Components Of Tensor Research Articles

Dimensionality reduction beyond neural subspaces with slice tensor component analysis.

Recent work has argued that large-scale neural recordings are often well described by patterns of coactivation across neurons. Yet the view that neural variability is constrained to a fixed, low-dimensional subspace may overlook higher-dimensional structure, including stereotyped neural sequences or slowly evolving latent spaces. Here we argue that task-relevant variability in neural data can also cofluctuate over trials or time, defining distinct 'covariability classes' that may co-occur within the same dataset. To demix these covariability classes, we develop sliceTCA (slice tensor component analysis), a new unsupervised dimensionality reduction method for neural data tensors. In three example datasets, including motor cortical activity during a classic reaching task in primates and recent multiregion recordings in mice, we show that sliceTCA can capture more task-relevant structure in neural data using fewer components than traditional methods. Overall, our theoretical framework extends the classic view of low-dimensional population activity by incorporating additional classes of latent variables capturing higher-dimensional structure.

Iterative removal of sources to model the turbulent electromotive force

Abstract We describe a novel method to compute the components of dynamo tensors from direct magnetohydrodynamic (MHD) simulations. Our method relies upon an extension and generalisation of the standard Högbom CLEAN algorithm widely used in radio astronomy to systematically remove the impact of the strongest beams onto the corresponding image. This generalisation, called the Iterative Removal of Sources (IROS) method, has been adopted here to model the turbulent electromotive force (EMF) in terms of the mean magnetic fields and currents. Analogous to the CLEAN algorithm, IROS treats the time series of the mean magnetic field and current as beams that convolve with the dynamo coefficients which are treated as (clean) images to produce the EMF time series ( the dirty image). We apply this method to MHD simulations of galactic dynamos, to which we have previously employed other methods of computing dynamo coefficients such as the test-field method, the regression method, as well as local and non-local versions of the singular value decomposition (SVD) method. We show that our new method reliably recovers the dynamo coefficients from the MHD simulations. It also allows priors on the dynamo coefficients to be incorporated easily during the inversion, unlike in earlier methods. Moreover, using synthetic data, we demonstrate that it may serve as a viable post-processing tool in determining the dynamo coefficients, even when the power of additive noise to the EMF is twice as much the actual EMF.

Surface tension of aqueous electrolyte solutions. A thermomechanical approach.

We determine the surface tension of aqueous electrolyte solutions in contact with non-polar dielectric media using a thermomechanical approach, which involves deriving the stress tensor from the thermodynamic potential of an inhomogeneous fluid. To obtain the surface tension, we calculate both the normal and tangential pressures using the components of the stress tensor, recently derived by us [Y. A. Budkov and P. E. Brandyshev, J. Chem. Phys. 159, 174103 (2023)] within the framework of Wang's variational field theory. Using this approach, we derive an analytical expression for the surface tension in the linear approximation. At low ionic concentrations, this expression represents the classical Onsager-Samaras limiting law. By utilizing only one fitting parameter, which is related to the affinity of anions to the dielectric boundary, we successfully approximated experimental data on the surface tension of several aqueous electrolyte solutions. This approximation applies to both the solution-air and solution-dodecane interfaces, covering a wide range of electrolyte concentrations.

Physical properties and maximum allowable mass-radius relation of complexity-free compact stellar objects within modified $f(R,T)$ gravity formalism

