Types Of Structures Research Articles

MFIB 2.0: a major update of the database of protein complexes formed by mutual folding of the constituting protein chains

Abstract While the majority of proteins with available structures are able to fold independently and mediate interactions only after acquiring their folded state, a subset of the known protein complexes contains protein chains that are intrinsically disordered in isolation. The Mutual Folding Induced by Binding (MFIB) database collects and classifies protein complexes, wherein all constituent protein chains would be unstable/disordered in isolation but fold into a well-defined 3D complex structure upon binding. This phenomenon is often termed as cooperative folding and binding or mutual synergistic folding (MSF). Here we present a major update to the database: we collected and annotated hundreds of new protein complexes fulfilling the criteria of MSF, leading to an almost six-fold increase in the size of the database. Many novel features have also been introduced, such as clustering of the complexes based on structural similarity and domain types, assigning different evidence levels to each entry and adding the evidence coverage label that allowed us to include complexes of multi(sub)domain monomers with partial MSF. The MFIB 2.0 database is available at https://mfib.pbrg.hu.

Modulating the Supramolecular Assembly of α‐Cyclodextrin and Anderson‐type Polyoxometalate through Covalent Modifications

A series of unprecedented supramolecular complexes of covalently modified Anderson‐type polyoxometalates (POMs) and α‐cyclodextrins (α‐CDs) have been obtained and characterized in solid state by single‐crystal X‐ray diffraction, and in aqueous solution using various techniques including 1H DOSY, 2D NOESY 1H NMR, isothermal titration calorimetry (ITC), and electrospray ionization time‐of‐flight mass spectroscopy (ESI‐TOF‐MS). It has been demonstrated that the supramolecular assembly process can be modulated by different covalent modification modes of the Anderson POMs, giving rise to a new type of POM/α‐CD complexes featuring organic‐inorganic pseudo‐rotaxane structures, which are in good contrast to those of POM/γ‐CD with poly‐rotaxane type structures. Moreover, it is delighted to find that these pseudo‐rotaxanes of POM/α‐CD exhibit stable chirality in aqueous solution, which has not been accomplished in previously reported POM/γ‐CD assemblies.

Molecular Dynamics Simulation of High-Energy Beam Irradiation of SiO2 Single Crystal with Three Different Morphologies: No Free Surface, with a Free Surface, and Thin Films

Abstract High-energy beam irradiation of SiO2 crystal with the α-quartz structure was simulated by the molecular dynamics method. Three types of specimen structures were examined: a single crystal without a free surface, a single crystal with a free surface, and a thin film. After structural relaxation at room temperature, a cylindrical region with diameter of 3.0 nm was set at the center of the specimen and high-thermal energy of Seff = 0.1 – 2.5 keV/nm was added to that region, where Seff is the effective stopping power. Atomic motions were calculated by the molecular dynamics method using the large-scale atomic/molecular massively parallel simulator). Vashishta potential was used for the atomic interaction. The single crystal structure of α-quartz without the free surface was stable up to 1.0 keV/nm and it gradually became amorphous with increasing thermal energy. In contrast, the single crystal structure with the free surface was stable up to 0.5 keV/nm and was amorphous at higher thermal energy. In particular, the atomic structures for the thermal impact Seff ≥ 2.0 keV/nm had a facet-like structure at the impacted surface, which corresponds to actual experimental results. Nano-hole formation was observed in the irradiation process of the film structure.

New Rotating Coanda-Type Intake for Sediment-Laden Flows

Tyrolean and Coanda types of water intake structures are the most widely preferred bottom intake structures used to divert water from channels and river systems for various purposes, such as energy production, irrigation, and domestic use. Estimation of withdrawal water and filtered sediment amounts are important to obtain maximum efficiency from an intake structure. The purpose of this study is to numerically analyze a newly patented rotating Coanda-type intake system to achieve the highest Water Capturing Efficiency and Sediment Release Efficiency. The newly patented system (patent no: TR2021011306B) presented here incorporates Coanda screens with variable void ratios, autonomously selected based on the sediment characteristics of the inflowing water. The process of selecting and positioning the appropriate screen is fully automated, requiring no human intervention. The system rotates under motor control to place the suitable screen at the operational point based on the specific requirements of the incoming sediment-laden flow.

