The Veselov's discrete Neumann system is derived through nonlinearization of a discrete spectral problem.Based on the commutative relation between the Lax matrix and the Darboux matrix with finite genus potentials...The Veselov's discrete Neumann system is derived through nonlinearization of a discrete spectral problem.Based on the commutative relation between the Lax matrix and the Darboux matrix with finite genus potentials,a special solution is calculated with the help of the Baker-Akhiezer-Kriechever function.展开更多
Under an algebraic constraint between the potentials and the eigenfunctions, the Jaulent-Miodek eigenvalue problem is nonlinearized to be a completely integrable C. Neumann system (M2N-2, dp AND dq\M2N-2, H) in Liouvi...Under an algebraic constraint between the potentials and the eigenfunctions, the Jaulent-Miodek eigenvalue problem is nonlinearized to be a completely integrable C. Neumann system (M2N-2, dp AND dq\M2N-2, H) in Liouville sense with the Hamiltonian function H = 1/2[q, q][LAMBDA2q, q] - 1/2[LAMBDAq, q]2 + 1/2[p, p][q, q] - 1/2[q, p]2 where M2N-2 = TS(N-1) = {(q, p)\F = 1/2 ([q, q]-1 = 0, G = [q, p] = 0}展开更多
This paper is concerned with the existence of solutions to a class of p（x）-Kirchhoff-type systems under Neumann boundary condition. By Ekeland Variational Principle and the theory of the variable exponent Sobolev sp...This paper is concerned with the existence of solutions to a class of p（x）-Kirchhoff-type systems under Neumann boundary condition. By Ekeland Variational Principle and the theory of the variable exponent Sobolev spaces, we establish conditions ensuring the existence of solutions for the problem. Since the Poincare＇s inequality does not hold in the space W1,p（x）（Ω）, we shall prove the Poincare-Wirtinger＇s inequality in a subspace of W1,p（x）（Ω）.展开更多
We prove the existence of solutions of the system for nonlocal Neumann problems with Minkowski-curvature operator(r^(N−1) u′/√(p 1−u′^(2)))′=r^(N−1)f(r,u),r 2(0,1),u′(0)=0,u′(1)=∫_(0)^(-) u′(s)dg(s),where k,N...We prove the existence of solutions of the system for nonlocal Neumann problems with Minkowski-curvature operator(r^(N−1) u′/√(p 1−u′^(2)))′=r^(N−1)f(r,u),r 2(0,1),u′(0)=0,u′(1)=∫_(0)^(-) u′(s)dg(s),where k,N≥1 are integers,f:[0,1]×R^(k)→R^(k) is continuous and g:[0,1]→R^(k) is a function of bounded variation.Our proof is based on the perturbation method.展开更多
Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bound...Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L<sup>∞</sup>(X, μ) on X. We confirm that the commutative von Neumann algebras M⊂B(H), with H=L<sup>2</sup>(X, μ), are unitary equivariant to the maximal ideals of the commutative algebra C(X). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C(X) following its action on M(X) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B(H) .展开更多
Developing efficient neural network(NN)computing systems is crucial in the era of artificial intelligence(AI).Traditional von Neumann architectures have both the issues of"memory wall"and"power wall&quo...Developing efficient neural network(NN)computing systems is crucial in the era of artificial intelligence(AI).Traditional von Neumann architectures have both the issues of"memory wall"and"power wall",limiting the data transfer between memory and processing units[1,2].Compute-in-memory(CIM)technologies,particularly analogue CIM with memristor crossbars,are promising because of their high energy efficiency,computational parallelism,and integration density for NN computations[3].In practical applications,analogue CIM excels in tasks like speech recognition and image classification,revealing its unique advantages.For instance,it efficiently processes vast amounts of audio data in speech recognition,achieving high accuracy with minimal power consumption.In image classification,the high parallelism of analogue CIM significantly speeds up feature extraction and reduces processing time.With the boosting development of AI applications,the demands for computational accuracy and task complexity are rising continually.However,analogue CIM systems are limited in handling complex regression tasks with needs of precise floating-point(FP)calculations.They are primarily suited for the classification tasks with low data precision and a limited dynamic range[4].展开更多
