Asymptotic exactness of an a posteriori error estimator based on the equilibrated residual method

In this paper we analyze an a posteriori error estimator based on the equilibrated residual method. We prove that this estimator is asymptotically exact in the energy norm for regular solutions and meshes. Numerical examples are included to illustrate the theoretical results.     

On Sharp Aperture-Weighted Estimates for Square Functions

Let \(S_{\alpha ,\psi }(f)\) be the square function defined by means of the cone in \({\mathbb R}^{n+1}_{+}\) of aperture \(\alpha \) , and a standard kernel \(\psi \) . Let \([w]_{A_p}\) denote the \(A_p\) characteristic of the weight \(w\) . We show that for any \(1<p<\infty \) and \(\alpha \ge 1\) , $$\begin{aligned} \Vert S_{\alpha ,\psi }\Vert _{L^p(w)}\lesssim \alpha ^n[w]_{A_p}^{\max \left( \frac{1}{2},\frac{1}{p-1}\right) }. \end{aligned}$$ For each fixed \(\alpha \) the dependence on \([w]_{A_p}\) is sharp. Also, on all class \(A_p\) the result is sharp in \(\alpha \) . Previously this estimate was proved in the case \(\alpha =1\) using the intrinsic square function. However, that approach does not allow to get the above estimate with sharp dependence on \(\alpha \) . Hence we give a different proof suitable for all \(\alpha \ge 1\) and avoiding the notion of the intrinsic square function.     

Analog of Formula of Total Probability for Quantum Observables Represented by Positive Operator Valued Measures

We represent Born’s rule as an analog of the formula of total probability (FTP): the classical formula is perturbed by an additive interference term. In this note we consider practically the most general case: generalized quantum observables given by positive operator valued measures and measurement feedback on states described by atomic instruments. This representation of Born’s rule clarifies the probabilistic structure of quantum mechanics (QM). The probabilistic counterpart of QM can be treated as the probability update machinery based on the special generalization of classical FTP. This is the essence of the Växjö interpretation of QM: statistical realist contextual and local interpretation. We analyze the origin of the additional interference term in quantum FTP by considering the contextual structure of the two slit experiment which was emphasized by R. Feynman.     

On Nonlocality of Quantum Objects

We present an option of the experiment with a correlated pair of particles in the entangled state, which provides the effect of a change in the polarization for entangled photons, and demonstrate the reality of all different superposition states and the corresponding vector of the quantum system state; also we analyze possible consequences of this fact. We propose a quantum realism paradigm within the relational paradigm instead of the local realism concept disproved by the experiments on verifying the Bell inequalities. We analyze the results of experimental research of the Leggett inequality violation with respect to the verification of the adequacy of different kinds of nonlocal hidden variable theories and suggest a new way of their evaluation based on the study of the photon cross-correlation suppression after a beam splitter and preparation of quantum squeezed states. We show that the interpretation based on the nonlocal hidden variable theory is inconsistent.     

Semi-Lagrangian semi-implicit time-splitting scheme for a regional model of the atmosphere

Semi-Lagrangian semi-implicit (SLSI) method is currently one of the most efficient approaches for numerical solution of the atmosphere dynamics equations. In this research we apply splitting techniques in the context of a two-time-level SLSI scheme in order to simplify the treatment of the slow physical modes and optimize the solution of the elliptic equations related to implicit part of the scheme. The performed numerical experiments show the accuracy and computational efficiency of the scheme.     

Modelling of coupled cross-flow and in-line vortex-induced vibrations of flexible cylindrical structures. Part I: model description and validation

Nonlinear Dynamics - This paper is first of the two papers dealing with the nonlinear modelling and investigation of coupled cross-flow and in-line vortex-induced vibrations (VIVs) of flexible...     

Design of pincer complexes based on (methylsulfanyl)acetic/propionic acid amides with ancillary S‐ and N‐donors as potential catalysts and cytotoxic agents

Pincer complexes featuring readily tunable tridentate ligand frameworks comprise one of the most actively studied classes of organometallic and metal–organic compounds and find extensive use in catalysis, organic synthesis, materials science, and other fields of chemistry and allied disciplines. Currently growing attention is devoted to non‐classical ligand scaffolds, such as functionalized carboxamides, which offer multiple options for directed structural modifications. In this study, the reactions of (methylsulfanyl)acetyl and propanoyl chlorides with 2‐(aminomethyl)pyridine, 2‐(2‐aminoethyl)pyridine, 8‐aminoquinoline and 2‐(diphenylthiophosphoryl)aniline afford a series of new pincer‐type ligands based on functionalized carboxamides. The ligands obtained readily undergo direct cyclopalladation under the action of PdCl₂(NCPh)₂ in dichloromethane at room temperature, resulting in Pd(II) pincer complexes with N,N,S‐ and S,N,S‐donor sets. Importantly, some of the cyclopalladated derivatives can also be produced efficiently under solvent‐free conditions according to the approach recently developed by our group. The complexes obtained have been tested for cytotoxicity against several human cancer cell lines and catalytic activity in the model Suzuki reaction. The results have been compared to those for the related Pd(II) pincer complexes to define the main structure–activity relationships and to outline the most promising structures for further investigations.