Abstract This paper focuses on investigating the physical properties and predicted radii of compact stars generated by Tolman--IV complexity-free model within the background of modified gravity theory, especially $f(\mathcal{R},T)$-gravity theory, under complexity formalism for a spherically symmetric spacetime proposed by L. Herrera [Phys Rev D 97: 044010, 2018]. By solving the resulting set of differential equations, the explicit forms of the energy-momentum (EM) tensor components, including the density, radial pressure and tangential pressure, are obtained. The influence of the parameter $\chi$ on various physical properties of the star has been thoroughly investigated. The model undergoes a series of rigorous tests to determine its physical relevance. The findings indicate that the model exhibits regularity, stability and features a surface with vanishing pressure. The boundary of this surface is determined by carefully selecting the parameter space. The complexity method employed in $f(\mathcal{R},T)$ gravity offers an interesting approach for developing astrophysical models that are consistent with observable events as demonstrated by recent experiments. In this regard, we present a study that uses observational data from the GW190814 event, detected by the LIGO and Virgo observatories, to investigate the validity of the Tolman-IV model in $f(R, T)$ gravity. The analysis includes comparing the model's predictions with the observed characteristics of the compact object involved in the merger. In addition, data from two-millisecond pulsars, PSR J1614-2230 and PSR J0952-0607, are incorporated to further constrain theoretical theories. However, we present a diagram illustrating the relationship between the total mass and radius of the compact object candidates for different values of a parameter $\chi$.

Anisotropy ansatz for the axisymmetric Jeans equations

Abstract The Jeans equations do not form a closed system, and to solve them a parametrization relating the velocity moments is often adopted. For axisymmetric models, a phenomenological choice (the “b-ansatz”) is widely used for the relation between the vertical ($\sigma _z^2$) and radial ($\sigma _R^2$) components of the velocity dispersion tensor, thus breaking their identity present in two-integral systems. However, the way in which the ansatz affects the resulting kinematical fields can be quite complicated, so that the analysis of these fields is usually performed only after numerically computing them. We present here a general procedure to study the properties of the ansatz-dependent fields $\overline{v_{\varphi }^2}$, $\Delta =\overline{v_{\varphi }^2}- \sigma _z^2$ and $\Delta _R= \overline{v_{\varphi }^2}- \sigma _R^2$. Specifically, the effects of the b-ansatz can be determined before solving the Jeans equations once the behaviour over the (R, z)-plane of three easy-to-build ansatz-independent functions is known. The procedure also constrains the ansatz to exclude unphysical results (as a negative $\overline{v_{\varphi }^2}$). The method is illustrated by discussing the cases of three well-known galaxy models: the Miyamoto & Nagai and Satoh disks, and the Binney logarithmic halo, for which the regions and the constraints on the ansatz values can be determined analytically; a two-component (Miyamoto & Nagai plus logarithmic halo) model is also discussed.

Experimental study of DC Kerr effect of chalcogenide glass film by free space coupling method

Chalcogenide glass is an important nonlinear optical material that has attracted much attention in the areas of integrated photonics, supercontinuum sources, and all-optical switches in recent years. However, the current research mainly focuses on the nonlinear effect excited by light, and the research on its properties under the action of a DC field is still deficient. Here, a metal-cladding optical waveguide, which is composed of a chalcogenide glass film coated on a glass substrate, is presented to analyze the quadratic electro-optic (QEO) effect of the chalcogenide glass film. The DC Kerr coefficient and the whole components of the QEO tensor of the sample were experimentally determined by the free space coupling method.

High-speed photoelastic tomography for axisymmetric stress fields in a soft material: Temporal evolution of all stress components

This study presents a novel approach for reconstructing all stress components of the dynamic axisymmetric fields of a soft material using photoelastic tomography (PT) and a high-speed polarization camera. This study focuses on static and dynamic Hertzian contact as an example of transient stress field reconstructions. For the static Hertzian contact (a solid sphere pressed against a gel block), all stress components in the urethane gel, which has an elastic modulus of 47.4 kPa, were reconstructed by PT using the measured photoelastic parameters. The results were compared with theoretical solutions and showed good agreement. For the dynamic Hertzian contact (a sphere impacting gel), a high-speed polarization camera was used to reconstruct the transient stress field within the gel. PT was used to quantitatively measure the shear and axial stress waves and showed different propagation speeds on the substrate. The technique allowed the simultaneous measurement of stress fields ranging from O(10−1) to O(101) kPa during large deformations, demonstrating its accuracy in capturing rapidly changing stress tensor components in dynamic scenarios. The scaling laws of the calculated impact force agreed with theoretical predictions, validating the accuracy of PT for measuring dynamic axisymmetric stress fields in soft materials.