本文研究了在von Neumann代数框架下,投影运算后等价的保持情形,特别是在不同类型的von Neumann代数中投影等价的刻画及其性质。设E、F、N为von Neumann代数ℳ中的三个有限投影,且E~F。当NE=NF=0时,那么有N+E~N+F,N∧E~N∧F,N∨E~N∨F成...本文研究了在von Neumann代数框架下,投影运算后等价的保持情形,特别是在不同类型的von Neumann代数中投影等价的刻画及其性质。设E、F、N为von Neumann代数ℳ中的三个有限投影,且E~F。当NE=NF=0时,那么有N+E~N+F,N∧E~N∧F,N∨E~N∨F成立。当NE,NF时,那么有N∧E~N∧F,N∨E~N∨F成立。当EN,FN时,那么有N−E~N−F,N∧E~N∧F,N∨E~N∨F成立。本文进一步推至N为von Neumann代数ℳ中的无限投影,并且考虑了von Neumann代数ℳ中四个投影运算的情况。This paper studies the conditions of equivalents after projection operations under the framework of von Neumann algebra, especially the characterization and properties of projection equivalents in different types of von Neumann algebras. Let E,F,Nbe three finite projections in von Neumann algebra ℳ, and E~F. If NE=NF=0, then it follows that N+E~N+F, N∧E~N∧F, N∨E~N∨Fholds. If NE, NF, then it follows that N∧E~N∧F, N∨E~N∨Fholds. If EN, FN, then it follows that N−E~N−F, N∧E~N∧F, N∨E~N∨Fholds. This paper further extends these results to infinite projections Nin von Neumann algebras ℳand considers the case of four projection operations within von Neumann algebras ℳ.展开更多
设ℳ和N是无I1或I2型中心直和项的von Neumann代数,其单位元分别为I和I′。本文证明非线性双射Φ:ℳ→N混合Lie可乘,即Φ([ [ A,B ],C ]∗)=[ [ Φ(A),Φ(B) ],Φ(C) ]∗,∀A,B,C∈ℳ,当且仅当存在线性*-同构和共轭线性*-同构的直和Ψ:ℳ→N使...设ℳ和N是无I1或I2型中心直和项的von Neumann代数,其单位元分别为I和I′。本文证明非线性双射Φ:ℳ→N混合Lie可乘,即Φ([ [ A,B ],C ]∗)=[ [ Φ(A),Φ(B) ],Φ(C) ]∗,∀A,B,C∈ℳ,当且仅当存在线性*-同构和共轭线性*-同构的直和Ψ:ℳ→N使得Φ(A)=Φ(I)Ψ(A),∀A∈ℳ,其中Φ(I)∈N是可逆中心元且Φ(I)2=I′。该结论将因子von Neumann代数上的非线性混合Lie可乘双射的结果推广到无I1或I2型中心直和项的von Neumann代数。Let ℳand Nbe von Neumann algebras with no central summands of type I1or I2, Iand I′be the identities of them. This paper proves that a bijective map Φ:ℳ→Nis mixed Lie multiplicative, that is, Φ([ [ A,B ],C ]∗)=[ [ Φ(A),Φ(B) ],Φ(C) ]∗,∀A,B,C∈ℳif and only if Φ(A)=Φ(I)Ψ(A)for all A∈ℳ, where Ψ:ℳ→Nis a direct sum of a linear *-isomorphism and a conjugate linear *-isomorphism, Φ(I)is a central element in Nwith Φ(I)2=I′. The results about mixed Lie multiplicative maps on factor von Neumann algebras are generalized to von Neumann algebras with no central summands of type I1or I2.展开更多
本文在有限von Neumann代数的情形下应用广义奇异值的方法证明了一类迹函数的若干性质。特别地,我们将Hansen的主要结果推广至有限von Neumann代数的情形。In this paper, via the method of generalized singular values, we prove som...本文在有限von Neumann代数的情形下应用广义奇异值的方法证明了一类迹函数的若干性质。特别地,我们将Hansen的主要结果推广至有限von Neumann代数的情形。In this paper, via the method of generalized singular values, we prove some properties of a class of trace functions defined over finite von Neumann algebras. In particular, we extend the main results of Hansen to the context of finite von Neumann algebras.展开更多
Let M and N be two factor von Neumann algebras that their dimensions are large than 1,η≠-1 a non zero complex number and Φa(not necessary linear)bijection between two factor von Neumann algebras satisfying Φ(I)=I....Let M and N be two factor von Neumann algebras that their dimensions are large than 1,η≠-1 a non zero complex number and Φa(not necessary linear)bijection between two factor von Neumann algebras satisfying Φ(I)=I.For all A,B∈M,define by A■B=AB+BA the Jordan product of A and B,A·_(η)B=AB+ηBA^(*)the Jordan η-*-product of A and B,respectively.Let Φ and Φ^(-1)preserve the mixed Jordan triple η-*-products.It is proved that Φ is a linear *-isomorphism if η is not real and Φ is the sum of a linear *-isomorphism and a conjugate linear *-isomorphism if η is real.展开更多
