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Let ( X , d , μ ) (X,d, \mu ) be an Ahlfors n n -regular metric measure space. Let L \mathcal {L} be a non-negative self-adjoint operator on L 2 ( X ) L^2(X) with heat kernel satisfying Gaussian estimate. Assume that the kernels of the spectral multiplier operators F ( L ) F(\mathcal {L}) satisfy an appropriate weighted L 2 L^2 estimate. By the spectral theory, we can define the imaginary power operator L i s , s ∈ R \mathcal {L}^{is}, s\in \mathbb R , which is bounded on L 2 ( X ) L^2(X) . The main aim of this paper is to prove that for any p ∈ ( 0 , ∞ ) p \in (0,\infty ) , ‖ L i s f ‖ H L p ( X ) ≤ C ( 1 + | s | ) n | 1 / p − 1 / 2 | ‖ f ‖ H L p ( X ) , s ∈ R , \begin{equation*} \big \|\mathcal {L}^{is} f\big \|_{H^p_{\mathcal {L}}(X)} \leq C (1+|s|)^{n|1/p-1/2|} \|f\|_{H^p_{\mathcal {L}}(X)}, \quad s \in \mathbb {R}, \end{equation*} where H L p ( X ) H^p_\mathcal {L}(X) is the Hardy space associated to L \mathcal {L} , and C C is a constant independent of s s . Our result applies to sub-Laplaicans on stratified Lie groups and Hermite operators on R n \mathbb {R}^n with n ≥ 2 n\ge 2 .

Let $\mathcal {H}=-\Delta + |x|^2$ be the Hermite operator on $\mathbb R^n$ with $n\ge 2$. In this paper, we prove the boundedness of Schrödinger groups of fractional powers of $\mathcal {H}$ on Lebesgue and Hardy spaces. More precisely, we prove that (a) for $p \in (1,\infty )$, $\gamma>0$ and $\beta /\gamma \geq n|1/p -1/2|$, $$ \begin{align*} \big\|\mathcal{H}^{-\beta /2}e^{it \mathcal{H}^{\gamma /2}}f\big\|_{L^p(\mathbb R^n)} \leq C (1 + |t|)^{n|1/p-1/2|}\|f\|_{L^p(\mathbb R^n)}, \quad \forall t \in \mathbb{R}, \end{align*}$$and (b) for $p \in (0, 1]$, $\gamma>0$ and $\beta /\gamma \geq n(1/p -1/2)$, $$ \begin{align*} \big\|\mathcal{H}^{-\beta /2}e^{it \mathcal{H}^{\gamma /2}}f\big\|_{H^p_{\mathcal{H}}(\mathbb R^n)} \leq C (1 + |t|)^{n(1/p-1/2)}\|f\|_{H^p_{\mathcal{H}}(\mathbb R^n)}, \quad \forall t \in \mathbb{R}, \end{align*}$$where $H^p_{\mathcal {H}}(\mathbb R^n)$ is the Hardy space associated with the operator $\mathcal {H}$. These estimates improve related result of Thangavelu [22] and have some interesting applications.

Let $(M,\rho,\mu)$ be a metric measure space satisfying the doubling, reverse doubling and non-collapsing conditions, and $\mathscr{L}$ be a self-adjoint operator on $L^2 (M, d\mu)$ whose heat kernel $p_t (x,y)$ satisfy the small-time Gaussian upper bound, H\"{o}lder continuity and Markov property. In this paper, we give characterizations of inhomogeneous "classical" and "non-classical" Besov and Triebel-Lizorkin spaces associated to $\mathscr{L}$ in terms of continuous Littlewood-Paley and Lusin area functions defined by the heat semigroup, for complete range of indices. This extends related classical results for Besov and Triebel-Lizorkin spaces on $\mathbb{R}^n$ to more general setting, and extends corresponding results in [Trans. Amer. Math Soc. 367 (2015), 121-189] to complete range of indices., Comment: 31 pages

TTE is used to regulate the Li+-solvation sheath structure in the low-concentration electrolyte (LCE). This work demonstrates the decisive role of Li+-solvation sheath structure in stabilizing the interfaces between the LCEs and cathodes.

