≤1-｜｜f（z）｜｜^2/1-｜z｜^2,z∈D. The extremal problem is also discussed when p is an even number. This result extends some related results on Schwarz lemma.
基金supported by National Natural Science Foundation of China(No.60710002,60974044)Program for Changjiang Scholars and Innovative Research Team in University
文摘This note presents a set of new versions of Barbalat＇s lemma combining with positive （negative） definite functions.Based on these results,a set of new formulations of Lyapunov-like lemma are established.A simple example shows the usefulness of our results.
基金supported by the National Natural Science Foundation of China (90715011, 10672033 and 10590354) the National Key Basic Research and Development Program (2002CB412709) the Australia Research Council through the ARC International Fellowship Offered at University of Newcastle (LX0666274)
文摘On the basis of Hill＇s lemma for classical Cauchy continuum, a version of Hill＇s lemma for micro-macro homogenization modeling of heterogeneous Cosserat continuum is presented in the flame of average-field theory. The admissible boundary conditions required to prescribe on the representative volume element for the modeling are extracted and discussed to ensure the satisfaction of Hill-Mandel energy condition and the first-order average field theory.
基金The project supported in part by the National Natural Science Foundation of China(11671306)
文摘Let B2,p:= {z ∈ C2: |z1|2+ |z2|p< 1}（0 < p < 1）. Then, B2,p（0 < p < 1） is a non-convex complex ellipsoid in C2 without smooth boundary. In this article, we establish a boundary Schwarz lemma at z0 ∈ ?B2,p for holomorphic self-mappings of the non-convex complex ellipsoid B2,p, where z0 is any smooth boundary point of B2,p.
基金supported by the National Key Program for Transgenic Crop Cultivation (2016ZX09001-001)The CAAS Agricultural Science and Technology Innovation Program Cooperation and Innovation Mission (CAAS-XTCX2016)
文摘The Q gene in common wheat encodes an APETALA2(AP2) transcription factor that causes the free threshing attribute. Wheat spikelets bearing several florets are subtended by a pair of soft glumes that allow free liberation of seeds. In wild species, the glumes are tough and rigid,making threshing difficult. However, the nature of these "soft glumes", caused by the domestication allele Q is not clear. Here, we found that over expression of Q in common wheat leads to homeotic florets at glume positions. We provide phenotypic, microscopy, and marker genes evidence to demonstrate that the soft glumes of common wheat are in fact lemma-like organs, or so-called sterile-lemmas. By comparing the structures subtending spikelets in wheat and other crops such as rice and maize, we found that AP2 genes may play conserved functions in grasses by manipulating vestigial structures, such as floret-derived soft glumes in wheat and empty glumes in rice. Conversion of these seemingly vegetative organs to reproductive organs may be useful in yield improvement of crop species.
基金National Natural Science Foundation of China(No.60274058)
文摘The strict bounded real lemma for linear system with finite discrete jumps was considered. Especially, the case where D matrices in the system are not assumed to be zero was dealt. Several versions of the bounded real lemma are presented in terms of solution to Riccati differential equations or inequalities with finite discrete jumps. Both the finite and infinite horizon cases are considered. These results generalize the existed bounded real lemma for linear systems.
基金supported by the National Natural Science Foundation of China(11201199)the Scientific Research Foundation of Jinling Institute of Technology(Jit-b-201221)Qing Lan Project
文摘In this paper we prove a Schwarz-Pick lemma for the modulus of holomorphic mappings from the polydisk into the unit ball. This result extends some related results.
基金supported by the National Transgenic Research Program of China(2016ZX08010002)
文摘Rice florets are subtended by two sterile lemmas,whose origin and biological functions have not been studied extensively.Here we demonstrate that two putative transcription factors,LAX PANICLE1(LAX1)and FRIZZY PANICLE(FZP),synergistically control the development of sterile lemmas.Both LAX1 and FZP are previously known for their roles in panicle and floret development.Disruption of either LAX1 or FZP greatly reduces the number of floret development.We generated new lax1 mutants(lax1-c)using CRISPR/Cas9 gene editing technology.In addition to the expected lax panicle phenotypes,we noticed that a significant number of spikelets of lax1-c developed elongated sterile lemmas.Moreover,our characterization of lax1-RNAi plants also revealed sterile lemma phenotypes similar to lax1-c mutants.We isolated a weak allele of fzp(fzp-14)in a genetic screen for lax1–1 enhancers.The fzp-14 lax1–1 double mutants completely eliminated flower development.Interestingly,the isolated fzp-14 produced spikelets with elongated sterile lemmas.Furthermore,fzp-14 was haploid-insufficient in the lax1–1 background whereas fzp-14 heterozygous plants were indistinguishable from wild type plants.The lax1–1 fzp-14+/−also developed elongated sterile lemma as observed in lax1-c,lax1-RNAi,and fzp-14,suggesting that LAX1 and FZP synergistically control sterile lemma development.
