Symmetry, Volume 16, Issue 4 (April 2024) – 126 articles

Since the topological properties of a b-metric space (which generalizes the concept of a metric space) seem sometimes counterintuitive due to the fact that the “open” balls may not be open sets, we review some aspects of these spaces concerning compactness, metrizability, continuity and fixed points. After doing so, we introduce new types of contractivities that extend the concept of Banach contraction. We study some of their properties, giving sufficient conditions for the existence of fixed points and common fixed points. Afterwards, we consider some iterative schemes in quasi-normed spaces for the approximation of these critical points, analyzing their convergence and stability. We apply these concepts to the resolution of a model of integral equation of Urysohn type. In the last part of the paper, we refine some results about partial contractivities in the case where the underlying set is a strong b-metric space, and we establish some relations between mutual weak contractions and quasi-contractions and the new type of contractivity. Full article

This paper proposes a novel route optimization framework to solve the problem of instant pick-up and delivery for e-grocery orders. The proposed framework extends the traditional time-windowed package delivery problem. We demonstrate the effectiveness of our approach for this integrated problem using actual delivery data from HepsiJet, a leading e-commerce logistics provider in Turkey. We first employ several machine learning algorithms and simulations to investigate the capacity of the courier. Subsequently, a dynamic route planning workflow is executed with a highly specialized and novel routing algorithm. Our proposed heuristic approach considers combined fleet operations for delivering regular packages originating from a central depot and dynamic e-grocery orders picked up at local supermarkets and delivered to the customers. The heuristic algorithm constitutes k-opt and node transfer operation variations customized for this integrated problem. We report the performance of our approach in problem instances from the literature and instances from HepsiJet’s daily operations, which we also publicly share as new route optimization problem instances. Our results suggest that, despite the more complex nature of the integrated problem, our proposed algorithm and solution framework produce more efficient and cost-effective solutions that offer additional business opportunities for companies such as HepsiJet. The computational analyses reveal that implementing our proposed approach yields significant efficiency gains and cost reductions for the company, with a distance reduction of over 30%, underscoring our approach’s effectiveness in achieving substantial cost savings and enhanced efficiency through integrating two distinct delivery operations. Full article

$U\left(n\right)$ is a semi-direct product group characterized by nontrivial homomorphisms mapping $U\left(1\right)$ into the automorphism group of $SU\left(n\right)$. For $U\left(3\right)$, there are three nontrivial homomorphisms that induce three separate defining representations. In a toy model of $U\left(3\right)$ Yang–Mills (endowed with a suitable inner product) coupled to massive fermions, this renders three distinct covariant derivatives acting on a single matter field. Employing a $\mathrm{mod}3$ permutation induced by a large gauge transformation acting on the defining representation vector space, the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields possessing the same $U\left(3\right)$ quantum numbers. One can interpret this as three species of matter fields in the defining representation. Full article

Researchers in cognitive science have long been interested in modeling human perception using statistical methods. This requires maneuvers because these multiple dimensional data are always intertwined with complex inner structures. The previous studies in cognitive sciences commonly applied principal component analysis (PCA) to truncate data dimensions when dealing with data with multiple dimensions. This is not necessarily because of its merit in terms of mathematical algorithm, but partly because it is easy to conduct with commonly accessible statistical software. On the other hand, dimension reduction might not be the best analysis when modeling data with no more than 20 dimensions. Using state-of-the-art techniques, researchers in various research disciplines (e.g., computer vision) classified data with more than hundreds of dimensions with neural networks and revealed the inner structure of the data. Therefore, it might be more sophisticated to process human perception data directly with neural networks. In this paper, we introduce the multi-dimensional data analysis platform (MuDAP), a powerful toolbox for data analysis in cognitive science. It utilizes artificial intelligence as well as network analysis, an analysis method that takes advantage of data symmetry. With the graphic user interface, a researcher, with or without previous experience, could analyze multiple dimensional data with great ease. Full article

In this paper, we introduce the novel concept of generalized distance denoted as $J_{\theta }$ and call it an extended b-generalized pseudo-distance. With the help of this generalized distance, we define a generalized point to set distance $J_{\theta }(\mathfrak{u},{\mathcal{H}}^{★}),$ a generalized Hausdorff type distance and a $P^{J_{\theta }}$-property of a pair $({\mathcal{H}}^{★},{\mathcal{K}}^{★})$ of nonempty subsets of extended b-metric space $({\mathfrak{U}}^{★},{\rho }_{\theta }).$ Additionally, we establish several best proximity point theorems for multi-valued contraction mappings of Nadler type defined on b-metric spaces and extended b-metric spaces. Our findings generalize numerous existing results found in the literature. To substantiate the introduced notion and validate our main results, we provide some concrete examples. Full article