Herein, the introduction of hexagonal LaNiO3 on the surface of LiNi0.8Co0.15Al0.05O2 provides a protective and adaptable ultrathin layer to stabilize the microstructure of Ni-rich layered cathode materials, enhancing the electrochemical and thermal stability.

An O3 type Li0.6[Li0.2Mn0.8]O2 lithium-rich material has a high reversible capacity due to the synergistic oxidation and reduction of anion and cation. However, the anion oxidation reaction that compensates the charge leads to a partial release of oxygen and the collapse of the structure inevitably. Here, we improve the structural stability of Li0.6[Li0.2Mn0.8]O2 by simultaneously introducing Al ions and B ions. Al ions and B ions randomly occupy octahedral and tetrahedral positions, hindering the migration of Mn ions and expanding the unit cell, resulting in a stable structure and promoting Li+ migration. The co-doped sample has better electrochemical performance than the bare material, and the capacity retention increases from 62.48% to 82.48% after 80 cycles at 0.1C rate, and still provides a capacity of 226 mAh g−1 between 2 and 4.8 V.

LiMn2 O4 (LMO) cathodes suffer from limited cycle life, resulting from Mn dissolution and side reactions between electrode and electrolyte. In this study, Sr-modified LMO is prepared by using a simple strategy. The nature and position of large-radius Sr ions are investigated, alongside their influence on the structural stability of the bulk. SrMnO3 (SMO) is found to be enriched at grain boundaries of LMO, with Mn-O-Sr bonds forming at the SMO/LMO interface. Furthermore, stable SMO alleviates the migration of Mn ions in LMO associated with structural integrity and suppresses side reactions between the electrode and electrolyte. The modified LMO cathodes maintain their structural integrity and display improved rate performance and cycling stability under harsh conditions. Remarkably, the discharge capacity of a Sr-modified LMO||Li half-cell maintains 94.8 % at 25 °C and 79.6 % at 55 °C after 500 cycles. Consequently, enrichment of strontium at grain boundaries presents a promising strategy for developing cathodes for long-term use.

LiNi0.8Co0.1Mn0.1O2 cathode material for lithium-ion battery exhibits high capacity, but it suffers from interfacial side reactions and structural/thermodynamic instability, which leads to capacity reduction and safety problems. Cubic brick (Ni0.8Co0.1Mn0.1)C2O4·2H2O particles with micron size are synthesized by co-precipitation method. The oxalic precursor is sintered with lithium hydroxide to obtain cubic mono-dispersion LiNi0.8Co0.1Mn0.1O2 micrometer particles. Structural stability, cycling performance, rate capability and compacting density of the cubic mono-dispersion material are investigated. Conventional spherical and irregular mono-dispersion LiNi0.8Co0.1Mn0.1O2 are also prepared for comparison. The results reveal that the cubic mono-dispersion LiNi0.8Co0.1Mn0.1O2 dramatically enhances the structural stability and cycling performance at a little cost of capacity and rate capability.

