The formation possibility of (Hf0.2Zr0.2Ta0.2Nb0.2Ti0.2)C high‐entropy ceramic (HHC‐1) was first analyzed by the first‐principles calculations, and then, it was successfully fabricated by ...hot‐pressing sintering technique at 2073 K under a pressure of 30 MPa. The first‐principles calculation results showed that the mixing enthalpy and mixing entropy of HHC‐1 were −0.869 ± 0.290 kJ/mol and 0.805R, respectively. The experimental results showed that the as‐prepared HHC‐1 not only had an interesting single rock‐salt crystal structure of metal carbides but also possessed high compositional uniformity from nanoscale to microscale. By taking advantage of these unique features, it exhibited extremely high nanohardness of 40.6 ± 0.6 GPa and elastic modulus in the range from 514 ± 10 to 522 ± 10 GPa and relatively high electrical resistivity of 91 ± 1.3 μΩ·cm, which could be due to the presence of solid solution effects.
Density functional theory calculations are robust tools to explore the mechanical properties of pristine structures at their ground state but become exceedingly expensive for large systems at finite ...temperatures. Classical molecular dynamics (CMD) simulations offer the possibility to study larger systems at elevated temperatures, but they require accurate interatomic potentials. Herein the authors propose the concept of first‐principles multiscale modeling of mechanical properties, where ab initio level of accuracy is hierarchically bridged to explore the mechanical/failure response of macroscopic systems. It is demonstrated that machine‐learning interatomic potentials (MLIPs) fitted to ab initio datasets play a pivotal role in achieving this goal. To practically illustrate this novel possibility, the mechanical/failure response of graphene/borophene coplanar heterostructures is examined. It is shown that MLIPs conveniently outperform popular CMD models for graphene and borophene and they can evaluate the mechanical properties of pristine and heterostructure phases at room temperature. Based on the information provided by the MLIP‐based CMD, continuum models of heterostructures using the finite element method can be constructed. The study highlights that MLIPs were the missing block for conducting first‐principles multiscale modeling, and their employment empowers a straightforward route to bridge ab initio level accuracy and flexibility to explore the mechanical/failure response of nanostructures at continuum scale.
A robust concept of first‐principles multiscale modeling of mechanical properties based on machine‐learning interatomic potentials conveniently trainable over short ab initio datasets is proposed. It is shown that mechanical/failure responses of complex nanostructures at continuum scale and high temperatures can be explored with the precision of sophisticated first‐principles calculations, affordable computational cost, and without the need for empirical data.
The aim of this article is to formulate some new uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions. Firstly, we derive an analogue of Pitt's inequality for ...the continuous shearlet transforms, then we formulate Beckner's uncertainty principle via two approaches: one based on a sharp estimate from Pitt's inequality and the other from the classical Beckner inequality in the Fourier domain. In continuation, a version of the logarithmic Sobolev inequality having a dual relation with Beckner's inequality is obtained. In sequel, the Nazarov's uncertainty principle is also derived for the continuous shearlet transforms in arbitrary space dimensions. The article concludes with the formulation of certain new local type uncertainty principles for the continuous shearlet transforms in arbitrary space dimensions.
Despite the accumulation of substantial cognitive science research relevant to education, there remains confusion and controversy in the application of research to educational practice. In support of ...a more systematic approach, we describe the Knowledge‐Learning‐Instruction (KLI) framework. KLI promotes the emergence of instructional principles of high potential for generality, while explicitly identifying constraints of and opportunities for detailed analysis of the knowledge students may acquire in courses. Drawing on research across domains of science, math, and language learning, we illustrate the analyses of knowledge, learning, and instructional events that the KLI framework affords. We present a set of three coordinated taxonomies of knowledge, learning, and instruction. For example, we identify three broad classes of learning events (LEs): (a) memory and fluency processes, (b) induction and refinement processes, and (c) understanding and sense‐making processes, and we show how these can lead to different knowledge changes and constraints on optimal instructional choices.
In this paper, we study a few versions of the uncertainty principle for the short‐time Fourier transform on the lattice Zn×Tn$\mathbb {Z}^n \times \mathbb {T}^n$. In particular, we establish the ...uncertainty principle for orthonormal sequences, Donoho–Stark's uncertainty principle, Benedicks‐type uncertainty principle, Heisenberg‐type uncertainty principle, and local uncertainty inequality for this transform on Zn×Tn$\mathbb {Z}^n \times \mathbb {T}^n$. Also, we obtain the Heisenberg‐type uncertainty inequality using the k$k$‐entropy of the short‐time Fourier transform on Zn×Tn$\mathbb {Z}^n \times \mathbb {T}^n$.
The quadratic‐phase Fourier transform (QPFT) is a recent addition to the class of Fourier transforms and embodies a variety of signal processing tools including the Fourier, fractional Fourier, ...linear canonical, and special affine Fourier transform. In this article, we formulate several classes of uncertainty principles for the QPFT. Firstly, we formulate the Heisenberg's uncertainty principle governing the simultaneous localization of a signal and the corresponding QPFT. Secondly, we obtain some logarithmic and local uncertainty inequalities such as Beckner and Sobolev inequalities for the QPFT. Thirdly, we study several concentration‐based uncertainty principles, including Nazarov's, Amrein–Berthier–Benedicks's, and Donoho–Stark's uncertainty principles. Finally, we conclude the study with the formulation of Hardy's and Beurling's uncertainty principles for the QPFT.
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