Transverse impact performance and finite element analysis of three‐dimensional woven tubular composite with different structures

AbstractIn this study, experiments combined with finite element analysis (FEA) were used to investigate the damage mechanism of transverse low‐velocity impacts on three‐dimensional woven tubular composites (3DWTC) with different structures. The damage morphology after impact was observed using three‐dimensional microscopy and scanning electron microscopy (SEM). Three mesoscale models were constructed based on the authentic structure of the 3DWTC. Additionally, the effects of the structure type on the impact resistance, stress distribution, and damage morphology of 3DWTC were studied. The impact resistance of the shallow cross‐linked (SCL) structure is the strongest, according to the data. The structure that resists impacts the least is the through orthogonal (TO) structure. The greater the straightness of the warps in the structure, the density of the weft, the volume fraction, the greater the cooperative load‐bearing capacity of the warps and wefts, and the greater the speed of stress propagation. The wefts and inner warps are subjected to tensile forces at the point of impact. The outer warps are subjected to compression. The stresses are distributed in a cross‐shaped pattern. The straightened warp and weft parts of the TO serve a key structural role in load‐bearing. The warps of the SCL and the shallow‐crossed curved joint (SCCJ) structure serve a key structural role in the load‐bearing capacity.Highlights The impact resistance of 3DWTCs was investigated. The tensile state, stress propagation rate of 3DWTC were analyzed using the mesoscale model. The main load‐bearing components are revealed. The differences in damage morphology of 3DWTCs are analyzed.

How to combat stigma surrounding mental health disorders: a scoping review of the experiences of different stakeholders

BackgroundThe stigma associated with mental health disorders (MHDs) results in delayed help-seeking, limited access to health services, suboptimal treatment, poor treatment outcomes, and an increased risk of human rights violations. This scoping review summarizes qualitative research on the lived experiences of different stakeholders regarding strategies and interventions to combat stigma for people with MHDs.MethodsThis study was a six-step scoping review using the Arksey and O’Malley framework. A comprehensive search of the following electronic databases was conducted to identify relevant records: PubMed, Scopus, Web of Science (WoS) and Google Scholar, as well as a manual search of the reference lists. All steps, including screening of eligible studies, data extraction, and analysis, were performed independently by multiple reviewers, with disagreements resolved by discussion. The data were synthesized based on the for-content synthesis guidelines.ResultsA total of 25 studies were included in this review of the 32,976 initial identified citations. The included studies were from all countries (low, middle, and high income), stigmatized disorders (e.g., schizophrenia, bipolar disorder, etc.) and target populations (e.g., people with MHDs and their families, health care providers, the general community, and students and school members). The thematic synthesis revealed six types of interventions and strategies and 17 themes related to reducing stigma in patients on MHDs. Strategies and interventions were classified by patient (self-stigma), family (family stigma), healthcare professionals’ stigma, workplace stigma, public/societal stigma, and structural type of stigma (institutional stigma).ConclusionsThis review contributes new evidence that should be considered in future interventions and policies to reduce stigma against MHDs. Multilevel and multistakeholder strategies and interventions are needed to reduce the stigmatization of MHDs.

Three-Dimensional Multi-Morphology TPMS-based Microchannel Design Method for Conformal Cooling in Injection Molding

Abstract Triply Periodic Minimal Surface (TPMS) possesses diverse morphological characteristics, such as pore sizes, porosity, and structural types. Integrating TPMS-based microchannels into micro-cellular cooling structures is advantageous for designing and controlling fluid characteristics within mold cooling channels. However, it is still difficult to design multi-morphology TPMS-based cooling microchannels that conform to the external shapes of injection molds. This work proposes a three-dimensional (3D) multi-morphology TPMS-based design method that transforms 3D constraints into a combination of two-dimensional (2D) constraints. Firstly, a beta growth algorithm based on closed-loop constraints is proposed to transition different morphologies smoothly on the plane. Subsequently, a transition optimization algorithm along the normal direction of the plane is introduced to smoothly transition multi-morphology TPMS-based microchannels in all directions. With this layered approach, multi-morphology microchannels can be obtained with first-order geometric continuity under complex shape constraints. Finally, several TPMS-based conformal cooling structures are designed for an automotive hood cover. The results of finite element simulations show that the cooling structures generated by the proposed method have a better cooling effect than the conformal channels. It can be concluded that multi-morphology TPMS-based structures perform better by contrast with conformal channels as the average temperature of the cooling surface decreases by 8.48K, and the standard deviation of temperature distribution decreases by 24.65%.

