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”.     

A posteriori error analysis of a quadratic finite volume method for nonlinear elliptic problems

In this article, we construct and analyze a residual-based a posteriori error estimator for a quadratic finite volume method (FVM) for solving nonlinear elliptic partial differential equations with homogeneous Dirichlet boundary conditions. We shall prove that the a posteriori error estimator yields the global upper and local lower bounds for the norm error of the FVM. So that the a posteriori error estimator is equivalent to the true error in a certain sense. Numerical experiments are performed to illustrate the theoretical results.     

The coupled-cluster formalism – a mathematical perspective

ABSTRACT

The Coupled-Cluster (CC) theory is one of the most successful high precision methods used to solve the stationary Schrödinger equation. In this article, we address the mathematical foundation of this theory with focus on the advances made in the past decade. Rather than solely relying on spectral gap assumptions (non-degeneracy of the ground state), we highlight the importance of coercivity assumptions – Gårding type inequalities – for the local uniqueness of the CC solution. Based on local strong monotonicity, different sufficient conditions for a local unique solution are suggested. One of the criteria assumes the relative smallness of the total cluster amplitudes (after possibly removing the single amplitudes) compared to the Gårding constants. In the extended CC theory the Lagrange multipliers are wave function parameters and, by means of the bivariational principle, we here derive a connection between the exact cluster amplitudes and the Lagrange multipliers. This relation might prove useful when determining the quality of a CC solution. Furthermore, the use of an Aubin–Nitsche duality type method in different CC approaches is discussed and contrasted with the bivariational principle.     

Local linear convergence analysis of Primal–Dual splitting methods

In this paper, we study the local linear convergence properties of a versatile class of Primal–Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these methods. More precisely, in our framework, we first show that (i) the sequences generated by Primal–Dual splitting methods identify a pair of primal and dual smooth manifolds in a finite number of iterations, and then (ii) enter a local linear convergence regime, which is characterized based on the structure of the underlying active smooth manifolds. We also show how our results for Primal–Dual splitting can be specialized to cover existing ones on Forward–Backward splitting and Douglas–Rachford splitting/ADMM (alternating direction methods of multipliers). Moreover, based on these obtained local convergence analysis result, several practical acceleration techniques are discussed. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from fields including signal/image processing, inverse problems and machine learning. The demonstration not only verifies the local linear convergence behaviour of Primal–Dual splitting methods, but also the insights on how to accelerate them in practice.     

Molecular structure,vibrational spectroscopic and chemical reactivity of nematogenic p-n-alkylbenzoic acids: A comparative computational analysis

Molecular structure and vibrational spectroscopic studies of higher homologous series nematogenic p-n-alkylbenzoic acids (nBAC) that have 6 (6BAC) and 7 (7BAC) carbon atoms in the alkyl chain have been investigated using the Density Functional Becke3-Lee-Yang-Parr (B3LYP) level with the basis set 6-31++G (d.p) and Hartree Fock (HF) with the same basis set. The observed vibrational spectra has been resolved and assigned in detail for comparision with both the molecules. These results indicate that DFT and HF values are slightly different at both the levels. A comparision of chemical reactivity such as HOMO (E_H), LUMO (E_L) energies, energy gap (E_g), ionization energy (I), electron affinity (A), electro negativity (χ), chemical hardness (η), electronic chemical potential (μ), electrophilicity index (ω), and softness (S) has been made. It has been observed that the decrement has occurred in the energy band gap value of isolated molecule with increment in alkyl chain length. This provides valuable information regarding enhancing the stability of liquid crystal materials by maintaining the conductivity.     

Direct Transformation of Arenols Based on C—O Activation

In the view of substrate availability, atomic efficiency and cost, directly using arenols as coupling partners in cross‐coupling, would be one of the most attractive goals. Up to date, many efforts have been made to activate the C—O bond of phenols with different strategies, for example, through in‐situ formed intermediates, through a catalytic reductive dearomatization‐condensation‐rearomatization sequence or catalytic deoxygenation. In this review, we summarized recent advances in cross‐couplings of arenols as the electrophiles via C—O activation.     

化学法改性煤沥青脱除苯并芘的研究

在管式反应器中采用苯甲酸、聚乙二醇、固体古马隆树脂（S）、液体古马隆树脂（L）为添加剂来降低煤沥青中有害物质苯并芘的含量，以期使得煤沥青可绿色化应用。采用紫外-可见分光光度计分析煤沥青中苯并芘含量。考察了反应温度、反应时间、添加剂添加量、催化剂等工艺条件对添加剂脱除煤沥青中苯并芘的影响。结果表明，不同工艺条件能降低煤沥青中苯并芘的含量。在优化条件下，不同添加剂对苯并芘脱除率由高到低依次为：液体古马隆树脂、聚乙二醇、苯甲酸和固体古马隆树脂。分析其反应机理，这与催化剂的酸性相关，发生亲电取代反应。结果表明，液体古马隆树脂（L）在催化剂存在下对煤沥青中苯并芘脱除率可达73.0%，显示了良好的应用前景。