The construction of normalized biholomorphic convex mappings in the Reinhardt domain $D_p = \{ (z_1 ,z_2 , \cdots ,z_n ) \in \mathbb{C}^n :\left| {z_1 } \right|^{p_1 } + \left| {z_2 } \right|^{p_2 } + \cdots + \left| ...The construction of normalized biholomorphic convex mappings in the Reinhardt domain $D_p = \{ (z_1 ,z_2 , \cdots ,z_n ) \in \mathbb{C}^n :\left| {z_1 } \right|^{p_1 } + \left| {z_2 } \right|^{p_2 } + \cdots + \left| {z_n } \right|^{p_n } < 1\} $ , p j > 2, j = 1,2,?, n) of ? n is discussed. The authors set up a Decomposition Theorem for such mappings. As a special case, it is proved that, for each such mapping f, the first k-terms of the homogeneous expansion of its j-th component f j , j = 1, 2, ?, n, depends only on z j , where k is the number that satisfies k < min {p 1, p 2,?, p n ≤ k + 1. When p1,p2, ... ,pn → ∞ , this derives the Decomposition Theorem of normalized biholomorphic convex mappings in the polydisc which was gotten by T.J. Suffridge in 1970.展开更多
We establish a new type of the classical boundary Schwarz lemma for holomorphic self-mappings of the unit polydisk Dnin Cn. By using the Carath′eodory metric and Kobayashi metric of Dn, we obtain some properties of t...We establish a new type of the classical boundary Schwarz lemma for holomorphic self-mappings of the unit polydisk Dnin Cn. By using the Carath′eodory metric and Kobayashi metric of Dn, we obtain some properties of the complex Jacobian matrix Jf(p) at a boundary point p of Dnfor a holomorphic self-mapping f of Dn. Our results extend the classical Schwarz lemma at the boundary to high dimensions.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
In this paper we study the function , for z∈C. We derive a functional equation that relates G(z) and G(1−z) for all z∈C, and we prove: 1) that G and the Riemann zeta function ζ have exactly the same zeros in the cr...In this paper we study the function , for z∈C. We derive a functional equation that relates G(z) and G(1−z) for all z∈C, and we prove: 1) that G and the Riemann zeta function ζ have exactly the same zeros in the critical region D:= {z∈C:ℜz∈(0,1)};2) the Riemann hypothesis, i.e., that all of the zeros of G in D are located on the critical line := {z∈D:ℜz =1/2};and that 3) all the zeros of the Riemann zeta function located on the critical line are simple.展开更多
By using Schwarz alternating method, this paper presents asimplified alternating algorithm for the problems of two holes having arbitrary shapes and arrangements in an isotropic homogeneous linear elastic infinite reg...By using Schwarz alternating method, this paper presents asimplified alternating algorithm for the problems of two holes having arbitrary shapes and arrangements in an isotropic homogeneous linear elastic infinite region, and obtains stress and displacement fields for random times of iteration. After precision analysis it is found that the results for twenty times of iteration are of very high precision, and those with higher precision can be acquired if the iteration solving is further conducted. The comparison of the results from FEM further proves the reliability of the simplified alternating algorithm presented by this paper.展开更多
Recently,a Schwarz crystal structure with curved grain boundaries(GBs)constrained by twin-boundary(TB)networks was discovered in nanocrystalline Cu through experiments and atomistic simulations.Nanocrystalline Cu with...Recently,a Schwarz crystal structure with curved grain boundaries(GBs)constrained by twin-boundary(TB)networks was discovered in nanocrystalline Cu through experiments and atomistic simulations.Nanocrystalline Cu with nanosized Schwarz crystals exhibited high strength and excellent thermal stability.However,the grainsize effect and associated deformation mechanisms of Schwarz nanocrystals remain unknown.Here,we performed large-scale atomistic simulations to investigate the deformation behaviors and grain-size effect of nanocrystalline Cu with Schwarz crystals.Our simulations showed that similar to regular nanocrystals,Schwarz nanocrystals exhibit a strengthening-softening transition with decreasing grain size.The critical grain size in Schwarz nanocrystals is smaller than that in regular nanocrystals,leading to a maximum strength higher than that of regular nanocrystals.Our simulations revealed that the softening in Schwarz nanocrystals mainly originates from TB migration(or detwinning)and annihilation of GBs,rather than GB-mediated processes(including GB migration,sliding and diffusion)dominating the softening in regular nanocrystals.Quantitative analyses of simulation data further showed that compared with those in regular nanocrystals,the GB-mediated processes in Schwarz nanocrystals are suppressed,which is related to the low volume fraction of amorphous-like GBs and constraints of TB networks.The smaller critical grain size arises from the suppression of GB-mediated processes.展开更多
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 ...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.展开更多