Bi-Hamiltonian structures of KdV type, cyclic Frobenius algebrae and Monge metrics

Bi-Hamiltonian structures of KdV type, cyclic Frobenius algebrae and Monge metrics

Exact Solution of Nonlinear TVMD Based on Maximizing Damping Enhancement Effect for Acceleration Response Control

The tuned viscous mass damper (TVMD) exhibits superior control effect and energy dissipation capability compared to the viscous damper (VD). Existing literature has primarily focused on the control of absolute acceleration response in structures by linear TVMD, with limited research on nonlinear damper for controlling structural performance and efficiency. This study proposes a methodology for determining the optimal parameters of linear TVMD. The closed-form solution of the nonlinear TVMD is derived through stochastic linearization based on the linear TVMD, aiming to minimize the damping ratio of TVMD by utilizing the damping enhancement effect of the system. For lightly damped structures, the closed-form solution remains a reliable method for accurately approximating TVMD design parameters. Compared to traditional fixed-point theory, the TVMD developed through the proposed methodology demonstrates superior control performance, especially in terms of absolute acceleration, with a smaller damping ratio and minimal deviation from the optimal state of the TVMD. Additionally, the study investigated the influence of the damping index on the optimal parameters and the efficacy of the TVMD in vibration control. It is observed that reducing the damping index can improve structural performance, particularly in long-period structures with small mass ratios, although it may reduce the efficiency of the TVMD. Notably, TVMDs with lower mass ratios maintain higher efficiency regardless of the structural natural period. The damping index does not affect the optimal frequency ratio or the mean square response. Ultimately, the effectiveness of the optimization method is displayed through the examination of response spectra following 22 real seismic events, showcasing its adaptability across various structural types. The comparison of the root-mean-square response between the VD and TVMD across different periods shows consistent results, with a maximum deviation of only 10%. The TVMD is highly effective at regulating the structural response, excelling particularly in controlling the absolute acceleration of long-period structures, where it reduces maximum absolute acceleration by 20% more than displacement. The proposed exact solution rapidly and precisely identifies the design parameters of nonlinear TVMD by the desired specifications, thereby offering theoretical guidance for practical engineering applications.

High‐Entropy Electrolyte Driven by Multi‐Solvation Structures for Long‐Lifespan Aqueous Zinc Metal Pouch Cells

Aqueous zinc metal batteries (AZMBs) are promising for grid‐scale energy storage due to their low cost and high safety. However, poor stability and an unfavorable freezing point hinder their actual application. Herein, a ternary salts‐based high‐entropy electrolyte (HEE) composed of Zn0.2Na0.4Li0.4(ClO4)1.2·7H2O is proposed to address the above issues. The addition of perchlorate salts with different cations reduces the size of ion clusters, significantly increases the solvation structure species, and promotes the anion‐rich Zn2+ solvation structures, resulting in an enlarged electrochemical stability window, favorable viscosity and ionic conductivity, and low freezing point. Furthermore, characterization and calculations confirm that multiple types of solvation structures effectively increase the electrolyte entropy. As a consequence, the Zn/Zn symmetric cells in HEE can sustainably cycle for at least 1000 hours and 1500 hours under room and subzero temperatures, respectively. The Na0.33V2O5/Zn and polyaniline/Zn full cells can even last for 30000 and 20000 cycles without capacity decay at −20 °C, respectively. The pouch cells employing HEE deliver promising capacity and stability, even at high mass loading of active materials. This strategy of introducing multiple salts with different cations to construct a high‐entropy environment provides a facile approach for high‐performance and long‐lifespan AZMBs across a wide temperature range.

High-Entropy Electrolyte Driven by Multi-Solvation Structures for Long-Lifespan Aqueous Zinc Metal Pouch Cells.

