Hierarchical Transition Research Articles

Fabrication of Hierarchical Nanostructures featuring Amplified Asymmetry through Co‐assembly of Liquid Crystalline Block Copolymer and Chiral Amphiphiles

The widespread presence of hierarchical asymmetric structures in nature has sparked considerable interest because of their unique functionalities. These ingenious structures across multiple scales often emerge from the transfer and amplification of asymmetry from chiral molecules under various synergistic effects. However, constructing artificial chiral asymmetric structures, particularly in developing hierarchical multicomponent structures analogous to those formed in nature through synergistic non‐covalent interactions, still presents tremendous challenges. Herein, we propose a co‐assembly strategy to fabricate hierarchical chiral mesostructures by combining a liquid crystalline block copolymer (LC‐BCP) with a small molecular amphiphile containing chiral alanine or phenylalanine as a linker. Through a classic solvent‐exchange process, chiral amphiphiles embedded within LC‐BCP finely regulate the LC ordering effect and facilitate transfer and amplification of asymmetry. Consequently, various co‐assembled structures with significant hierarchical chirality features are obtained through synergetic effects. Remarkably, subtle alterations to the side groups of amino acids in the amphiphiles effectively adjust the hierarchical morphology transition. Moreover, the covalent bonding sequence of amino acids in the amphiphiles emerges as a critical factor governing the formation of hierarchical nanofibers and multilayered vesicles exhibiting a superhelical sense.

Fabrication of Hierarchical Nanostructures Featuring Amplified Asymmetry Through Co-Assembly of Liquid Crystalline Block Copolymer and Chiral Amphiphiles.

The widespread presence of hierarchical asymmetric structures in nature has sparked considerable interest because of their unique functionalities. These ingenious structures across multiple scales often emerge from the transfer and amplification of asymmetry from chiral molecules under various synergistic effects. However, constructing artificial chiral asymmetric structures, particularly in developing hierarchical multicomponent structures analogous to those formed in nature through synergistic non-covalent interactions, still presents tremendous challenges. Herein, we propose a co-assembly strategy to fabricate hierarchical chiral mesostructures by combining a liquid crystalline block copolymer (LC-BCP) with a small molecular amphiphile containing chiral alanine or phenylalanine as a linker. Through a classic solvent-exchange process, chiral amphiphiles embedded within LC-BCP finely regulate the LC ordering effect and facilitate transfer and amplification of asymmetry. Consequently, various co-assembled structures with significant hierarchical chirality features are obtained through synergetic effects. Remarkably, subtle alterations to the side groups of amino acids in the amphiphiles effectively adjust the hierarchical morphology transition. Moreover, the covalent bonding sequence of amino acids in the amphiphiles emerges as a critical factor governing the formation of hierarchical nanofibers and multilayered vesicles exhibiting a superhelical sense.

Impulsive maneuver strategy for multi-agent orbital pursuit-evasion game under sparse rewards

To address the subjectivity of dense reward designs for the orbital pursuit-evasion game with multiple optimization objectives, this paper proposes the reinforcement learning method with a hierarchical network structure to guide game strategies under sparse rewards. Initially, to overcome the convergence challenges in the reinforcement learning training process under sparse rewards, a hierarchical network structure is proposed based on the hindsight experience replay. Subsequently, considering the strict constraints imposed by orbital dynamics on spacecraft state space, the reachable domain method is introduced to refine the subgoal space in the hierarchical network, further facilitating the achievement of subgoals. Finally, by adopting the centralized training-layered execution approach, a complete multi-agent reinforcement learning method with the hierarchical network structure is established, enabling networks at each level to learn effectively in parallel within sparse reward environments. Numerical simulations indicate that, under the single-agent reinforcement learning framework, the proposed method exhibits superior stability in the late training stage and enhances exploration efficiency in the early stage by 38.89% to 55.56% to the baseline method. Under the multi-agent reinforcement learning framework, as the relative distance decreases, the subgoals generated by the hierarchical network transition from long-term to short-term, aligning with human behavioral logic.