More than 20 years ago,an old German man embarked on a quest to explore the suburbs of Shanghai and the villages of jiangsu Province in search of antique Chinese furmiture.He was not a mere antique collector or aficio...More than 20 years ago,an old German man embarked on a quest to explore the suburbs of Shanghai and the villages of jiangsu Province in search of antique Chinese furmiture.He was not a mere antique collector or aficionado of ancient furmiture,but a German scientist motivated by his profound love of Chinese culture.His name was Uli Schwarz(Fig.1),a distinguished German scientist who had dedicated the majority of his life to promoting cross-cultural scientific exchanges,an effort that left an indelible mark on the landscape of research collaborations between China and Germany.展开更多
In this note, we obtained an Young type inequality about the traces for semi-positivedefinite matrices, which easily gave the Holder type and Minkowski type inequalities on the same objects. We also got a PolyaSzego t...In this note, we obtained an Young type inequality about the traces for semi-positivedefinite matrices, which easily gave the Holder type and Minkowski type inequalities on the same objects. We also got a PolyaSzego type inequality as an answer to the converse problem of Bellman conjecture.展开更多
In this paper we study some nonoverlapping domain decomposition methods for solving a class of elliptic problems arising from composite materials and flows in porous media which contain many spatial scales. Our precon...In this paper we study some nonoverlapping domain decomposition methods for solving a class of elliptic problems arising from composite materials and flows in porous media which contain many spatial scales. Our preconditioner differs from traditional domain decomposition preconditioners by using a coarse solver which is adaptive to small scale heterogeneous features. While the convergence rate of traditional domain decomposition algorithms using coarse solvers based on linear or polynomial interpolations may deteriorate in the presence of rapid small scale oscillations or high aspect ratios, our preconditioner is applicable to multiple-scale problems without restrictive assumptions and seems to have a convergence rate nearly independent of the aspect ratio within the substructures. A rigorous convergence analysis based on the Schwarz framework is carried out, and we demonstrate the efficiency and robustness of the proposed preconditioner through numerical experiments which include problems with multiple-scale coefficients, as well problems with continuous scales.展开更多
New better estimates, which are given in terms of elementary functions, for the function r →（2/π）（1 -r2）κ（r）κ′（r） ＋ logr appearing in Hubner＇s sharp upper bound for the Hersch-Pfluger distortion functio...New better estimates, which are given in terms of elementary functions, for the function r →（2/π）（1 -r2）κ（r）κ′（r） ＋ logr appearing in Hubner＇s sharp upper bound for the Hersch-Pfluger distortion function are obtained. With these estimates, some known bounds for the Hersch-Pfluger distortion function in quasiconformal theory are improved, thus. improving the explicit quasiconformal Schwarz lemma and some known estimates for the solutions to the Ramanujan modular equations.展开更多
基金This work was supported by 973 Project, the National Natural Science Foundation of China (Grant No. 19871081) the Natural Science Foundation of Guangdong Province and Anhui Province.
文摘The construction of normalized biholomorphic convex mappings in the Reinhardt domain $D_p = \{ (z_1 ,z_2 , \cdots ,z_n ) \in \mathbb{C}^n :\left| {z_1 } \right|^{p_1 } + \left| {z_2 } \right|^{p_2 } + \cdots + \left| {z_n } \right|^{p_n } < 1\} $ , p j > 2, j = 1,2,?, n) of ? n is discussed. The authors set up a Decomposition Theorem for such mappings. As a special case, it is proved that, for each such mapping f, the first k-terms of the homogeneous expansion of its j-th component f j , j = 1, 2, ?, n, depends only on z j , where k is the number that satisfies k < min {p 1, p 2,?, p n ≤ k + 1. When p1,p2, ... ,pn → ∞ , this derives the Decomposition Theorem of normalized biholomorphic convex mappings in the polydisc which was gotten by T.J. Suffridge in 1970.
基金supported by National Natural Science Foundation of China(Grant Nos.11101139,11271124 and 11301136)Natural Science Foundation of Zhejiang Province(Grant No.LY14A010017)Natural Science Foundation of Hebei Province(Grant No.A2014205069)
文摘We establish a new type of the classical boundary Schwarz lemma for holomorphic self-mappings of the unit polydisk Dnin Cn. By using the Carath′eodory metric and Kobayashi metric of Dn, we obtain some properties of the complex Jacobian matrix Jf(p) at a boundary point p of Dnfor a holomorphic self-mapping f of Dn. Our results extend the classical Schwarz lemma at the boundary to high dimensions.
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.