Aqueous zinc metal batteries (AZMBs) are promising for grid-scale energy storage due to their low cost and high safety. However, poor stability and an unfavorable freezing point hinder their actual application. Herein, a ternary salts-based high-entropy electrolyte (HEE) composed of Zn0.2Na0.4Li0.4(ClO4)1.2·7H2O is proposed to address the above issues. The addition of perchlorate salts with different cations reduces the size of ion clusters, significantly increases the solvation structure species, and promotes the anion-rich Zn2+ solvation structures, resulting in an enlarged electrochemical stability window, favorable viscosity and ionic conductivity, and low freezing point. Furthermore, characterization and calculations confirm that multiple types of solvation structures effectively increase the electrolyte entropy. As a consequence, the Zn/Zn symmetric cells in HEE can sustainably cycle for at least 1000 hours and 1500 hours under room and subzero temperatures, respectively. The Na0.33V2O5/Zn and polyaniline/Zn full cells can even last for 30000 and 20000 cycles without capacity decay at -20 °C, respectively. The pouch cells employing HEE deliver promising capacity and stability, even at high mass loading of active materials. This strategy of introducing multiple salts with different cations to construct a high-entropy environment provides a facile approach for high-performance and long-lifespan AZMBs across a wide temperature range.

Applied Terminology in Geodiversity and Geotourism Activity: a Sustainable Conceptual Exercise

Objective: Promoting environmental geoconservation, geodiversity and local culture, namely through tours and research on these natural environments. Theoretical Framework:t is based on modern scientific definitions used in geology, geomorphology, paleontology, culture, and for educational and geotourism purposes. Method:The literature review significantly contextualizes knowledge and broad understanding, as well as advances in the terminology most used in geotourism activity, adopted in Brazil and European references in the area.It is representativeness: relational: the suitability of the geosite to illustrate a geological process or quality, which contributes significantly to the understanding of the theme, process, characteristic or (i) representation: geological context, (ii) integrity: related to the state of conservation of the geosite, (iii) rarity: number of geosites in the geological study area, (iv) scientific knowledge. Results and Discussion:The results obtained revealed that review studies of specialized terminology applied in geodiversity and geotourism activity play a significant role in interpreting unique places where most geosites occur, attracting tourists who are increasingly surprised by geosites. Research Implications:The implications of geosite terminology do not involve universal consensus, and there are several ways to scientifically describe a geosite in the context of a type of terrestrial relief, geological structure and minerals. This interpretation should be carried out by a specialist or by experienced people who are knowledgeable about the local specificities. Originality/Value: This study contributes to the literature by addressing the terminology applied in geoscience that encompasses geodiversity and geotourism. The relevance and potential of geosites for education, scientific interpretation of geological characteristics in the central region of the North Amazon.

Material Characteristics of Compressed Dry Masonry Made of Medium-Size Elements with Perlite Aggregate

Dry masonry is a type of construction that is nowadays used to a limited extent in the construction sector, including the housing sector. A lack of codified computational methods enabling engineers to design consciously is one of the factors limiting the development of dry walls. This article presents results from testing an innovative solution for dry masonry made of medium-size elements with expanded perlite aggregate. Material with this type of aggregate has a low bulk density (390 ± 10% kg/m3), which allows the production of large blocks and significantly reduces the value of the thermal conductivity coefficient λ = 0.084 ± 0.003 W/m·K. The results obtained were used to determine material parameters for designing a structure mainly exposed to vertical load. The important practical significance of the presented research results from the lack of provisions, specifications or standards allowing for the design, calculation and construction of dry masonry; it is not possible to analyse the behaviour of this type of structure and to design it consciously and safely. The presented research is therefore an important source of information on mechanical parameters essential for the design of structures and provides tools for this. As a result of the tests of nine panels, the mean compressive strength was determined (1.085 N/mm2), and then the procedure of “design assisted by testing” implemented into Eurocode was used to determine characteristic (0.873 N/mm2) and design compressive strength (0.565 N/mm2). Using the relationships σ-ε, an attempt was made to identify material models for the linear and non-linear analysis of the structure and for designing cross-sections. The material models were made considering increased non-linear deformations of a structure under low stresses (the compression toe) which are true and typical for dry masonry. A specific deformability of dry masonry, slightly different to that in masonry structures joined together with mortar, also affects the reduction factor for load-bearing capacity due to second-order effects. Reduction factors determined from true non-linear deformations were lower than the values specified by EC6 for masonry structures.

