In this paper,Schwarz-type lemmas for different classes of quaternion functions are obtained.Firstly,some properties of symmetric points are given.Secondly,the Schwarz-type lemma and the Schwarz-Pick-type theorem for ...In this paper,Schwarz-type lemmas for different classes of quaternion functions are obtained.Firstly,some properties of symmetric points are given.Secondly,the Schwarz-type lemma and the Schwarz-Pick-type theorem for quaternion regular functions are obtained.Finally,the Schwarz-type lemma for quaternion k-regular functions is derived.展开更多
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
基金supported by the Innovation Foundation of the School of Mathematical Sciences in Hebei Normal University in 2025(ycxzzbs202503)the NSF of Hebei Province(A2023205006,A2022208007,A2023205045,A2024208005)+2 种基金the Hebei Research Center of the Basic Discipline Pure Mathematics,the Key Development Foundation of Hebei Normal University(L2024ZD08)the NSFC(12431005)the Funding Project of Central Guidance for Local Scientific and Technological Development(246Z7608G).
文摘In this paper,Schwarz-type lemmas for different classes of quaternion functions are obtained.Firstly,some properties of symmetric points are given.Secondly,the Schwarz-type lemma and the Schwarz-Pick-type theorem for quaternion regular functions are obtained.Finally,the Schwarz-type lemma for quaternion k-regular functions is derived.
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.