This work presents a novel investigation that utilizes the integral operator $I_{p,\lambda }^n$ in the field of geometric function theory, with a specific focus on sandwich theorems. We obtained findings about the differential subordination and superordination of a novel formula for a generalized integral operator. Additionally, certain sandwich theorems were discovered. Full article

Taiwan’s auditors have suffered from processing excessive audit data, including drawing audit evidence. This study advances sampling techniques by integrating machine learning with sampling. This machine learning integration helps avoid sampling bias, keep randomness and variability, and target risker samples. We first classify data using a Naive Bayes classifier into some classes. Next, a user-based, item-based, or hybrid approach is employed to draw audit evidence. The representativeness index is the primary metric for measuring its representativeness. The user-based approach samples data symmetrically around the median of a class as audit evidence. It may be equivalent to a combination of monetary and variable samplings. The item-based approach represents asymmetric sampling based on posterior probabilities for obtaining risky samples as audit evidence. It may be identical to a combination of non-statistical and monetary samplings. Auditors can hybridize those user-based and item-based approaches to balance representativeness and riskiness in selecting audit evidence. Three experiments show that sampling using machine learning integration has the benefits of drawing unbiased samples; handling complex patterns, correlations, and unstructured data; and improving efficiency in sampling big data. However, the limitations are the classification accuracy output by machine learning algorithms and the range of prior probabilities. Full article

The aim of this paper is to study the coupled fixed point of a class of mixed monotone operators in the setting of a subordinate semimetric space. Using the symmetry between the subordinate semimetric space and a JS-space, we generalize the results of Senapati and Dey on JS-spaces. In this paper, we obtain some coupled fixed point results and support them with some examples. Full article

The aim of this work is to investigate osculating type ruled surfaces with a type 2-Bishop frame in $E^3$. We accomplish this by employing the symmetry of osculating curves. We examine osculating type ruled surfaces by taking into account the curvatures of the base curve. We investigate the geometric properties of these surfaces, focusing on their cylindrical and developable characteristics. Moreover, we calculate the Gaussian and mean curvatures and provide the requirements for the surface to be flat and minimal. We determine the requirements for the curves lying on this surface to be geodesic, asymptotic curves, or lines of curvature. Furthermore, relations between osculating type ruled surfaces with central tangent and central normal vectors are given. Finally, some examples of these surfaces are presented. Full article

Let $D=\left(V\right(D),A(D\left)\right)$ be a digraph of order n and let $r\in S\subseteq V\left(D\right)$ with $2\le \left|S\right|\le n$. A directed $(S,r)$-Steiner path (or an $(S,r)$-path for short) is a directed path P beginning at r such that $S\subseteq V\left(P\right)$. Arc-disjoint between two $(S,r)$-paths is characterized by the absence of common arcs. Let ${\lambda }_{S,r}^p\left(D\right)$ be the maximum number of arc-disjoint $(S,r)$-paths in D. The directed path k-arc-connectivity of D is defined as ${\lambda }_k^p\left(D\right)=min\{{\lambda }_{S,r}^p\left(D\right)\mid S\subseteq V\left(D\right),\left|S\right|=k,r\in S\}.$ In this paper, we shall investigate the directed path 3-arc-connectivity of Cartesian product ${\lambda }_3^p(G\square H)$ and prove that if G and H are two digraphs such that ${\delta }^0\left(G\right)\ge 4$, ${\delta }^0\left(H\right)\ge 4$, and $\kappa \left(G\right)\ge 2$, $\kappa \left(H\right)\ge 2$, then ${\lambda }_3^p(G\square H)\ge min\left\{2\kappa \left(G\right),2\kappa \left(H\right)\right\};$ moreover, this bound is sharp. We also obtain exact values for ${\lambda }_3^p(G\square H)$ for some digraph classes G and H, and most of these digraphs are symmetric. Full article