First-principles study of three isomers of the large birefringent crystal Sb4O4F4: electronic structure and optical properties

Abstract Metal oxyhalides are becoming an important branch of birefringent materials due to their rich structural types and stable physicochemical properties. In this work, the electronic structure and optical properties of three isomers of antimony oxyhalide Sb4O4F4 (I- Pna21), Sb4O4F4 (II- Pnma), and Sb4O4F4 (III- Pnma) were investigated using first principles method. The results show that the band gap of I and III reaches the ultraviolet region (4.10 eV (I), 3.87 eV (III)). In particular, I-III all exhibit large birefringence of 0.138–0.266 at 1064 nm. Born effective charge, and project density of states results show that the large birefringence of I-III is mainly from the stereochemically active lone pair electrons around the antimony atom. Finally, structure–property relationship studies of I-III indicate that different Sb-F bonds and structural arrangements are the key factors leading to the significantly different properties.

A novel computer vision and point cloud-based approach for accurate structural analysis of a tall irregular timber structure

Wood material has been widely used in critical heritage structures such as pagodas, totem poles, and large-scale sculptures. Conducting rigorous structural analysis is crucial to protect these structures in high-seismic regions. Typically, these types of structures are modeled using an equivalent finite element (FE) model which used simple cylinder with a constant cross section equal to the average of the top and bottom cross section. However, simplified equivalent FE models may not accurately consider the irregularities and complexities of these structures. In this paper, advanced computer vision and point cloud techniques were adopted to accurately and rapidly construct a FE model of a 30-meter irregular timber sculpture. This was achieved using video scans, computer vision-aided 3D reconstruction, point cloud processing, and mesh to solid element conversion. The refined FE model was used to conduct capacity check, mesh sensitivity study, pushover analysis, and response spectrum analysis. The results of the refined FE model were compared to an equivalent FE model. The results show: 1) the proposed numerical modeling methodology for structural analysis can efficiently and accurately measure the dimension of the irregular sculpture up to 98.2 % accuracy; 2) the lateral stiffnesses of the 30-meter irregular sculpture vary significantly (up to 42.6 %) from one direction to the other; 3) the equivalent FE model overestimated the shear and moment capacities by 20.6 % and 13.2 %, respectively; 4) on average, the equivalent FE model overestimated the shear and moment demands by 8.9 % and 5.5 %, respectively. Overall, the proposed application has demonstrated a fast, economical and accurate method to conduct seismic evaluation and design for irregular structures.

Atypical phase transition, twinning and ferroelastic domain structure in bis(ethylenediammonium) tetrabromozincate(II) bromide, [NH3(CH2)2NH3]2[ZnBr4]Br2

Single-crystal growth, differential thermal analysis (DTA), derivative thermogravimetry (DTG), differential scanning calorimetry (DSC), X-ray structural studies and polarized microscopy observations of bis(ethylenediammonium) tetrabromozincate(II) bromide [NH3(CH2)2NH3]2[ZnBr4]Br2 are presented. A reversible phase transition is described. At room temperature, the complex crystallizes in the monoclinic system. In some cases, the single crystals are twinned into two or more large domains of ferroelastic type with domain walls in the (100) crystallographic plane. DTA and DTG measurements show chemical stability of the crystal up to ∼538 K. In the DSC studies, a reversible isostructural phase transition was revealed at ∼526/522 K on heating/cooling run, respectively. Optical observation on the heating run reveals that at the phase transition the plane of twinning (domain wall) does not disappear and additionally the appearance of a new domain structure of ferroelastic type with domain walls in the planes (101), (101), (100) and (001) is observed. The domain structure pattern is preserved after cooling to the room-temperature phase and the symmetry of this phase is unchanged.

Structural, electronic, optical, mechanical, and phononic bulk properties of tetragonal trirutile antimonate MSb2O6 (M = Fe, Co, Ni, or Zn): a first-principles prediction for optoelectronic applications