Gradient variations in rhizospheric soil exchangeable cations across a forest-steppe transect

The forest-steppe ecotone (FSE) is vulnerable to climate change and valuable to predict ecosystem succession. Although species diversity covaries with climatic factors and changes gradually across FSE transects, aligning with “Whittaker’s hierarchy theory”, whether soil nutrients follow a similar distribution pattern remains unclear. We studied the ecological gradient changes in effective cation-exchange capacity (ECEC, sum of exchangeable Ca2+, Mg2+, K+ and Na+) in the near-neutral-pH rhizosphere and bulk soils along the north–south zonal transect of the Hulunbuir FSE, China. The 230-km transect includes hierarchical transitions from two closed forests through two FSE to four meadow steppes. For both rhizosphere and bulk soils, soil ECEC were the highest in meadow steppes followed by closed forests and FSE area. ECEC were inversely correlated with soil water content but positively with soil pH. Therefore, the distribution of ECEC along the forest-steppe transect did not conform to a hierarchical distribution and decoupled from vegetation transitions, possibly due to the fact that the distribution of soil pH, plant uptake and leaching break the uniformity of ECEC variation along the gradient. In closed forests and FSE ecosystems, the ECEC and exchangeable Ca2+ in rhizosphere soils were higher than those in bulk soil, indicating that rhizosphere processes may alleviate cation limitation in these ecosystems. The study substantially improves our understanding of soil nutrient mobilization and soil fertility across forest-steppe ecotones which helps projecting ecosystem succession. It further highlights the critical role of rhizosphere processes in modulating hierarchical vegetation transition along environmental gradients and provides avenues for future research opportunities.

Tissue-scale in vitro epithelial wrinkling and wrinkle-to-fold transition

Although epithelial folding is commonly studied using in vivo animal models, such models exhibit critical limitations in terms of real-time observation and independent control of experimental parameters. Here, we develop a tissue-scale in vitro epithelial bilayer folding model that incorporates an epithelium and extracellular matrix (ECM) hydrogel, thereby emulating various folding structures found in in vivo epithelial tissue. Beyond mere folding, our in vitro model realizes a hierarchical transition in the epithelial bilayer, shifting from periodic wrinkles to a single deep fold under compression. Experimental and theoretical investigations of the in vitro model imply that both the strain-stiffening of epithelium and the poroelasticity of ECM influence the folded structures of epithelial tissue. The proposed in vitro model will aid in investigating the underlying mechanism of tissue-scale in vivo epithelial folding relevant to developmental biology and tissue engineering.

Low temperature aging induced conjugated polymers aggregation by UV–Vis spectroscopy

AbstractMultiscale structural evolution of conjugated polymer (CP) solutions from disorder to aggregation via environmental perturbations is a universal phenomenon in condensed matter physics. Here, we take poly(3‐hexylthiophene‐2,5‐diyl) (P3HT) as a model material and use Ultraviolet–Visible (UV–Vis) spectroscopy to study the hierarchical structure evolution during prolonged aging at relatively low temperature. Using in‐situ UV–Vis spectroscopy and the Franck‐Condon (FC) method, the dynamics of multiscale structural transitions in P3HT were elucidated. Hierarchical structural details can be extracted from the UV–Vis spectra, including the overall percentage of aggregates, conjugation length, degree of H‐/J‐type molecular packing, and site energetic disorder. Through covariance analysis of conformational disorder to aggregation transitions of conjugated polymers obtained under multiple conditions, we find that the trends of the above four key parameters are not consistent, which informs the unique evolution of the degree of ordering at different length scales within the aggregates. This work facilitates the scrutiny of the hierarchical structural transition from multi‐length scale perspective.

Retardation factor scaling for contaminant transport in fractured media

This paper delves into the key factors governing contaminant retardation and the scaling of retardation factor for contaminant transport in naturally fractured media. It employs a geostatistical approach, utilizing hierarchical transition probability and covariance models, to characterize multimodal reactive mineral facies. Subsequently, scale-dependent models are developed to illustrate the transport behaviors of reactive contaminants under equilibrium sorption conditions and quantify the spatial and temporal variations of the effective retardation factor. These models incorporate various factors, including the structural characteristics of reactive minerals (the volume proportions and lengths of different facies types), fracture characteristics (the means and variances of fracture apertures), and the adsorption capacity of contaminants on minerals (sorption coefficients (ln Kd)). The results from these Lagrangian-based models are compared and validated against experimental data of solute transport in a single natural fracture. These findings underscore the remarkable agreement between the derived scale-dependent analytical expressions and the experimental results. Additionally, a global sensitivity analysis was performed using the Monte Carlo-based Sobol’ indices method to understand the influential factors governing contaminant retardation. The study reveals that the primary influential factor is the sorption capacity of contaminants on minerals, while fracture and reactive mineral structural characteristics play a secondary role. These insights provide significant information for understanding the reactive transport processes and the environmental impact of transport parameters in fractured media.