A microstructure-based parameterization of the effective anisotropic elasticity tensor of snow, firn, and bubbly ice

Abstract. Quantifying the link between microstructure and effective elastic properties of snow, firn, and bubbly ice is essential for many applications in cryospheric sciences. The microstructure of snow and ice can be characterized by different types of fabrics (crystallographic and geometrical), which give rise to macroscopically anisotropic elastic behavior. While the impact of the crystallographic fabric has been extensively studied in deep firn, the present work investigates the influence of the geometrical fabric over the entire range of possible volume fractions. To this end, we have computed the effective elasticity tensor of snow, firn, and ice by finite-element simulations based on 391 X-ray tomography images comprising samples from the laboratory, the Alps, Greenland, and Antarctica. We employed a variant of Eshelby's tensor that has been previously utilized for the parameterization of thermal and dielectric properties of snow and utilized Hashin–Shtrikman bounds to capture the nonlinear interplay between density and geometrical anisotropy. From that we derive a closed-form parameterization for all components of the (transverse isotropic) elasticity tensor for all volume fractions using two fit parameters per tensor component. Finally, we used the Thomsen parameter to compare the geometrical anisotropy to the maximal theoretical crystallographic anisotropy in bubbly ice. While the geometrical anisotropy clearly dominates up to ice volume fractions of ϕ≈0.7, a thorough understanding of elasticity in bubbly ice may require a coupled elastic theory that includes geometrical and crystallographic anisotropy.

Friction tensor for an ellipsoid enclosed in a rarefied gas

This research renders calculations of rotranslational friction and diffusion tensors in a rarefied gas, considering both specular and diffuse reflection off ellipsoidal particles. Calculations were conducted in a spherical polar coordinate system, revealing the dynamic nature of these off-diagonal tensors. The tensor elements were determined analytically, intending to compute the surface integrals. All diagrams, and pictures of two-dimensional surfaces were obtained by the numerical integration process in MATLAB. Additionally, we performed calculations without using the time parameter and time derivatives in the relevant relations. Deviations from the state of spherical symmetry were observed in the results for an ellipsoid and show that the tensors are dynamic quantities. Therefore, it is necessary to provide average values. We established that the non-zero elements of the friction tensor for an ellipsoid represent the directions of linear momentum transfer. Also, the non-zero components of the diffusion tensor signify the extent of fluctuations in relevant momentum transfers in corresponding directions.

Uniaxially drawn poly (L-lactic acid)/ poly (vinylidene fluoride) polymer blend as a promising piezoelectric material: Hysteresis and resonance spectroscopy study

This study explores the potential for piezoelectricity in a blend of polyvinylidene fluoride (PVDF) and poly-L-lactic acid (PLLA) prepared using the solution casting method. Dielectric frequency measurements across a range of 102 to 106 Hz revealed piezoelectric resonance within a broader dielectric relaxation regime. Analysis of the resonance spectra for length and width vibrational modes enabled the determination of the piezoelectric tensor components of the PVDF/PLLA blend. The blend exhibited enhanced ferroelectric polarization reversal, remnant polarization, and coercive field compared to pristine PVDF samples. The piezoelectric constants d31 and d14 (d36 of the blend) were determined. Mechanical drawing and poling techniques were employed to generate multiple piezoelectric constants, allowing for efficient energy harvesting from various directions. This flexible blend offers promising prospects for versatile energy conversion applications.