文摘In this paper we study the function , for z∈C. We derive a functional equation that relates G(z) and G(1−z) for all z∈C, and we prove: 1) that G and the Riemann zeta function ζ have exactly the same zeros in the critical region D:= {z∈C:ℜz∈(0,1)};2) the Riemann hypothesis, i.e., that all of the zeros of G in D are located on the critical line := {z∈D:ℜz =1/2};and that 3) all the zeros of the Riemann zeta function located on the critical line are simple.
基金the National Natural Science Foundation of China (Grant No. 49772166).
文摘By using Schwarz alternating method, this paper presents asimplified alternating algorithm for the problems of two holes having arbitrary shapes and arrangements in an isotropic homogeneous linear elastic infinite region, and obtains stress and displacement fields for random times of iteration. After precision analysis it is found that the results for twenty times of iteration are of very high precision, and those with higher precision can be acquired if the iteration solving is further conducted. The comparison of the results from FEM further proves the reliability of the simplified alternating algorithm presented by this paper.
基金the financial support from National Natural Science Foundation of China (Grants Nos.12325203,91963117,and 11921002)。
文摘Recently,a Schwarz crystal structure with curved grain boundaries(GBs)constrained by twin-boundary(TB)networks was discovered in nanocrystalline Cu through experiments and atomistic simulations.Nanocrystalline Cu with nanosized Schwarz crystals exhibited high strength and excellent thermal stability.However,the grainsize effect and associated deformation mechanisms of Schwarz nanocrystals remain unknown.Here,we performed large-scale atomistic simulations to investigate the deformation behaviors and grain-size effect of nanocrystalline Cu with Schwarz crystals.Our simulations showed that similar to regular nanocrystals,Schwarz nanocrystals exhibit a strengthening-softening transition with decreasing grain size.The critical grain size in Schwarz nanocrystals is smaller than that in regular nanocrystals,leading to a maximum strength higher than that of regular nanocrystals.Our simulations revealed that the softening in Schwarz nanocrystals mainly originates from TB migration(or detwinning)and annihilation of GBs,rather than GB-mediated processes(including GB migration,sliding and diffusion)dominating the softening in regular nanocrystals.Quantitative analyses of simulation data further showed that compared with those in regular nanocrystals,the GB-mediated processes in Schwarz nanocrystals are suppressed,which is related to the low volume fraction of amorphous-like GBs and constraints of TB networks.The smaller critical grain size arises from the suppression of GB-mediated processes.
基金supported by the National Natural Science Foundation of China(12071161,11971165)supported by the National Natural Science Foundation of China(11971042)the Natural Science Foundation of Zhejiang Province(Z24A010005)。
文摘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.
文摘More than 20 years ago,an old German man embarked on a quest to explore the suburbs of Shanghai and the villages of jiangsu Province in search of antique Chinese furmiture.He was not a mere antique collector or aficionado of ancient furmiture,but a German scientist motivated by his profound love of Chinese culture.His name was Uli Schwarz(Fig.1),a distinguished German scientist who had dedicated the majority of his life to promoting cross-cultural scientific exchanges,an effort that left an indelible mark on the landscape of research collaborations between China and Germany.
文摘In this note, we obtained an Young type inequality about the traces for semi-positivedefinite matrices, which easily gave the Holder type and Minkowski type inequalities on the same objects. We also got a PolyaSzego type inequality as an answer to the converse problem of Bellman conjecture.
基金Supported by STATOIL under the VISTA programSupported in part by a grant from National Science Foundation under the contract DMS-0073916by a grant from Army Research Office under the contract DAAD19-99-1-0141.
文摘In this paper we study some nonoverlapping domain decomposition methods for solving a class of elliptic problems arising from composite materials and flows in porous media which contain many spatial scales. Our preconditioner differs from traditional domain decomposition preconditioners by using a coarse solver which is adaptive to small scale heterogeneous features. While the convergence rate of traditional domain decomposition algorithms using coarse solvers based on linear or polynomial interpolations may deteriorate in the presence of rapid small scale oscillations or high aspect ratios, our preconditioner is applicable to multiple-scale problems without restrictive assumptions and seems to have a convergence rate nearly independent of the aspect ratio within the substructures. A rigorous convergence analysis based on the Schwarz framework is carried out, and we demonstrate the efficiency and robustness of the proposed preconditioner through numerical experiments which include problems with multiple-scale coefficients, as well problems with continuous scales.
基金Supported by National 973 Project of China(2006CB708304)National Natural Science Foundation of China(10771195)
文摘New better estimates, which are given in terms of elementary functions, for the function r →（2/π）（1 -r2）κ（r）κ′（r） ＋ logr appearing in Hubner＇s sharp upper bound for the Hersch-Pfluger distortion function are obtained. With these estimates, some known bounds for the Hersch-Pfluger distortion function in quasiconformal theory are improved, thus. improving the explicit quasiconformal Schwarz lemma and some known estimates for the solutions to the Ramanujan modular equations.