Abstract This work does an extensive analysis of the optoelectronic and mechanical properties of the tri rutile structure type 3d transition metal Antimonate MSb2O6 (M = Fe, Co, Ni, or Zn), for the first time, utilizing the pseudo-potential plane wave approach within the density functional theory framework. When calculating the structural, optical, and mechanical properties, the exchange–correlation interactions were studied using the GGA-PBE functional, whereas when computing electronic, it is analyzed using the HSE06 hybrid functional. The equilibrium lattice parameters exhibit good agreement with the available experimental results. The electronic properties were estimated using the GGA-PBE and HSE06 functionals. Based on the calculated electronic properties with the GGA-PBE functional, the FeSb2O6, CoSb2O6, and NiSb2O6 materials exhibit metallic behavior with energy gap values of 0 eV, while ZnSb2O6 is a semiconductor with a narrow direct band gap (Γ–Γ) of 0.5 eV. Furthermore, the computed band gaps using the HSE06 functional are 0 eV, 0 eV, 1 eV (Γ–Γ), and 4 eV (Γ–Γ) for FeSb2O6, CoSb2O6, NiSb2O6, and ZnSb2O6, respectively. Density of states diagrams were used to gain deeper insights into the characteristics of the energy bands. The optical properties of these compounds, such as the dielectric function, energy loss function, conductivity, reflectivity, refractive index, and absorption coefficient were investigated over the energy range of 0 to 40 eV. The materials exhibited a high absorption coefficient and a significantly low reflectivity within the UV–vis energy spectrum. The negative cohesive energy Ecoh implies the chemical (thermodynamic) stability of the trirutile MSb2O6 (M = Fe, Co, Ni, or Zn). Mechanical stability is confirmed by applying the Born stability criteria using elastic constants (Cij). The absence of imaginary frequencies in the phonon spectrum calculations confirms the dynamic stability of the studied compounds. These results are consistent with previous experimental research on these materials in photocatalysis and gas sensor applications. On the other hand, these compounds possess exceptional high and broad optical absorption UV range, making them suitable for use in next-generation ultraviolet photodetectors.

Hydrodynamic analysis of a round-ended caisson in submerging

ABSTRACT Flow around a round-ended caisson in the submerging process is investigated using the k − ω SST turbulent model and the volume of fluid (VOF) method in the present study. Six types of vortex structures around the round-ended caisson are identified, i.e. the neck vortex in front of the cylinder near the free surface, the attached vortex below the free end surface, the tip vortex originating from the free end, the spanwise vortex on the side surface, the U-shaped vortex behind the cylinder and the trailing vortex in the wake. The influence of the free surface, free end and bottom boundary on vortexes are analysed. The physical generation mechanism of vorticity on the free end surface is discussed. Additionally, the drag coefficient increases with Fr when Fr < 1 and decreases with Fr when 1 ≤ Fr ≤ 1.8, and the run-up of the free surface increases with increasing Fr.

Humidity Sensing Using Polymers: A Critical Review of Current Technologies and Emerging Trends

In the post-pandemic era, human demand for a healthy lifestyle and a smart society has surged, leading to vibrant growth in the field of flexible electronic sensor technology for health monitoring. Flexible polymer humidity sensors are not only capable of the real-time monitoring of human respiration and skin moisture information but also serve as a non-contact human–machine interaction method. In addition, the development of moist-electric generation technology is expected to break free from the traditional reliance of flexible electronic devices on power equipment, which is of significant importance for the miniaturization, reliability, and environmentally friendly development of flexible devices. Currently, flexible polymer humidity sensors are playing a significant role in the field of wearable electronic devices and thus have attracted considerable attention. This review begins by introducing the structural types and working principles of various humidity sensors, including the types of capacitive, impedance/resistive, frequency-based, fiber optic, and voltage-based sensors. It mainly focuses on the latest research advancements in flexible polymer humidity sensors, particularly in the modification of humidity-sensitive materials, sensor fabrication, and hygrosensitivity mechanisms. Studies on material composites including different types of polymers, polymers combined with porous nanostructured materials, polymers combined with metal oxides, and two-dimensional materials are reviewed, along with a comparative summary of the fabrication and performance mechanisms of related devices. This paper concludes with a discussion on the current challenges and opportunities faced by flexible polymer humidity sensors, providing new research perspectives for their future development.

Complete conditional type structures

Hierarchies of conditional beliefs (Battigalli and Siniscalchi 1999) play a central role for the epistemic analysis of solution concepts in sequential games. They are modelled by type structures, which allow the analyst to represent the players' hierarchies without specifying an infinite sequence of conditional beliefs. Here, we study type structures that satisfy a “richness” property, called completeness. Friedenberg (2010) shows that, under specific conditions, a complete type structure with ordinary beliefs represents all hierarchies. This paper shows that Friedenberg's result can be extended to type structures with conditional beliefs. As an ancillary result of independent interest, we provide a construction of the “canonical” space of hierarchies of conditional beliefs which generalizes the one in Battigalli and Siniscalchi (1999).