Surface crack effect on frequency and vibration mode switching of solar-powered aerospace structures

In the engineering design of modern wing panels and their optimal control system, the first step is to find and predict their dynamic characteristics. These characteristics are dependent on material properties, panel geometry, and boundary conditions. In addition, possible fracture existence, imperfections and environmental conditions may unpredictably change the frequency and modes shape of the panel and then the parameters of a control system. In the present paper, the effect of surface fracture depth and orientation on frequency and vibration mode switching of perovskite solar cells-based aircraft wing panels is investigated for the first time. The PSC (perovskite solar cell) panel is made of a polymer substrate reinforced with even-distributed graphene platelets (GPLs) and thin solar layers to enhance the overall electromechanical performance of the whole structure. Equations of motion of the panel are obtained by the Hamilton approach based on the properties of Reddy's third-order shear deformation theory and linear components of the Green-Lagrange strain tensor. The novel aspect is a line-spring model of surface crack by Cartmell et al. adapted to the model of the PSC panel under consideration. The discretization of the established equations of motion for the cracked panel is carried out by the element-free IMLS-Ritz method with mesh-free features. Validation of the obtained solutions is carried out by comparisons with results from previous studies for cracked laminated plates.First, material and geometric properties effect on vibration characteristics of the PSC structure was investigated. Next, partial surface crack depth and orientation angle are examined to show changes in frequency and vibration modes shape of the panel. It is shown, that there is a mode swapping between the second and third modes when transitioning from an intact panel to a cracked one. As the crack length increases, higher-order mode variations become more noticeable, potentially leading to their swapping or the excitation of new ones. Additionally, the findings demonstrate that the surface crack orientation angle significantly impacts the higher-order modes of the panel. When the orientation angle is 90°, the first six modes align with those of the intact plate. Mode shapes and frequency shifts are significant for crack orientation angles between 0° and 90°.

HYSCORE and QM/MM Studies of Second Sphere Variants of the Type 1 Copper Site in Azurin: Influence of Mutations on the Hyperfine Couplings of Remote Nitrogens.

Secondary coordination sphere (SCS) interactions have been shown to play important roles in tuning reduction potentials and electron transfer (ET) properties of the Type 1 copper proteins, but the precise roles of these interactions are not fully understood. In this work, we examined the influence of F114P, F114N, and N47S mutations in the SCS on the electronic structure of the T1 copper center in azurin (Az) by studying the hyperfine couplings of (i) histidine remote Nε nitrogens and (ii) the amide Np using the two-dimensional (2D) pulsed electron paramagnetic resonance (EPR) technique HYSCORE (hyperfine sublevel correlation) combined with quantum mechanics/molecular mechanics (QM/MM) and DLPNO-CCSD calculations. Our data show that some components of hyperfine tensor and isotropic coupling in N47SAz and F114PAz (but not F114NAz) deviate by up to ∼±20% from WTAz, indicating that these mutations significantly influence the spin density distribution between the CuII site and coordinating ligands. Furthermore, our calculations support the assignment of Np to the backbone amide of residue 47 (both in Asn and Ser variants). Since the spin density distributions play an important role in tuning the covalency of the Cu-Scys bond of Type 1 copper center that has been shown to be crucial in controlling the reduction potentials, this study provides additional insights into the electron spin factor in tuning the reduction potentials and ET properties.

Relations between Reynolds stresses and their dissipation rates during premixed flame–wall interaction within turbulent boundary layers

A direct numerical simulation (DNS) database for head-on quenching of premixed flames propagating across turbulent boundary layers representative of friction Reynolds numbers, Reτ, of 110 and 180 has been utilized to analyze the interrelation between Reynolds stresses and their dissipation rates during flame–wall interaction. The Reynolds stresses and their dissipation rates exhibit significant deviations from the corresponding non-reacting flow profiles within the flame brush and in the burned gas region. This behavior is prominent for the components in the wall-normal direction because the mean direction of flame normal acceleration due to thermal expansion aligns with the wall-normal direction in this configuration. The anisotropy of Reynolds stresses and their dissipation rate tensors have been found to be qualitatively similar, but the anisotropic behavior weakens with increasing Reτ. However, the components of the anisotropy tensors of Reynolds stresses and viscous dissipation rate are not related according to a linear scaling, and thus, the models based on this assumption do not successfully capture the viscous dissipation rate components obtained from the DNS data. By contrast, a model, which includes the invariants of the anisotropy tensor of Reynolds stresses and satisfies the limiting conditions, has been found to capture especially the diagonal components of the viscous dissipation rate tensor more successfully for both non-reacting and reacting cases considered in this work. However, the quantitative prediction of this model suffers for the components in the wall-normal direction for lower values of Reτ, but the performance of this model improves with an increase in Reτ.