All-solid-state Li-ion batteries (ASSLBs) are promising systems to power electronic devices and electric vehicles for their high-energy density and safety. PEO-based composite electrolytes containing lithium salts and various fillers have been regarded as the most attractive solid electrolytes for ASSLBs due to their high interfacial compatibility with electrodes, high mechanical flexibility, and easy fabrication. For PEO/lithium salt/ceramic composite electrolytes, both “ceramic-in-polymer” and “polymer in ceramic” can be applied to solid-state batteries. “Ceramic-in-polymer” can provide high ionic conductivity but narrow electrochemical window; “polymer in ceramic” can provide wide electrochemical window but low ionic conductivity. In order to combine the high conductivity of “ceramic in polymer” with the wide electrochemical window of “polymer in ceramic,” we designed multilayer PEO/ Li6.4La3Zr1.4Ta0.6O12(LLZTO) composite electrolytes with low content of LLZTO (10wt %) in the inside layer and high content of LLZTO (40wt %) in the surface layer (PCEs-40–10-40) by a simple solution casting method. The PCEs-40–10-40 showed high ionic conductivity(4.61 × 10–4 $${\mathrm{S}\cdot \mathrm{cm}}^{-1}$$ ) and a wide electrochemical window(> 5.1 V) at 60 ℃. Compared with monolayer composite electrolytes, solid-state LiFePO4|Li batteries with multilayer composite electrolytes showed much better cycling stability (capacity retention of 98.8% after 200 cycles) at 0.5C and 60 ℃. This work shall give practical insights for the applications of PEO-based composite electrolytes.

Chromic oxide (Cr2O3) is one of the most important chromic salts and is widely used in industrial applications. Traditionally, Cr2O3 is prepared from chromium trioxide (CrO3) or sodium dichromate (Na2Cr2O7), both of which are manufactured from chromite ore. However, traditional manufacturing processes are time-consuming, inefficient, and energy-consuming, and they can catheuse severe environmental pollution by discharging large quantities of solid residues containing Cr in the form of Cr(VI) compounds. These compounds are also harmful to human health. Therefore, developing an environmentally friendly production process is of great concern to the chromic oxide industry. In this study, a new method for producing Cr2O3 without hexavalent chromium pollution was investigated. In this novel process, 97% of the total Cr—in the form of chromium sulfate (Cr2(SO4)3)—and Fe are leached from ferrochrome alloy using sulfuric acid. The concentration of Fe in the Cr2(SO4)3 solution can be reduced to 2 mg/L after precipitating with oxalic acid and extracting with di (2-ethylhexyl) phosphate. Cr2O3 is then produced by precipitating Cr from the Cr2(SO4)3 solution with sodium hydroxide, followed by calcination of the precipitated Cr(OH)3 at 500°C. Using this method, Cr2O3 with a purity of up to 99.3% was obtained in this study. Throughout the process, Cr existed only in the form of Cr(III); thus, the pollution caused by hexavalent Cr was eliminated, and environmentally friendly production of Cr2O3 was achieved.

The large-scale and efficient synthesis of MnPO4·H2O is an urgent problem to be solved. In this paper, self-assembled dumbbell-shaped MnPO4·H2O particles were successfully synthesized just using KMnO4 and commercial concentrated H3PO4, and then employed as the precursor to prepare LiMnPO4/C composites. The XRD, SEM, TEM, and electrochemical performance tests were used to investigate MnPO4·H2O and the corresponding LiMnPO4/C. The formation mechanism of MnPO4·H2O was discussed. The results showed that temperature and water content have a significant effect on the yield, purity, and morphology of MnPO4·H2O. The synthesized LiMnPO4/C with reduced carbon content at 650 °C delivered a discharge capacity of 109.3 mA h g-1 at 1 C and 104.3 mA h g-1 at 2 C. This green and efficient synthesis method of MnPO4·H2O provides a new idea for the large-scale preparation of LiMnPO4/C cathode materials.

Let $$M^{(i)}$$ , $$i=1,2,\ldots , n$$ , be the boundaries of unbounded domains $$\Omega ^{(i)}$$ of finite type in $$\mathbb {C}^2$$ , and let $$\Box _b^{(i)}$$ be the Kohn Laplacian on $$M^{(i)}$$ . In this paper, we study multivariable spectral multipliers $$m(\Box _b^{(1)},\ldots , \Box _b^{(n)})$$ acting on the Shilov boundary $${\widetilde{M}}=M^{(1)} \times \cdots \times M^{(n)}$$ of the product domain $$\Omega ^{(1)}\times \cdots \times \Omega ^{(n)}$$ . We show that if a function $$m(\lambda _1, \ldots ,\lambda _n)$$ satisfies a Marcinkiewicz-type smoothness condition defined using Sobolev norms, then the spectral multiplier operator $$m(\Box _b^{(1)}, \ldots , \Box _b^{(n)})$$ is a product Calderon–Zygmund operator of Journe type.

