A generalization of Banach's lemma and its applications to perturbations of bounded linear operators

Let X be a Banach space and let P:X→X be a bounded linear operator.Using an algebraic inequality on the spectrum of P,we give a new sufficient condition that guarantees the existence of(I-P)^(-1) as a bounded linear operator on X,and a bound on its spectral radius is also obtained.This generalizes the classic Banach lemma.We apply the result to the perturbation analysis of general bounded linear operators on X with commutative perturbations.

A Finite-Time Convergent Analysis of Continuous Action Iterated Dilemma

Dear Editor,In this letter, a finite-time convergent analysis of continuous action iterated dilemma(CAID) is proposed. In traditional evolutionary game theory, the strategy of the player is binary(cooperation or defection), which limits the number of strategies a player can choose from.

THE BOUNDARY SCHWARZ LEMMA AND THE RIGIDITY THEOREM ON REINHARDT DOMAINS B_(p)^(n) OF C^(n)

By introducing the Carathéodory metric,we establish the Schwarz lemma at the boundary for holomorphic self-mappings on the unit p-ball B_(p)^(n) of C^(n).Furthermore,the boundary rigidity theorem for holomorphic self-mappings defined on B_(n)^(p) is obtained.These results cover the boundary Schwarz lemma and rigidity result for holomorphic self-mappings on the unit ball for p=2,and the unit polydisk for p=∞,respectively.

Rice AGL1 determines grain size and sterile lemma identity

The grass spikelet is a unique inflorescence structure that determines grain size.Although many genetic factors have been well characterized for grain size and glume development,the underlying molecular mechanisms in rice are far from established.Here,we isolated rice gene,AGL1 that controlled grain size and determines the fate of the sterile lemma.Loss of function of AGL1 produced larger grains and reduced the size of the sterile lemma.Larger grains in the agl1 mutant were caused by a larger number of cells that were longer and wider than in the wild type.The sterile lemma in the mutant spikelet was converted to a rudimentary glume-like organ.Our findings showed that the AGL1(also named LAX1)protein positively regulated G1 expression,and negatively regulated NSG1 expression,thereby affecting the fate of the sterile lemma.Taken together,our results revealed that AGL1 played a key role in negative regulation of grain size by controlling cell proliferation and expansion,and supported the opinion that rudimentary glume and sterile lemma in rice are homologous organs.

National Security Dilemma in the Digital Era and the Building of a Community with a Shared Future in Cyberspace

A review of the history of human development shows that every major technological revolution brings new challenges to national security.With the advent of the information technology revolution represented by the Internet,cybersecurity has become an issue of paramount importance,and the digital intelligence revolution with digital technology innovations such as big data and cloud computing has further expanded the connotation and extension of cybersecurity.

A Proof of Brouwer’s Fixed Point Theorem Using Sperner’s Lemma

This article offers a simple but rigorous proof of Brouwer’s fixed point theorem using Sperner’s Lemma.The general method I have used so far in the proof is mainly to convert the n-dimensional shapes to the corresponding case under the Sperner’s Labeling and apply the Sperner’s Lemma to solve the question.

New versions of Barbalat's lemma with applications

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.

A NOTE ON SCHWARZ LEMMA FOR THE MODULUS OF HOLOMORPHIC MAPPINGS ON D

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.

A version of Hill's lemma for Cosserat continuum

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.

THE SCHWARZ LEMMA AT THE BOUNDARY OF THE NON-CONVEX COMPLEX ELLIPSOIDS

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 ￡2?, where zq is any smooth boundary point of B2,p.

The soft glumes of common wheat are sterile-lemmas as determined by the domestication gene Q

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.

Bounded Real Lemma for Generalized Linear System with Finite Discrete Jumps

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.

A SCHWARZ-PICK LEMMA FOR THE MODULUS OF HOLOMORPHIC MAPPINGS FROM THE POLYDISK INTO THE UNIT BALL

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.

Synergistic roles of LAX1 and FZP in the development of rice sterile lemma

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.

On necessity proof of strict bounded real lemma for generalized linear systems with finite discrete jumps

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.

A SCHWARZ LEMMA FOR HARMONIC FUNCTIONS IN THE REAL UNIT BALL

We establish a precise Schwarz lemma for real-valued and bounded harmonic functions in the real unit ball of dimension n. This extends Chen's Schwarz-Pick lemma for real-valued and bounded planar harmonic mapping.

Wiener's lemma:localization and various approaches

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.

A BOUNDARY SCHWARZ LEMMA FOR PLURIHARMONIC MAPPINGS FROM THE UNIT POLYDISK TO THE UNIT BALL

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.

The Taxonomy and the SEM Observation on Tubercles on Lemma of Yihuang Wild Rice

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.

Smulyan Lemma and Differentiability of the Support Function

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.