基金Supported by the National Natural Science Foundation of China under Grant No. 10971200
文摘The Veselov's discrete Neumann system is derived through nonlinearization of a discrete spectral problem.Based on the commutative relation between the Lax matrix and the Darboux matrix with finite genus potentials,a special solution is calculated with the help of the Baker-Akhiezer-Kriechever function.
文摘Under an algebraic constraint between the potentials and the eigenfunctions, the Jaulent-Miodek eigenvalue problem is nonlinearized to be a completely integrable C. Neumann system (M2N-2, dp AND dq\M2N-2, H) in Liouville sense with the Hamiltonian function H = 1/2[q, q][LAMBDA2q, q] - 1/2[LAMBDAq, q]2 + 1/2[p, p][q, q] - 1/2[q, p]2 where M2N-2 = TS(N-1) = {(q, p)\F = 1/2 ([q, q]-1 = 0, G = [q, p] = 0}
基金Supported by the National Natural Science Foundation of China(Grant No.11261052)
文摘This paper is concerned with the existence of solutions to a class of p（x）-Kirchhoff-type systems under Neumann boundary condition. By Ekeland Variational Principle and the theory of the variable exponent Sobolev spaces, we establish conditions ensuring the existence of solutions for the problem. Since the Poincare＇s inequality does not hold in the space W1,p（x）（Ω）, we shall prove the Poincare-Wirtinger＇s inequality in a subspace of W1,p（x）（Ω）.
基金Supported by the National Natural Science Foundation of China(Grant Nos.11901464,12361040)the National Science Foundation of Gansu Province(Grant Nos.20JR10RA100,21JR1RA230)the Department of Education University Innovation Fund of Gansu Province(Grant Nos.2022A-218,2021A-006)。
文摘We prove the existence of solutions of the system for nonlocal Neumann problems with Minkowski-curvature operator(r^(N−1) u′/√(p 1−u′^(2)))′=r^(N−1)f(r,u),r 2(0,1),u′(0)=0,u′(1)=∫_(0)^(-) u′(s)dg(s),where k,N≥1 are integers,f:[0,1]×R^(k)→R^(k) is continuous and g:[0,1]→R^(k) is a function of bounded variation.Our proof is based on the perturbation method.
文摘Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L<sup>∞</sup>(X, μ) on X. We confirm that the commutative von Neumann algebras M⊂B(H), with H=L<sup>2</sup>(X, μ), are unitary equivariant to the maximal ideals of the commutative algebra C(X). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C(X) following its action on M(X) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B(H) .
文摘Developing efficient neural network(NN)computing systems is crucial in the era of artificial intelligence(AI).Traditional von Neumann architectures have both the issues of"memory wall"and"power wall",limiting the data transfer between memory and processing units[1,2].Compute-in-memory(CIM)technologies,particularly analogue CIM with memristor crossbars,are promising because of their high energy efficiency,computational parallelism,and integration density for NN computations[3].In practical applications,analogue CIM excels in tasks like speech recognition and image classification,revealing its unique advantages.For instance,it efficiently processes vast amounts of audio data in speech recognition,achieving high accuracy with minimal power consumption.In image classification,the high parallelism of analogue CIM significantly speeds up feature extraction and reduces processing time.With the boosting development of AI applications,the demands for computational accuracy and task complexity are rising continually.However,analogue CIM systems are limited in handling complex regression tasks with needs of precise floating-point(FP)calculations.They are primarily suited for the classification tasks with low data precision and a limited dynamic range[4].