Turbulence Closure With Small, Local Neural Networks: Forced Two‐Dimensional and β‐Plane Flows

AbstractWe parameterize sub‐grid scale (SGS) fluxes in sinusoidally forced two‐dimensional turbulence on the β‐plane at high Reynolds numbers (Re ∼25,000) using simple 2‐layer convolutional neural networks (CNN) having only O(1000) parameters, two orders of magnitude smaller than recent studies employing deeper CNNs with 8–10 layers; we obtain stable, accurate, and long‐term online or a posteriori solutions at 16× downscaling factors. Our methodology significantly improves training efficiency and speed of online large eddy simulations runs, while offering insights into the physics of closure in such turbulent flows. Our approach benefits from extensive hyperparameter searching in learning rate and weight decay coefficient space, as well as the use of cyclical learning rate annealing, which leads to more robust and accurate online solutions compared to fixed learning rates. Our CNNs use either the coarse velocity or the vorticity and strain fields as inputs, and output the two components of the deviatoric stress tensor, Sd. We minimize a loss between the SGS vorticity flux divergence (computed from the high‐resolution solver) and that obtained from the CNN‐modeled Sd, without requiring energy or enstrophy preserving constraints. The success of shallow CNNs in accurately parameterizing this class of turbulent flows implies that the SGS stresses have a weak non‐local dependence on coarse fields; it also aligns with our physical conception that small‐scales are locally controlled by larger scales such as vortices and their strained filaments. Furthermore, 2‐layer CNN‐parameterizations are more likely to be interpretable.

First principles study on the elastic properties of two-dimensional Janus ZrXY (X/Y = Cl, Br, and I, X ≠ Y)

In recent years, a novel two-dimensional semiconductor ZrX2 (X = Cl, Br, and I) has been found to have excellent optoelectronic properties and has attracted much attention. In this paper, the elastic properties of two-dimensional Janus ZrXY (X/Y = Cl, Br, and I, X ≠ Y) are studied by first principles, including elastic parameters, such as elastic tensor components, Young’s modulus, Poisson’s ratio, and stability. Research has found that the lattice parameters of two-dimensional Janus ZrXY are still influenced by the atomic radius. The Janus structure shows excellent dynamic stability both before and after its construction. Based on the elastic theory, the mechanical stability of the two-dimensional Janus ZrXY was proved indirectly. After constructing the two-dimensional Janus ZrXY structure with different planes, the elastic tensor component increases to a certain extent, and Young’s modulus and Poisson’s ratio increase, but the anisotropy of Young’s modulus and Poisson’s ratio decreases significantly.

On Generalized Kenmotsu Manifolds as Hypersurfaces of Vaisman-Gray Manifolds

In this paper, we conclude that the hypersurfaces of Vaisman-Gray manifolds have generalized Kenmotsu structures under some conditions for the Lee form, Kirichenko's tensors and the second fundamental form of the immersion of the hypersurface into the manifold of Vaisman-Gray class. Moreover, the components of the second fundamental form are determined when the foregoing hypersurfaces have generalized Kenmotsu structures or any special kind of it or Kenmotsu structures, such that some of these components are vanish. Also, some components of Lee form and some components of some Kirichenko's tensors in the Vaisman--Gray class are equal to zero. On the other hand, the minimality of totally umbilical, totally geodesic hypersurfaces of Vaisman--Gray manifolds with generalized Kenmotsu structures are investigated. In addition, we deduced that the hypersurface of Vaisman--Gray manifold that have generalized Kenmotsu structure is totally geodesic if and only if it is totally umbilical and some components of Lee form are constants.