For radially extended Bessel modes, the helicity density distributions of magnetic fields associated with relativistic electron vortex beams are investigated for first time in the literature. The form of the distribution is defined by the electron beam’s cylindrically symmetric density flux, which varies with the winding number ℓ and the electron spin. Different helicity distributions are obtained for different signs of the winding number $\pm \ell$, confirming the chiral nature of the magnetic fields associated with the electron vortex beam. The total current helicity for the spin-down state is smaller than that of the spin-up state. The different fields and helicities associated with opposite winding numbers and/or spin values will play an important role in the investigation of the interaction of relativistic electron vortices with matter and especially chiral matter. A comparison of the calculated quantities with the corresponding ones in the case of non-relativistic spin-polarized electron beams is performed. Full article

A long-standing conjecture due to R. Fox states that the coefficients of the Alexander polynomial of an alternating knot exhibit a trapezoidal pattern. In other words, these coefficients increase, stabilize, then decrease in a symmetric way. A stronger version of this conjecture states that these coefficients form a log-concave sequence. This conjecture has been recently highlighted by J. Huh as one of the most interesting problems on log-concavity of sequences. In this expository paper, we shall review the various versions of the conjecture, highlight settled cases and outline some future directions. Full article

In recent years, with the rapid development of deep learning technology, a large number of excellent convolutional neural networks (CNNs) have been proposed, many of which are based on improvements to classical methods. Based on the Inception family of methods, depthwise separable convolution was applied to Xception to achieve lightweighting, and Inception-ResNet introduces residual connections to accelerate model convergence. However, existing improvements for the Inception module often neglect further enhancement of its receptive field, while increasing the receptive field of CNNs has been widely studied and proven to be effective in improving classification performance. Motivated by this fact, three effective expansion modules are proposed in this paper. The first expansion module, Inception expand (Inception-e) module, is proposed to improve the classification accuracy by concatenating more and deeper convolutional branches. To reduce the number of parameters for Inception e, this paper proposes a second expansion module—Equivalent Inception-e (Eception) module, which is equivalent to Inception-e in terms of feature extraction capability, but which suppresses the growth of the parameter quantity brought by the expansion by effectively reducing the redundant convolutional layers; on the basis of Eception, this paper proposes a third expansion module—Lightweight Eception (Lception) module, which crosses depthwise convolution with ordinary convolution to further effectively reduce the number of parameters. The three proposed modules have been validated on the Cifar10 dataset. The experimental results show that all these extensions are effective in improving the classification accuracy of the models, and the most significant effect is the Lception module, where Lception (rank = 4) on the Cifar10 dataset improves the accuracy by 1.5% compared to the baseline model (Inception module A) by using only 0.15 M more parameters. Full article

Thermo-oxidative stability testing plays a critical role in accurately predicting shelf life. These tests are performed in real time and under stress conditions, where degradation processes are accelerated by increasing storage conditions. In this study, high-performance liquid chromatography (HPLC) analyses were performed to evaluate the degradation of resveratrol in nutraceutical tablets as a function of time under different storage conditions in terms of temperature and relative humidity (RH), namely 25 °C/60% RH, 30 °C/65% RH, and 40 °C/75% RH. The latter is an accelerated test and is used to estimate shelf life for long-term storage. Resveratrol is present in both pure form and as a solid dispersion on magnesium dihydroxide microparticles (Resv@MDH). Degradation kinetic constants were determined at 25 °C, 30 °C, and 40 °C, and the Arrhenius behavior of the kinetic constants as a function of temperature was verified. The main results of this work are as follows: (i) the stability of resveratrol in nutraceutical tablets is affected by temperature; (ii) the dependence of the kinetic constants on temperature does not follow the Arrhenius equation, determining an overestimation of the degradation rate at 25 °C; in this regard a modified version of the Arrhenius equation that takes into account the deviation from linearity has been used to estimate the dependence of the kinetic constant on the temperature. These results suggest that accelerated testing does not provide a general model for predicting the shelf life of foods and dietary supplements. The reason may be due to possible matrix effects that result in different degradation mechanisms depending on the temperature. In this regard, symmetry relationships in the kinetics of chemical reactions resulting from microscopic reversibility and their relationship to the deviation from the Arrhenius equation are discussed. However, further research is needed to characterize the degradation mechanisms at different temperatures. The results of these studies would allow accurate prediction of food degradation to improve food safety and risk management and reduce food waste. In addition, knowledge of stability processes is necessary to ensure the maintenance of physiological processes by dietary supplements. Full article