Opinion formation over dynamic hierarchical networks with acquaintances and strangers: A genetic variation based double-mechanism framework

Inspired by the psychological phenomenon that agents generally adopt different opinions or action-update mechanisms when faced with different types of neighbors, we propose a novel double-mechanism framework over dynamic hierarchical networks to fill this gap. First, a novel multi-attribute genetic variation-based leader-influencer-follower (LIF) dynamic hierarchical network is developed. Second, we propose a double-mechanism framework in which a synchronous asymmetric Deffuant–Weisbuch model with opinion memory effect (MSADW) and a continuous opinion and discrete action (CODA) model with action memory effect (MCODA) are built, achieving the co-evolution of the agent’s attribute and the social network with acquaintances and strangers, respectively. Finally, the experimental results show that the memory effect can effectively weaken the bounded confidence rule and behavioral preference, accelerating the consensus of group opinions or actions. In addition, all agents attempt to obtain a higher out-degree during the co-evolution process to realize the hierarchical transition. The fitted hierarchical partition functions provide a basis for adjusting social structure from the inverted T-shaped to the pyramid-shaped structure and finally to the olive-shaped structure, which has insightful social interpretations.

Mode transition prescribed performance control of connected hybrid electric vehicles based on driver intention fuzzy prediction

A novel hierarchical mode transition control (NHMTC) algorithm utilizing traffic information is investigated for realizing a superior dynamic and ride comfort performance of a connected hybrid electric vehicle (CHEV). The upper level includes the target vehicle velocity planning by model predictive control (MPC) considering less vehicle jerk based on the vehicle-to-infrastructure (V2I) information and the driving intention prediction by a type-2 fuzzy neural network (T2FNN) model using the planned target velocity. The lower level involves the automatic clutch system’s prescribed performance fixed-time controller (PPFTC) design and the desired velocity of the clutch engagement calculated by the predicted driving intention to improve the clutch position trajectory’s tracking accuracy and stability under the actual mode transition process. The comparison results with existing schemes under a co-simulation environment demonstrate the advantage and effectiveness of the designed NHMTC method.

Uniform one-dimensional hierarchical CoOx-N-C feather duster breaking the activity-stability trade-off for hydrogenation reactions

Carbon-based non-precious metal/oxide catalysts always face the dilemma of a trade-off between high activity and long-term stability in hydrogenation reactions, detrimental to their practical applications. Herein we propose a unique and uniform feather duster-type catalyst as a promising concept to solve this activity-stability trade-off. The feather duster is composed of both exposed and embedded active sites, that can facilitate substrate adsorption/activation and can be resistant against site leaching during the non-neutral aqueous phase reactions. We constructed a uniform one-dimensional hierarchical transition metal oxide-containing N-doped carbon (TMO-N-C), especially for CoOx-N-C (x = 1.0 or 1.3), based on a controlled depolymerization-reassembly strategy before morphology-conserved pyrolysis. The uniform, thermally stable feather duster-type precursor was induced in view of the interaction between cobalt 2-methylimidazolate framework (ZIF-67) polyhedron and zinc-trimesic acid (ZnBTC) fiber in water. And subsequent pyrolysis leads to a uniform CoOx-N-C nanoarchitecture with numerous nanorods planted on a single fiber. The resulting CoOx-N-C proves to be highly efficient for 4-nitrophenol reduction, delivering a high activity factor of 7.32 s−1 g−1 with exceptional long life over 10 cycles, thus ranking CoOx-N-C one of the best transition metal oxide catalysts for hydrogenation reactions. This research overcomes the dilemma that either high activity contributed by surface active sites or superior stability provided by embedded ones, offering valuable guidance for rational design of efficient metal/oxide catalysts.