Study of the distribution of residual stresses across the surface layer depth following machining through probing hole drilling and digital image correlation

The article aims to study the distribution of residual stresses across the surface layer depth following the machining of test specimens (free orthogonal cutting and burnishing). The machining by cutting and burnishing of 45 steel specimens was carried out using a milling machine with numerical control under varying machining factors. For a comparative analysis, the processes were also modeled via the finite element method using identical experimental values of geometric parameters and machining modes. In order to obtain residual stress distributions across the surface layer depth, probe holes having depths of 0.5; 0.75; 1; 1.5; and 2 mm were drilled. By differentiating the displacement of specimen surface particles, measured using digital image correlation, radial strains around the probe holes were determined. Then, these values were used to determine the residual stress components for each probe hole, and the averaged values of each residual stress component were calculated using the calculation algorithm presented in this article. After burnishing the specimen with a force of 3400 N, the test value of the σx component within the depth range (from the surface) of 0.5–0.75 mm amounted to -250 MPa. Model and experimental distributions of residual stress tensor components across the surface layer depth were obtained following machining via two methods. The experimental values of residual stresses were found to have good convergence with each other and with model distributions at depths up to 1 mm from the machined surface at a drill diameter of 1.7 mm. The proposed approach provides a means to obtain the residual stress distribution across the surface layer depth by drilling probe holes of different depths and estimating radial strains on the specimen surface using the digital image correlation method.

Some New Apsects of Quantum Gravity

We have proposed the quantization of the gravitational field in a synchronous reference frame taking as independent position fields, the six spatial components of the metric tensor. The Einstein-Hilbert Lagrangian is quadratic in the space time derivatives of these metric tensor components and hence in particular, the momentum fields become linear functions of the space-time derivatives of the position fields. It is this fact that gives a simple form to the Hamiltonian density of the gravitational field in a synchronous frame, this simple form of the Hamiltonian being a quadratic function of the momentum fields with a shift that is linear in the spatial derivatives of the metric, very much like the Hamiltonian of a non-relativistic particle moving in a vector potential. Differential equations for the gravitational field propagator are derived and we explain how approximations to this propagator can be derived and used to deduce the graviton propagator corrections caused by nonlinear interactions of the graviton field with itself. We explain how this corrected graviton propagator can be used to deduce how much mass the graviton acquires due to these self-interactions of cubic and higher order. We then consider the important problem involving the coupling of a nonlinear field theory described by its Lagrangian density to a quantum noisy bath and explain how the resulting Hamiltonian of the field plus bath can be used to derive the Hudson-Parthasarathy noisy Schrodinger equation (HPS) which is a quantum stochastic differential equation for the joint unitary evolution of the field interacting with the noisy bath. We explain this in the context of gravity coupled to a noisy bath like a noisy electromagnetic field. The HPS equation contains linear as well as quadratic terms in the white bath noise with the linear terms representing quantum annihilation and creation/quantum Brownian motion process differentials and the quadratic terms representing quantum conservation/Poisson processes differentials. Finally we explain how using Feynman path integrals for fields for evaluating the quantum effective action produced by higher order cumulants of the current field, we can calculate corrections to the quantum effective action produced by higher order cumulants of the current field and hence demonstrate how gauge symmetries of the classical action get broken when we pass over to the quantum effective action with additional symmetry breaking terms produced by the presence of higher order cumulants of the current. This kind of approximate symmetry breaking is known to give masses to massless particles or more generally, corrections to the masses of already massive particles and we illustrate this idea in the context of interactions of the gravitational field with a random electromagnetic field being regarded as the current. This interaction is the standard Maxwell action used in general relativity. The drawback of our approach to quantum gravity is that is its not diffeomorphic invariant since we have chosen our frame to be always synchronous. Further work on how one can incorporate interactions of the gravitational field with a random non-Abelian gauge field is in progress which becomes important because it generates non only quadratic but also cubic and fourth degree terms in the gauge field when it interacts with gravity.