This paper introduces a multidisciplinary conceptual perspective encompassing artificial intelligence (AI), artificial general intelligence (AGI), and cybernetics, framed within what we call the formalism of generalized neuromorphism. Drawing from recent advancements in computing, such as neuromorphic computing and spiking neural networks, as well as principles from the theory of open dynamical systems and stochastic classical and quantum dynamics, this formalism is tailored to model generic networks comprising abstract processing events. A pivotal aspect of our approach is the incorporation of the memory space and the intrinsic non-Markovian nature of the abstract generalized neuromorphic system. We envision future computations taking place within an expanded space (memory space) and leveraging memory states. Positioned at a high abstract level, generalized neuromorphism facilitates multidisciplinary applications across various approaches within the AI community. Full article

In this paper, we investigate the necessary and sufficient conditions for solving a dual split quaternion matrix equation $AXB = C$, and present the general solution expression when the solvability conditions are met. As an application, we delve into the necessary and sufficient conditions for the existence of a Hermitian solution to this equation by using a newly defined real representation method. Furthermore, we obtain the solutions for the dual split quaternion matrix equations $AX = C$ and $XB = C$. Finally, we provide a numerical example to demonstrate the findings of this paper. Full article

Background: The midface plays an important role in the judgment of symmetry. However, studies on three-dimensional analyses of midface asymmetry are limited. This study investigated the characteristics of midface asymmetry in skeletal Class III malocclusion patients through three-dimensional analysis. Methods: Sixty-eight adult subjects with skeletal Class III malocclusion were included and divided into mandible symmetry and asymmetry groups. The prevalence of recognizable malar asymmetry and the deviation of anterior nasal spine (ANS) were examined. The relation between midface and mandible asymmetry were investigated with Spearman correlation. The difference in distance of landmarks to reference planes were compared between the two groups using Mann–Whitney U test (p < 0.05). Results: The overall prevalence of malar asymmetry was 7.35% and of ANS deviation was 38.24%. In subjects with chin deviated to the right, there was a moderate negative correlation between chin deviation and difference of zygion and zygomatic process to mid-sagittal plane. The absolute value of difference in the glenoid fossa was significantly greater in female asymmetry subjects. Conclusions: The prevalence of midface asymmetry is not low. The more severely the chin is shifted, the greater asymmetrical position of the zygoma and glenoid fossa was found. Therefore, pre-surgical case-by-case evaluation of the midface region is essential for understanding the midface skeletal characteristics of Class III patients with chin deviation, thereby providing patients with realistic expectations and optimizing surgical outcomes and patient satisfaction. Full article

In the recent literature, various aspects of the transfer of quantum states by spin chains have been thoroughly investigated. Part of the existing study is devoted to the problem of optimal control, with the goal of achieving a highly reliable information/state transfer for a given time T. In general, achieving this goal is not an easy task in the case of (open) quantum systems. Various approaches have been developed and applied, including Krotov’s method to study the problem. It is a gradient-based method used here to study the problem of state transfer control in central spin models. Our results show that with Krotov’s method, it is possible to find an optimal control form that allows for very-high-fidelity state transfer in the central spin models we have developed. Our results will be of interest for a better understanding of the non-trivial effects of the classical world on the quantum world, which have been discussed in the form of various new effects, including the Epstein effect, in the recent literature. Full article

The reliability of circuit systems is primarily affected by cascading failures due to their complex structural and functional coupling. Causes of cascading failure during circuit operation include the continuous degradation process of components and external random shocks. Circuit systems can exhibit asymmetric structural changes and functional loss during cascading failure propagation due to the coupling of degradation and shock and their uncertainty effects. To tackle this issue, this paper abstracts the circuit into an impedance network and constructs a component failure behavior model that considers the correlation between degradation and shock. The interactions between soft and hard failure processes among different components are discussed. Two types of cascading failure propagation processes are described: slow propagation associated with continuous degradation and damage shock, and fast propagation due to fatal shock. Based on this, a cascading failure simulation algorithm is developed. This article presents a case study to demonstrate the proposed models and to analyze the reliability of a typical circuit system. Full article

Due to the operation of live-line working robots at elevated heights for precision tasks, a suitable visual assistance system is essential to determine the position and distance of the robotic arm or gripper relative to the target object. In this study, we propose a method for distance measurement in live-line working robots by integrating the YOLOv5 algorithm with binocular stereo vision. The camera’s intrinsic and extrinsic parameters, as well as distortion coefficients, are obtained using the Zhang Zhengyou calibration method. Subsequently, stereo rectification is performed on the images to establish a standardized binocular stereovision model. The Census and Sum of Absolute Differences (SAD) fused stereo matching algorithm is applied to compute the disparity map. We train a dataset of transmission line bolts within the YOLO framework to derive the optimal model. The identified bolts are framed, and the depth distance of the target is ultimately calculated. And through the experimental verification of the bolt positioning, the results show that the method can achieve a relative error of 1% in the proximity of positioning. This approach provides real-time and accurate environmental perception for symmetrical structural live-line working robots, enhancing the stability of these robots. Full article