文摘本文研究了在von Neumann代数框架下,投影运算后等价的保持情形,特别是在不同类型的von Neumann代数中投影等价的刻画及其性质。设E、F、N为von Neumann代数ℳ中的三个有限投影,且E~F。当NE=NF=0时,那么有N+E~N+F,N∧E~N∧F,N∨E~N∨F成立。当NE,NF时,那么有N∧E~N∧F,N∨E~N∨F成立。当EN,FN时,那么有N−E~N−F,N∧E~N∧F,N∨E~N∨F成立。本文进一步推至N为von Neumann代数ℳ中的无限投影,并且考虑了von Neumann代数ℳ中四个投影运算的情况。This paper studies the conditions of equivalents after projection operations under the framework of von Neumann algebra, especially the characterization and properties of projection equivalents in different types of von Neumann algebras. Let E,F,Nbe three finite projections in von Neumann algebra ℳ, and E~F. If NE=NF=0, then it follows that N+E~N+F, N∧E~N∧F, N∨E~N∨Fholds. If NE, NF, then it follows that N∧E~N∧F, N∨E~N∨Fholds. If EN, FN, then it follows that N−E~N−F, N∧E~N∧F, N∨E~N∨Fholds. This paper further extends these results to infinite projections Nin von Neumann algebras ℳand considers the case of four projection operations within von Neumann algebras ℳ.
文摘设ℳ和N是无I1或I2型中心直和项的von Neumann代数,其单位元分别为I和I′。本文证明非线性双射Φ:ℳ→N混合Lie可乘,即Φ([ [ A,B ],C ]∗)=[ [ Φ(A),Φ(B) ],Φ(C) ]∗,∀A,B,C∈ℳ,当且仅当存在线性*-同构和共轭线性*-同构的直和Ψ:ℳ→N使得Φ(A)=Φ(I)Ψ(A),∀A∈ℳ,其中Φ(I)∈N是可逆中心元且Φ(I)2=I′。该结论将因子von Neumann代数上的非线性混合Lie可乘双射的结果推广到无I1或I2型中心直和项的von Neumann代数。Let ℳand Nbe von Neumann algebras with no central summands of type I1or I2, Iand I′be the identities of them. This paper proves that a bijective map Φ:ℳ→Nis mixed Lie multiplicative, that is, Φ([ [ A,B ],C ]∗)=[ [ Φ(A),Φ(B) ],Φ(C) ]∗,∀A,B,C∈ℳif and only if Φ(A)=Φ(I)Ψ(A)for all A∈ℳ, where Ψ:ℳ→Nis a direct sum of a linear *-isomorphism and a conjugate linear *-isomorphism, Φ(I)is a central element in Nwith Φ(I)2=I′. The results about mixed Lie multiplicative maps on factor von Neumann algebras are generalized to von Neumann algebras with no central summands of type I1or I2.
文摘本文在有限von Neumann代数的情形下应用广义奇异值的方法证明了一类迹函数的若干性质。特别地,我们将Hansen的主要结果推广至有限von Neumann代数的情形。In this paper, via the method of generalized singular values, we prove some properties of a class of trace functions defined over finite von Neumann algebras. In particular, we extend the main results of Hansen to the context of finite von Neumann algebras.
文摘Let M and N be two factor von Neumann algebras that their dimensions are large than 1,η≠-1 a non zero complex number and Φa(not necessary linear)bijection between two factor von Neumann algebras satisfying Φ(I)=I.For all A,B∈M,define by A■B=AB+BA the Jordan product of A and B,A·_(η)B=AB+ηBA^(*)the Jordan η-*-product of A and B,respectively.Let Φ and Φ^(-1)preserve the mixed Jordan triple η-*-products.It is proved that Φ is a linear *-isomorphism if η is not real and Φ is the sum of a linear *-isomorphism and a conjugate linear *-isomorphism if η is real.