Experimental measurement of spatio-temporally resolved energy dissipation rate in turbulent Rayleigh–Bénard convection

We report a home-built velocity-gradient-tensor-resolved particle image velocimetry (VGTR-PIV) system which spatio-temporally resolves all components of the velocity gradient tensor. This technique is applied to the paradigmatic turbulent Rayleigh–Bénard convection system in a cylindrical cell at three representative positions, i.e. centre, side and bottom regions. The VGTR-PIV system allows us to directly measure, for the first time, the spatio-temporally resolved energy dissipation rate and enstrophy in turbulent thermal convection. In the experiment, the Rayleigh number $Ra$ varied in the range $2 \times 10^8 \leqslant Ra \leqslant 8 \times 10^9$ and the Prandtl number $Pr$ was fixed at $Pr = 4.34$ . Compared with the fully resolved energy dissipation rate $\varepsilon$ , the pseudo-dissipation provides the best estimate within $3\,\%$ , the planar (two-dimensional) surrogate has a larger relative error and the one-dimensional surrogate leads to the largest error. The power-law scalings of the time-averaged energy dissipation rate with the Rayleigh number follow $\langle \varepsilon _c \rangle _t / (\nu ^3 H^{-4}) = 9.86 \times 10^{-6} Ra^{1.54 \pm 0.02}$ , $\langle \varepsilon _s \rangle _t / (\nu ^3 H^{-4}) = 9.26 \times 10^{-3} Ra^{1.25 \pm 0.02}$ and $\langle \varepsilon _b \rangle _t / (\nu ^3 H^{-4}) = 2.70 \times 10^{-2} Ra^{1.23 \pm 0.02}$ in the centre, side and bottom regions, respectively where $\nu$ is dynamic viscosity and $H$ is cell height. These scaling relations, along with our earlier measured time-averaged energy dissipation rate at the bottom wall surface $\langle \varepsilon _w \rangle _t / (\nu ^3 H^{-4}) = 9.65 \times 10^{-2} Ra^{1.25 \pm 0.02}$ (J. Fluid Mech., vol. 947, 2022, A15), provide important constraints against which theoretical models may be tested. For the centre and side locations in the convection cell, the probability density functions (p.d.f.s) of the energy dissipation rate and enstrophy both follow a stretched exponential distribution. For the bottom region, the p.d.f.s of dissipation and enstrophy exhibit a stretched exponential distribution outside the viscous boundary layer and an exponential distribution inside the viscous boundary layer. It is also found that extreme events with high dissipation are the most intermittent in the side region, whereas the bottom region is less intermittent than the cell centre.

A glaserite-like ternary molybdate K(Mg0.5Zr0.5)(MoO4)2: Synthesis, thermal expansion, and ionic conductivity

A ternary molybdate K(Mg0.5Zr0.5)(MoO4)2 was successfully synthesized using the sol-gel method and traditional solid-phase reactions technique. The synthesis time reduced from 200 to 15 h and smaller particles (20–3 μm) were reached by the sol-gel method. Phase purity and crystal structure were confirmed by X-ray diffraction (XRD) analysis with Rietveld refinement. The surface morphology was characterized using a scanning electron microscope (SEM). Using the powder XRD data recorded over 30–550 °C, the principal components of the thermal expansion tensor were determined. K(Mg0.5Zr0.5)(MoO4)2 can be classified as a high-expansion material (αV = 160 × 10−6 °С−1 at 500°С). At elevated temperatures, the compound exhibits significant ionic conductivity, with values reaching 4.2·10−3 S/cm at 650°С, and oxygen ions are the probable charge carriers with activation energy Ea = 0.51 eV.