There are deep historical connections between symmetry, harmonic functions, and stratified sets. In this article, we prove an analog of the removable singularity theorem for bounded harmonic functions on stratified sets. The harmonic functions are understood in the sense of the soft Laplacian. The result can become one of the main technical components for extending the well-known Poincaré–Perron’s method of proving the solvability of the Dirichlet problem for the soft Laplacian. Full article

Presently, active detectors are widely used to detect mines, providing high accuracy. However, the principle of the operation of active detectors can lead to the explosion of hidden mines. The novelty of this work is the development of the morphology of a neural network for the classification of mines made of different materials (metallic, semi-metallic, plastic) with high accuracy (99.23%), based on a vector of input features with the following components: the value of the output voltage of the FLC-100 magnetic field sensor, which measures magnetic field anomalies in the vicinity of mines with an accuracy of 10⁻¹⁰–10⁻⁴ Tesla; six different soil types, depending on the humidity; and the height at which the magnetic field sensor is located above the mine. Due to the fact that mines, when made of different materials (metallic, semi-metallic, plastic), have different magnetic properties, the neural network method of mine classification, based on the sensor data regarding anomalies of the magnetic field in the vicinity of mines, allows the classification of mines made of different materials. The accuracy of mine classification was assessed with two-layer and three-layer neural networks on various metrics (confusion matrix, ROC curves, accuracy–loss curves), using ADAM, RMSprop, and SGD optimisers, and analyses and comparisons were then carried out. The impact of asymmetry in the neuron number and the types of activation functions in the first and second hidden layers on the values of the accuracy and loss metrics was studied. In particular, it was established that the asymmetry of the number of neurons in the first and second hidden layers relative to the plane of symmetry between the hidden layers has a significant effect on the accuracy of the model (decrease in accuracy by 25%), while the loss function, when the symmetry of the neurons number in the hidden layers is violated, increases to a maximum of 50%. Full article

The content of this study combines city safety, optimum excavation situation, mining, geology, and civil engineering principles. Tunnel boring machines (TBM) are the most commonly used machines in the excavation of urban tunnels. These machines prevent the inward movement of the tunnel face and control the amount of settlement formed on the ground by applying pressure to the tunnel face. The most important question here is to determine the amount of pressure to be applied to the tunnel face. There are many widely accepted formulas used in the calculation of the face pressure and these formulas generally attempt to limit the settlements on the ground by using parameters such as groundwater level, overburden thickness, physical and mechanical properties of the surrounding rocks, etc. In this study, a new formula was developed. This new formula calculates the face pressure required to be applied by EPB to the tunnel face in order to prevent damage to a structure located on the route and within the area to be affected by tunnel excavation, instead of only preventing settlements on the surface. In the formula, produced within the scope of this study, in addition to other studies, 3D distances of the structure to which the deformation limitation will be made to prevent damage is also one of the parameters. Full article

In this paper, we obtain an equivalent condition to the existence of a common fixed point of a given family of nonexpansive mappings defined on a Hadamard space. Moreover, if the space is bounded, we show that the generating process of the approximate sequence by a specific projection method will stop in finite steps if there is no common fixed point. It is a significant advantage to reveal the nonexistence of a common fixed point in a finite time. Full article

The objective of this work is to derive the structure of Minkowski spacetime using a Hermitian spin basis. This Hermitian spin basis is analogous to the Pauli spin basis. The derived Minkowski metric is then employed to obtain the corresponding Lorentz factors, potential Lie algebra, effects on gamma matrices and complex representations of relativistic time dilation and length contraction. The main results, a discussion of the potential applications and future research directions are provided. Full article

In this research article, we propose a new matrix iterative method with a convergence order of five for computing the sign of a complex matrix by examining the different patterns and symmetry of existing methods. Analysis of the convergence of the method was explored on a global scale, and attraction basins were demonstrated. In addition to this, the asymptotic stability of the scheme was explored.Then, an algorithm for determing thegeneralized eigenvalues for the case of regular matrix pencils was investigated using the matrix sign computation. We performed a series of numerical experiments using numerous matrices to confirm the usefulness and superiority of the proposed method. Full article