Extended depth of field (EDoF) microscopy has emerged as a powerful solution to greatly increase the access into neuronal populations in table-top imaging platforms. Here, we present EDoF-Miniscope, which integrates an optimized thin and lightweight binary diffractive optical element (DOE) onto the gradient refractive index (GRIN) lens of a head-mounted fluorescence miniature microscope, i.e. “miniscope”. We achieve an alignment accuracy of 70 μm to allow a 2.8X depth-of-field extension between the twin foci. We optimize the phase profile across the whole back aperture through a genetic algorithm that considers the primary GRIN lens aberrations, optical property of the submersion media, and axial intensity loss from tissue scattering in a Fourier optics forward model. Compared to other computational miniscopes, our EDoF-Miniscope produces high-contrast signals that can be recovered by a simple algorithm and can successfully capture volumetrically distributed neuronal signals without significantly compromising the speed, signal-to-noise, signal-to-background, and maintain a comparable 0.9-μm lateral spatial resolution and the size and weight of the miniature platform. We demonstrate the robustness of EDoF-Miniscope against scattering by characterizing its performance in 5-μm and 10-μm beads embedded in scattering phantoms. We demonstrate that EDoF-Miniscope facilitates deeper interrogations of neuronal populations in a 100-μm thick mouse brain sample, as well as vessels in a mouse brain. Built from off-the-shelf components augmented by a customizable DOE, we expect that this low-cost EDoF-Miniscope may find utility in a wide range of neural recording applications.

Recent experiments suggested that adenosine triphosphate (ATP) can regulate liquid-liquid phase separation (LLPS) of various proteins and inhibit protein aggregations at its physiological concentration, which is highly correlated with the nonspecific interactions of ATP to a wide variety of proteins. However, the mechanism underlying the general binding capability of ATP largely remains unclear. In this work, we used molecular dynamics simulation to study the binding of ATPs to three proteins with distinct net charges: TDP-43 NTD (-7 e), TAF15-RRM (0 e), HWEL (+8 e). Negatively charged ATP exhibits a strong trend to accumulate around all of these proteins. While only a fraction of the accumulated ATPs directly binds to the limited regions of the protein surface, additional ATPs indirectly bind to proteins by aggregating into ATP clusters. Hence, the proportion of the directly bound ATPs in the clusters as well as their binding regions can be adjusted in response to different proteins, which makes ATP well adapted to a variety of proteins. Moreover, our results suggest that ATP tightly binds to Arg with high affinity, and Arg dominates the direct binding of ATP. Meanwhile, Arg also affects the self-association of accumulated ATPs. The size of the ATP cluster is effectively regulated by the distribution of Arg. Considering the ubiquity of Arg in proteins, our findings are helpful to understand the general binding capability of ATP.

This work reports a study of the extraction of iron (III) deeply from the chromium sulphate solution in which most of the iron was precipitated with oxalic acid. Di(2-ethylhexyl)phosphoric acid (P2...

Topography measurement is essential for surface characterization, semiconductor metrology, and inspection applications. To date, performing high-throughput and accurate topography remains challenging due to the trade-off between field-of-view (FOV) and spatial resolution. Here we demonstrate a novel topography technique based on the reflection-mode Fourier ptychographic microscopy, termed Fourier ptychograhpic topography (FPT). We show that FPT provides both a wide FOV and high resolution, and achieves nanoscale height reconstruction accuracy. Our FPT prototype is based on a custom-built computational microscope consisting of programmable brightfield and darkfield LED arrays. The topography reconstruction is performed by a sequential Gauss-Newton-based Fourier ptychographic phase retrieval algorithm augmented with total variation regularization. We achieve a synthetic numerical aperture (NA) of 0.84 and a diffraction-limited resolution of 750 nm, increasing the native objective NA (0.28) by 3×, across a 1.2 × 1.2 mm2 FOV. We experimentally demonstrate the FPT on a variety of reflective samples with different patterned structures. The reconstructed resolution is validated on both amplitude and phase resolution test features. The accuracy of the reconstructed surface profile is benchmarked against high-resolution optical profilometry measurements. In addition, we show that the FPT provides robust surface profile reconstructions even on complex patterns with fine features that cannot be reliably measured by the standard optical profilometer. The spatial and temporal noise of our FPT system is characterized to be 0.529 nm and 0.027 nm, respectively.