MOCVD Growth of Hierarchical Nanostructured MoS2: Implications for Reactive States as the Large-Area Film

Large-area hierarchical grown transition metal dichalcogenide (TMD) nanostructures as out-of-plane directional layered TMD nanocrystals on a substrate could be considerably useful in electrochemical applications because of the high density of edge states. In this study, layered MoS2 nanocrystals less than 50 nm in diameter were synthesized by overlapping with each other as vertical growth in the form of a film with an area of 10 mm × 40 μm using metal–organic chemical vapor deposition (MOCVD) for the lateral growth of TMDs. Notably, in the spatial distribution, a singular point where the vertical growth is promoted was found through optical dark-field data and spatially resolved micro-Raman spectroscopy. The accelerated vertical growth region distinguished by the singular point is due to the layer-dependent dielectric constant of MoS2, which affects the surface energy and changes the conditions of vapor-phase deposition. In addition, with these phenomena, we discuss a way for expanding the hierarchical vertically grown MoS2 region on a substrate through analytical methods based on hydromechanics.

Self-supervised Dual Hypergraph learning with Intent Disentanglement for session-based recommendation

Existing works on session-based recommendation have shown the advantage in enhancing the prediction ability of recommendation with various deep learning techniques. However, the following challenges need to be addressed: (1) the hierarchy of item transition patterns is overlooked; (2) existing works fail to distinguish various factors of item transition within a single session for disentangling user intents. To cope with the above challenges, we propose a novel session based recommendation model called Self-supervised Dual Hypergraph learning with Intent Disentanglement model (SDHID). Specifically, we first propose a disentangled capsule hypergraph convolutional channel for ne-grained intent learning to capture the intra-session pattern. Accordingly, we introduce the hypergraph and capsule networks in disentangling to learn the item embedding for different factors, and then the representation of the intra-session pattern is obtained by aggregating item embedding with attention weights. Moreover, we build a novel dual–primal hypergraph convolutional channel by mapping the hypergraph to a dual–primal graph for learning the item transition pattern of inter-session. In addition, the above two channels are combined into a self-supervised contrastive learning framework by maximizing mutual information between the learned session representations. We unify the recommendation and the self-supervised tasks under a primary and auxiliary learning framework. The combined optimization of two tasks leads to a hierarchical joint learning item transition for intra- and inter-session. Extensive experiments on real datasets show that the proposed model outperforms several state-of-the-art models.

Geographical Detection Analysis and Spatiotemporal Disparity Characteristics of the Coupling Coordination Development between Urbanization and the Eco-Environment

The study of the coupling coordinated relationship between urbanization and the eco-environment is important to promote regional high-quality development. This paper measured the coupling coordination degree between urbanization and the eco-environment from 2008 to 2020, using the coupling coordination model, depicted its spatiotemporal characteristics with ArcGIS, and explored its driving factors and impacts of the interaction between pairwise factors on it with geo-detection methods. The results show that the coupling coordination level becomes increasingly improved, but there exist widening spatial disparities, and in 2020, its spatial distribution presents characteristics of “decreasing from north to south” and “increasing from west to east”. According to their evolutionary characteristics of coupling coordination categories, cities can be classified into four types: “transition from basic misalignment to primary coordination”, “transition from primary coordination to intermediate coordination”, “transition from primary coordination to senior coordination”, “no hierarchical transition”. Currently, driving factors of industrial structure, opening to the outside world and technological innovation significantly affect the coupling coordination degree, and the interaction between pairwise factors is enhanced. Policies should be implemented in categories to improve the coupling coordination level. Regional cooperation, exchanges, and interconnection of resources and factors should be strengthened to jointly build a coordinated regional development pattern.

Nickel Cobalt LDH/Graphene Film on Nickel-Foam-Supported Ternary Transition Metal Oxides for Supercapacitor Applications