基金supported by National Natural Science Foundation of China(Nos.61004017 and 60974103)
文摘This paper derives the bounded real lemmas corresponding to L∞norm and H∞norm(L-BR and H-BR) of fractional order systems. The lemmas reduce the original computations of norms into linear matrix inequality(LMI) problems, which can be performed in a computationally efficient fashion. This convex relaxation is enlightened from the generalized Kalman-YakubovichPopov(KYP) lemma and brings no conservatism to the L-BR. Meanwhile, an H-BR is developed similarly but with some conservatism.However, it can test the system stability automatically in addition to the norm computation, which is of fundamental importance for system analysis. From this advantage, we further address the synthesis problem of H∞control for fractional order systems in the form of LMI. Three illustrative examples are given to show the effectiveness of our methods.
文摘Jordan's lemma can be used for a wider range than the original one. The extended Jordan's lemma can be described as follows. Let f(z) be analytic in the upper half of the z plane (Imz≥0), with the exception of a finite number of isolated singularities, and for P>o, if then where z=Rei and CR is the open semicircle in the upper half of the z plane.With the extended Jordan's lemma one can find that Laplace transform and Fourier transform are a pair of integral transforms which relate to each other.
基金This work was supported by the National Natural Science Foundation of China (No. 60274058).
文摘Some preliminary results on strict bounded real lemma for time-varying continuous linear systems are proposed, where uncertainty in initial conditions, terminal cost and extreme of the cost function are dealt with explicitly. Based on these results, a new recursive approach is proposed in the necessity proof of strict bounded real lemma for generalized linear system with finite discrete jumps.
基金the Basic Research Foundation of Xi'an University Architecture Technology(JC0620)the Youth Science and Technology Foundation of Xi'an University of Architecture and Technology(QN0736)
文摘A. Robinson＇s sequential lemma is extended to nets in general topological space, and obviously the case of nets in ^＊R is its corollary. As its application, the paper proves a property about topology of uniform convergence.
基金supported by Basic Science Research Program through the National Research Foundation of Korea(NRF)funded by the Ministry of Education,Science and Technology(2013R1A1A2005402)National Science Foundation(DMS-1109063)
文摘Matrices and integral operators with off-diagonal decay appear in numerous areas of mathematics including numerical analysis and harmonic analysis, and they also play impor-tant roles in engineering science including signal processing and communication engineering. Wiener’s lemma states that the localization of matrices and integral operators are preserved un-der inversion. In this introductory note, we re-examine several approaches to Wiener’s lemma for matrices. We also review briefly some recent advances on localization preservation of operations including nonlinear inversion, matrix factorization and optimization.
基金Supported by the Natural and Science Foundation of China(61379001,61771001)
文摘In this article, we present a Schwarz lemma at the boundary for pluriharmonic mappings from the unit polydisk to the unit ball, which generalizes classical Schwarz lemma for bounded harmonic functions to higher dimensions. It is proved that if the pluriharmonic mapping f ∈ P（Dn, BN） is C1＋α at z0 ∈ ErDn with f（0） = 0 and f（z0） = ω0∈BN for any n,N ≥ 1, then there exist a nonnegative vector λf =（λ1,0,…,λr,0,…,0）T∈R2 nsatisfying λi≥1/（22 n-1） for 1 ≤ i ≤ r such that where z’0 and w’0 are real versions of z0 and w0, respectively.
基金supported by the National Natural Science Foundation of China(30560073)the Key Technologies R&D Program of Jiangxi Province,China(200210100210).
文摘To distinguish the taxonomy of Yihuang wild rice found in Yihuang County, Jiangxi Province, China in 1999, the botanical characteristics of Yihuang wild rice were retrieved by genus, and species; at the same time, the tubercle characters on lemma in Yihuang wild rice, O. rufipogon, 0. officinalis, 0. meyeriana, and O. granulata, were also observed by scanning electron microscopy. The retrieval result by genus showed that Yihuang wild rice is Leersia Soland. Ex Swartz., not Oryza L., Hygroryza Nees and Chikusichloa Koidz. The retrieval result by species showed that Yihuang wild rice is L. sayanuka Ohwi, not L. hexandra Swartz., L. japonica （Makino） Honda and L. oryzoides （L.） Swartz. The results of the SEM observation on tubercles on lemma showed that the tubercle characters on lemma of Yihuang wild rice belongs to ＂inlaid- silica-peak＂ type, which is different from the three types of wild rice existing in China, O. rufipogon, O. officinalis, O. Meyeriana, and also O. granulata, but is similar to L. tisseranti, thereby gives a proof of the above retrieval results. It could be concluded that Yihuang wild rice does not belong to Oryza L., but to Leersia sayanuka Ohwi of Leersia Soland. Ex Swartz.
文摘The purpose of this paper is to verify the Smulyan lemma for the support function, and also the Gateaux differentiability of the support function is studied on its domain. Moreover, we provide a characterization of Frechet differentiability of the support function on the extremal points.
文摘A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in R^n＋1, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane Xn＋1 =constant in case M satisfies： for any two points （X^1, ^↑Xn＋1）, （X^1, Xn＋1） on M, with Xn＋1 〉^↑Xn＋1, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part I dealt with corresponding one dimensional problems.
文摘This paper is to extend the Poincar’e Lemma for differential forms in a bounded, convex domain [1] in Rn to a more general domain that, we call, is deformable to every point in itself. Then we extend the homotopy operator T in [1] to the domain defromed to every point of itself.
