数学知识的表征及其学习策略

根据数学知识的符号性与严谨性等特点提出数学知识所独有的六种表征形式 :命题表征、符号表征、算子表征、图式表征、心智映象表征及产生式表征 ,并指明了数学知识的学习策略 .

应用图示表征方法 提高解决问题能力

《数学课程标准(2011年版)》中提出"四能",最终以"解决问题"为教学目标.本文通过调查,获得当下影响学生解决问题能力有哪些因素.在此基础上提出应用图示表征方法的教学策略实践,总结出图示表征的三种方法,即几何图表征、符号表征及算子表征等直观教学策略,对培养学生解决问题能力有重要实践意义.

Relationship Between Wave Function and Corresponding Wigner Function Studied in Entangled State Representation

By using the explicit form of the entangled Wigner operator and the entangled state representation we derive the relationship between wave function and corresponding Wigner function for bipartite entangled systems. The technique of integration within an ordered product （IWOP） of operators is employed in our discussions.

Generating Generalized Bessel Equations by Virtue of Bose Operator Algebra and Entangled State Representations

With the help of Bose operator identities and entangled state representation and based on our previous work.[Phys. Lett. A 325 （2004） 188] we derive some new generalized Bessel equations which also have Bessel function as their solution. It means that for these intricate higher-order differential equations, we can get Bessel function solutions without using the expatiatory power-series expansion method.

Continuous Multipartite Entangled Representation and Its Wigner Operator and Marginal Distribution

By extending the EPR bipartite entanglement to multipartite case, we briefly introduce a continuous multipartite entangled representation and its canonical conjugate state in the multi-mode Fock space, analyze their Schmidt decompositions and give their entangling operators. Furthermore, based on the above analysis we also find the n-mode Wigner operator. In doing so we may identify the physical meaning of the marginal distribution of the Wigner function.

Quantum computational advantage via 60-qubit 24-cycle random circuit sampling

To ensure a long-term quantum computational advantage,the quantum hardware should be upgraded to withstand the competition of continuously improved classical algorithms and hardwares.Here,we demonstrate a superconducting quantum computing systems Zuchongzhi 2.1,which has 66 qubits in a two-dimensional array in a tunable coupler architecture.The readout fidelity of Zuchongzhi 2.1 is considerably improved to an average of 97.74%.The more powerful quantum processor enables us to achieve larger-scale random quantum circuit sampling,with a system scale of up to 60 qubits and 24 cycles,and fidelity of FXEB=(3·66±0·345)×10^(-4).The achieved sampling task is about 6 orders of magnitude more difficult than that of Sycamore[Nature 574,505(2019)]in the classic simulation,and 3 orders of magnitude more difficult than the sampling task on Zuchongzhi 2.0[arXiv:2106.14734(2021)].The time consumption of classically simulating random circuit sampling experiment using state-of-the-art classical algorithm and supercomputer is extended to tens of thousands of years(about 4·8×104years),while Zuchongzhi 2.1 only takes about 4.2 h,thereby significantly enhancing the quantum computational advantage.

The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert C~*-module l_2(A)

We characterize A-linear symmetric and contraction module operator semigroup{Tt}t∈R+L(l2(A)),where A is a finite-dimensional C-algebra,and L(l2(A))is the C-algebra of all adjointable module maps on l2(A).Next,we introduce the concept of operator-valued quadratic forms,and give a one to one correspondence between the set of non-positive definite self-adjoint regular module operators on l2(A)and the set of non-negative densely defined A-valued quadratic forms.In the end,we obtain that a real and strongly continuous symmetric semigroup{Tt}t∈R+L(l2(A))being Markovian if and only if the associated closed densely defined A-valued quadratic form is a Dirichlet form.

Linear operators and positive semidefiniteness of symmetric tensor spaces

We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.

Characterization of Lie multiplicative isomorphisms between nest algebras

Let N and M be nests on Banach spaces X and Y over the real or complex field F, respectively, with the property that if M ∈ M such that M_ =M, then M is complemented in Y. Let AlgN and AlgM be the associated nest algebras. Assume that Ф ： AlgN → AlgM is a bijective map. It is proved that, if dim X = ∞ and if there is a nontrivial element in N which is complemented in X, then Ф is Lie multiplicative （i.e. Ф（[A, B]） = [Ф（A）, Ф（B）] for all A, B ∈ AlgN） if and only if Ф has the form Ф（A） = TAT^-1 ＋ τ（A） for all A ∈ AlgAN or Ф（A） = -TA^＊T^-1 ＋ τ（A） for all A ∈ AlgN, where T is an invertible linear or conjugate linear operator and τ ： AlgN →FI is a map with τ（[A, B]） = 0 for all A, B ∈ AlgN. The Lie multiplicative maps are also characterized for the case dim X 〈 ∞.