von Neumann’s trace inequality for Hilbert–Schmidt operators

von Neumann’s inequality in matrix theory refers to the fact that the Frobenius scalar product of two matrices is less than or equal to the scalar product of the respective singular values. Moreover, equality can only happen if the two matrices share a joint set of singular vectors, and this latter part is hard to find in the literature. We extend these facts to the separable Hilbert space setting, and provide a self-contained proof of the “latter part”.     

线性方程组通解并行数值方法

1预备知识 线性方程组消息传递MIMD算法是20世纪90年代至今的活跃课题，但尚无此问题 的通用有效算法发表.用!.}.}表示矩阵列分块划分，记AX=b为[AI司.定理1、2是文 {l]、[s]成果综述和推广. 定理i[‘]设有线性方程组!e}己l，e〔尺”“m，d〔R”“‘，rank(e)=r.当且仅当rank(!     

Multi-cracks problem for piezoelectric materials strip subjected to dynamic loading

This paper analyses and models the dynamic interaction among permeable multi-cracks in a piezoelectric strip under anti-plane shear waves by the Schmidt method. The Fourier transform is applied and then two pairs of triple integral equations can be solved using the Schmidt method. The results show that the stress and the electric displacement intensity factors of cracks depend on not only the crack length and the piezoelectric coefficient, but also the thickness of the piezoelectric strip, the distance between multi-cracks and the frequency of incident wave.     

Synthesis of vinyl-substituted oxiranes and their transformation to dihydrofurans

The reaction of dimethylsulfonium methylide with the carbonyl function of 2-arylidene-cyclohexanones obtained from 2-methylcyclohexanone affords the corresponding spirooxiranes. The application of this reaction to 2,2-dimethyl-6-(5-methylfurfurylidene)cyclohexanone and 2,2-dimethyl-6-(5-chlorofurfurylidene)cyclohexanone leads to substituted 4,4-dimethyl-1-furyl-1,-3,4,5,6,7-hexahydroisobenzofurans.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 8, pp. 1439–1443, August, 1994.     

A straightforward preparation of amino–polystyrene resin from Merrifield resin

Azidomethyl–polystyrene, obtained by nucleophilic substitution of chloromethyl–polystyrene, undergoes a Schmidt rearrangement when treated with trifluoromethanesulfonic acid, affording amino–polystyrene. To assess its loading and reactivity the resin is used as a support for the preparation of triazene-linked amine.     

Tensor products of coherent configurations

A Cartesian decomposition of a coherent configuration ✗ is defined as a special set of its parabolics that form a Cartesian decomposition of the underlying set. It turns out that every tensor decomposition of ✗ comes from a certain Cartesian decomposition. It is proved that if the coherent configuration ✗ is thick, then there is a unique maximal Cartesian decomposition of ✗; i.e., there is exactly one internal tensor decomposition of ✗ into indecomposable components. In particular, this implies an analog of the Krull–Schmidt theorem for the thick coherent configurations. A polynomial-time algorithm for finding the maximal Cartesian decomposition of a thick coherent configuration is constructed.     

TheSemmler-Wolff aromatization andSchmidt reaction applied to some pyrido[2′,3′:3,4]pyrazolo[5,1-c][1,2,4]benzotriazines

Pyrido[2,3:3,4]pyrazolo[5,1-c][1,2,4]benzotriazin-4(1H)ones were transformed via their oximes in aSemmler-Wolff aromatization process in the tetracyclic heteroaromatic amines4 or bySchmidt reaction into a mixture of the same amine4 and a ring enlarged lactam3. Syntheses of some halo pyrazolo[3,4-b]pyridines and a photochemical transformation of 3-azidopyrazolo[3,4-b]pyridine are also described.

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Über dieSemmler-Wolff- undSchmidt-Reaktion einiger Pyrido[2,3:3,4]pyrazolo[5,1-c][1,2,4]benzotriazine

Zusammenfassung Pyrido[23:3,4]pyrazolo[5,1-c][1,2,4]benzotriazin-4(1H)one werden über Oxime in einerSemmler-Wolff-Reaktion in die tetracyklischen aromatischen Amine4 umgewandelt. In einerSchmidt-Reaktion wurden dieselben Ketone in ein Gemisch aus Amin4 und Lactam3 übergeführt. Synthesen von halogensubstituierten Pyrazolo[3,4-b]pyridinen und photochemische Umwandlung von 3-Azidopyrazolo[3,4-b]pyridin werden beschrieben.

     

Steady-state and Hopf bifurcations in the Langford ODE and PDE systems

This paper is concerned with the Langford ODE and PDE systems. For the Langford ODE system, the existence of steady-state solutions is firstly obtained by Lyapunov–Schmidt method, and the stability and bifurcation direction of periodic solutions are established. Then for the Langford PDE system, the steady-state bifurcations from simple and double eigenvalues are intensively studied. The techniques of space decomposition and implicit function theorem are adopted to deal with the case of double eigenvalue. Finally, by the center manifold theory and the normal form method, the direction of Hopf bifurcation and the stability of spatially homogeneous and inhomogeneous periodic solutions for the PDE system are investigated.     

Two-electron entanglement in elliptically deformed quantum dots

Entropic entanglement measures of a two-dimensional system of two Coulombically interacting particles confined in an anisotropic harmonic potential are discussed in dependence on the anisotropy and the interaction strength. The harmonic approximation appears exact in the strong interaction limit, allowing determination of the asymptotic expression for the linear entropy. Entanglement properties are dramatically influenced by the anisotropy of the confining potential in the strong-correlation regime.