Layered double hydroxides (LDHs) of transition metals have attained significant attention for supercapacitor applications due to their excellent charge storage, low internal resistance, and superior electrochemical stability. Here, a nanocomposite of reduced graphene oxide/nickel cobalt layered double hydroxide (rGO/NiCo LDH) on the surface of nickel foam (NF) containing hierarchical nickel cobalt copper transition metal oxides (TMOs) is prepared through two-step processes of electrochemical and coprecipitation methods. The TMOs/rGO/NiCo LDH nanocomposite is characterized by X-ray diffraction (XRD), Fourier transform infrared (FTIR), Raman, X-ray energy-dispersive (EDS), and X-ray photoelectron (XPS) spectroscopies as well as by transmission electron microscopy (TEM), field-emission scanning electron microscopy (FE-SEM), elemental mapping analysis, nitrogen adsorption/desorption, and contact angle measurements. The supercapacitive behavior of the electrodes has been investigated through cyclic voltammetry (CV), galvanostatic charge–discharge (GCD) measurements, and electrochemical impedance spectroscopy (EIS). The study has shown that the synergetic effect and the electrochemical properties have considerably improved when the layered double hydroxides are synthesized in the presence of rGO. The TMOs/rGO/NiCo LDH nanocomposite exhibits an excellent specific capacitance of 2763 F g–1 at a current density of 1 A g–1 and a stability of 85% after 3000 GCD cycles at a current density of 24 A g–1. Also, a TMOs/rGO/NiCo LDH//rGO asymmetric supercapacitor device is constructed with an aqueous KOH electrolyte, which shows a capacitance of 244 F g–1 at a current density of 1 A g–1. The device attains the highest energy density of 34 Wh kg–1 and power density of 2500 W kg–1, with an excellent cycling stability of 100% after 3000 GCD cycles at a current density of 10 A g–1.

Assessing hierarchical corporate sustainability transition practices under uncertainty: An approach in the port and shipping industry in Southeast Asia

Assessing hierarchical corporate sustainability transition practices under uncertainty: An approach in the port and shipping industry in Southeast Asia

Interface Engineering on Amorphous/Crystalline Hydroxides/Sulfides Heterostructure Nanoarrays for Enhanced Solar Water Splitting

Developing highly efficient and stable noble-metal-free electrocatalysts for water splitting is critical for producing clean and sustainable energy. Here, we design a hierarchical transition metal hydroxide/sulfide (NiFe(OH)x-Ni3S2/NF) electrode with dual heterointerface coexistence using a cation exchange-induced surface reconfiguration strategy. The electrode exhibits superior electrocatalytic activities, achieving low overpotentials of 55 mV for hydrogen evolution and 182 mV for oxygen evolution at 10 mA cm-2. Furthermore, the assembled two-electrode system requires voltages as low as 1.55 and 1.62 V to deliver industrially relevant current densities of 500 and 1000 mA cm-2, respectively, with excellent durability for over 200 h, which is comparable to commercial electrolysis. Theoretical calculations reveal that the hierarchical heterostructure increases the electronic delocalization of the Fe and Ni catalytic centers, lowering the energy barrier of the rate-limiting step and promoting O2 desorption. Finally, by implementing the catalysts in a solar-driven water electrolysis system, we demonstrate a record and durable solar-to-hydrogen (STH) conversion efficiency of up to 20.05%. This work provides a promising strategy for developing low-cost and high-efficiency bifunctional catalysts for a large-scale solar-to-hydrogen generation.

Tuning the Architecture of Hierarchical Porous CoNiO2 Nanosheet for Enhanced Performance of Li-S Batteries

As the desired components and crystal structure of a transition metal oxide catalyst are selected, architecture is a dominating factor affecting its electrocatalytic performance for applications in lithium-sulfur (Li-S) batteries. Nano-compounds with a hollow architecture are undoubtedly the ideal catalysts for enhancing cathodic performance for more exposed active sites and shortened path lengths than are other architectures. Additionally, the internal stress in hollow architecture is favorable for further performance enhancement, due to its regulation effects of driving the d-band center of the transition metal in the active sites to migrate toward the Fermi level, which will promote the chemical adsorption and catalytic conversion of the polysulfides (PSs). To this point, we select hierarchical porous dual transition metal oxide CoNiO2 nano-boxes (CoNiO2(B)) as the conceptual model; meanwhile, CoNiO2 nano-flakes (CoNiO2(F)) with identical stoichiometry and crystal structure are also analyzed as a comparison. Li-S batteries based on CoNiO2(B) deliver superior energy storage features, including a reversible discharge capacity of 1232 mAh g−1 at 0.05 C and a stable cycle performance with decay rate of 0.1% each cycle even after 300 cycles at 1 C. This research presents an alternative scheme for booting the performance of Li-S batteries.