Establishing a scientifically effective systemic financial risk early warning model is of great significance for prudently mitigating systemic financial risks and enhancing the efficiency of financial supervision. Based on the measurement of systemic financial risk and the network sentiment index of 47 financial institutions, this study adopted the “decomposition–reconstruction–integration” approach, utilizing techniques such as extreme-point symmetric empirical mode decomposition (ESMD), empirical mode decomposition (EMD), variational mode decomposition (VMD), hierarchical clustering, fast independent component analysis (FastICA), attention mechanism, bidirectional long short-term memory neural network (BiLSTM), support vector regression (SVR), and their combination, to construct a systemic financial risk prediction model. The empirical results demonstrate that decomposing and reconstructing relevant indicators before predicting systemic financial risks can enhance prediction accuracy. Among the proposed models, the ESMD-HFastICA-BiLSTM-Attention model exhibits superior performance in systemic financial risk early warning. Full article

With the increasing expansion of power systems, there is a growing trend towards active distribution networks for decentralized power generation and energy management. However, the instability of distributed renewable energy introduces complexity to power system operation. The active symmetry and balance of power systems are becoming increasingly important. This paper focuses on the characteristics of distributed resources and under-frequency load shedding, and a coordinated operation and control strategy based on the rapid adjustment of energy storage power is proposed. The characteristics of various controllable resources are analyzed to explore the rapid response capabilities of energy storage. The energy storage types are categorized based on the support time, and the final decision is achieved with power allocation and adjustment control of the energy storage system. Additionally, a comprehensive control strategy for under-frequency load shedding and hierarchical systems is provided for scenarios with insufficient active support. The feasibility of the proposed model and methods is verified via a multi-energy system case. Full article

The processing and comprehension of numerical information in natural language represent pivotal focal points of scholarly inquiry. Across diverse applications spanning text analysis to information retrieval, the adept management and understanding of the numerical content within natural language are indispensable in achieving task success. Specialized encoding and embedding techniques tailored to numerical data offer an avenue toward improved performance in tasks involving masked prediction and numerical reasoning, inherently characterized by numerical values. Consequently, treating numbers in text merely as words is inadequate; their numerical semantics must be underscored. Recent years have witnessed the emergence of a range of specific encoding methodologies designed explicitly for numerical content, demonstrating promising outcomes. We observe similarities between the Transformer architecture and CPU architecture, with symmetry playing a crucial role. In light of this observation and drawing inspiration from computer system theory, we introduce a floating-point representation and devise a corresponding embedding module. The numerical representations correspond one-to-one with their semantic vector values, rendering both symmetric regarding intermediate transformation methods. Our proposed methodology facilitates the more comprehensive encoding and embedding of numerical information within a predefined precision range, thereby ensuring a distinctive encoding representation for each numerical entity. Rigorous testing on multiple encoder-only models and datasets yielded results that stand out in terms of competitiveness. In comparison to the default embedding methods employed by models, our approach achieved an improvement of approximately $3.8\%$ in Top-1 accuracy and a reduction in perplexity of approximately $0.43$. These outcomes affirm the efficacy of our proposed method. Furthermore, the enrichment of numerical semantics through a more comprehensive embedding contributes to the augmentation of the model’s capacity for semantic understanding. Full article

Until now, the comprehensive structural analysis of single crystals of zeolite ECR-1, an aluminosilicate with the EON topology, has been hindered owing to the submicron dimensions of the obtained crystals. Additionally, this zeolite, which is characterized by a topology comprising alternating periodic building units of MAZ and MOR layers, exhibits stacking faults that impede accurate refinement through the Rietveld method. In this report, we present, for the first time, the structure of ECR-1 elucidated by studying a nanocrystal with a significantly reduced number of stacking faults. The sample used was synthesized hydrothermally using trioxane as the organic structure-directing agent. The structure determination was conducted using precession electron diffraction (PED) at 103 K. Partial dehydration occurred owing to the high vacuum conditions in the TEM sample chamber. From the dynamical refinement (R_obs = 0.097), 8.16 Na⁺ compensating cations were localized on six distinct crystallographic sites, along with approximately four water molecules per unit cell. Furthermore, a canonical Monte Carlo computational study was conducted to compare the experimental cationic distribution and location of water molecules with the simulation. Full article