Titanium and boron are simultaneously introduced into LiNi0.8Co0.1Mn0.1O2 to improve the structural stability and electrochemical performance of the material. X-ray diffraction studies reveal that Ti4+ ion replaces Li+ ion and reduces the cation mixing; B3+ ion enters the tetrahedron of the transition metal layers and enlarges the distance of the [LiO6] layers. The co-doped sample has spherical secondary particles with elongated and enlarged primary particles, in which Ti and B elements distribute uniformly. Electrochemical studies reveal the co-doped sample has improved rate performance (183.1 mAh·g-1 at 1 C and 155.5 mAh·g-1 at 10 C) and cycle stability (capacity retention of 94.7% after 100 cycles at 1 C). EIS and CV disclose that Ti and B co-doping reduces charge transfer impedance and suppresses phase change of LiNi0.8Co0.1Mn0.1O2.

A feasible, environment-friendly and economical method to regenerate a graphite anode from discarded lithium-ion batteries is carried out. The regenerated graphite (G-A-T-SGT@C) was obtained after the treatment of scrapped graphite (SGT) via pyrolyzation, acid leaching, graphitization and coating by amorphous carbon. As a result, G-A-T-SGT@C has negligible impurities, smaller specific surface area, reduced pore defects and much higher degree of graphitization compared with SGT, which will enhance the electrochemical performance of G-A-T-SGT@C. Consequently, lithium-ion batteries with the G-A-T-SGT@C anode exhibited a high initial coulombic efficiency of 90.64%, a high specific capacity of 344 mA h g−1 at 0.2C, excellent rate capability and eximious cycling performance for more than 250 cycles at 0.5C, which is also outstanding compared with the performance of lithium-ion batteries using commercial graphite as the anode.

PEO-based composite electrolytes are one of the most practical electrolytes in all-solid batteries (ASSBs). To achieve the perspective of ASSBs with high energy density, PEO based composite electrolytes should match high-voltage cathodes, but the stability of cathode interface is a challenging issue. Herein, we investigate the electrochemical performance of single-crystal LiNi0.6Mn0.1Co0.3O2 (SC-NMC613) in ASSBs with PEO-based electrolytes, and compared the electrochemical performance of SC-NMC613 with conventional polycrystalline LiNi0.6Mn0.1Co0.3O2 (PC-NMC613) systematically. The Li+ diffusion coefficient of SC-NMC613 is 2-4 times higher than that of PC-NCM613 in ASSBs. SC-NCM613 shows high initial specific capacity (152 mAh•g−1) at 0.5 C, stable cycling performance (100 cycles, retention of 108.9 mAh•g−1), and exceptional rate capability even without surface modification, while PC-NCM613 shows low initial specific capacity (137.5 mAh•g−1) at 0.5 C, poor cycling performance (100 cycles, retention of 90.1 mAh•g−1), and poor rate capability in ASSBs. The severe cycling degradation of the PC-NCM613 is mainly attributed to the structural deterioration inside particles and the formation of rock-salt phase. The SC-NCM613 particles observably alleviate the generation of intergranular cracks and decelerate the formation of rock-salt phase. SC-NCM cathodes in ASSBs with PEO-based electrolyte are beneficial to the stability of cathode interface. Therefore, developing single-crystal Ni-rich cathodes represents a promising strategy for both high energy density and power density of ASSBs.