Novel properties of stearic acid / MXene - Graphene oxide shape - Stabilized phase change material: Ascended phase transition temperature and hierarchical transition

To develop novel materials with active photothermal conversion and latent heat storage properties, the shape-stabilized phase change material was fabricated via self-assembly method employing stearic acid (SA) as heat storage medium, graphene oxide and MXene as hybrid matrix (GM) and photothermal conversion agent. The morphology, structure, properties and phase transition behavior were investigated by various techniques. The results shown that SA/GM has sandwich-like structure and SA is confined between matrix nanosheets in multiple crystallographic forms. Due to the superposition of host-guest interaction and intrinsic van der Waals force between nanosheets, the primary melting temperature of SA/GM climbs 27.7 °C approximately than that of SA, as well as the total latent heat decreases by 28.55%–30.85% compared with theoretical values. The confinement effect increases the crystallization activation energy of confined SA, and leads to the formation of non-phase transition layer, transitional confinement layer and proximate bulk layer. However, the above observations are applicable to fatty acids/GM, only the suppressed latent heat occurs in all GM-matrix composites. In addition, the SA/GM7 has excellent photothermal conversion efficiency, as high as 93.34%, and gradient latent heat storage capacity, 45.96 J g −1 at 74.43 °C and 91.19 J g −1 at 92.37 °C. Stearic acid/Mxene - graphene oxide (SA/GM) composite was fabricated though self-assembly method. The confined SA exists in multiple crystal forms and shows abnormal properties, including the ascended phase transition temperature, hierarchical phase transition and the formation of confined layer. The SA/GM presents excellent photothermal conversion efficiency and gradient heat storage/release performances. • SA/GM was self-assembled by the inherent forces of matrix nanosheets and host-guest interaction. • The main melting temperature of SA/GM increased by 27.7 °C approximately. • Non-phase change layer, transitional confinement layer and proximate bulk layer exist in SA/GM. • Ascended transition temperature and stratified phase transition are applicable to fatty acids/GM.

Mixed metal chalcogenide shell with carbon interlayer wrapped 3D NiCo 2 O 4 nanowire array: A hierarchical self‐supported electrode for high‐performance supercapacitors

The expansion of effective mass transportation and elevated electrochemical active sites are cogent strategies for designing hierarchical transition metal-based heterostructures electrodes in high-performance supercapacitors (SC). In this study, we fabricated a self-supported three-dimensional NiCoO@NC@NiCoMnSe core-shell nanowire array (NWA) on Ni foam via a two-step hydrothermal process followed by vapor phase selenification. Fabricated multivalent heterostructured core-shell NWA electrode exhibited a high areal capacity of 0.761 mA h cm−2 (2.73 C cm−2) at 2 mA cm−2 with cyclic stability of 83.1% after 5000 cycles at 30 mA cm−2. An assembled HSC device with NiCoO@NC@NiCoMnSe as a positive electrode and activated carbon (AC) as a negative electrode exhibited a high volumetric energy density of 3.87 mWh cm−3 at a volumetric power density of 20.17 mW cm−3 and further retained an energy density of 2.64 Wh cm−3 at a power density of 201.5 W cm−3. The HSC exhibited high-capacity retention of 98.2% after 10 000 cycles at 6 mA cm−2, evidence of the high durability of the rich redox-active heterostructured electrode. A thin carbon nanowall between the core and shell interface enriched the mass electron transfer, reduced the impedance, and promoted its structural durability, resulting in the heterostructure's superior electrochemical performance.

Hierarchical Symbol Transition Entropy: A Novel Feature Extractor for Machinery Health Monitoring

This article develops a novel collaborative health monitoring framework based on hierarchical symbol transition entropy (HSTE) and 2-D-extreme learning machine (2-D-ELM) without fusion. In the proposed framework, a novel metric called symbol transition entropy (STE) is first presented to evaluate the dynamical complexity of time series through multistep transition and the joint probability distribution of the symbol state and its transition state. Compared with the existing entropy algorithms, STE has better robustness and captures more detailed dynamical changes. Subsequently, a new feature representation method called HSTE is proposed by combining STE with the hierarchical analysis. The two-order tensor features can be constructed for multichannel data by stacking HSTE values extracted from each single-channel data. Finally, 2-D-ELM is incorporated to identify the extracted two-order tensor features without vectorization. The feasibility of the proposed schemes is verified through simulation and experimental studies, and the final results confirm that the developed schemes have better performance than the existing entropy-based collaborative fault diagnosis methods.