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16 pages, 4411 KB  
Article
Interpretable Deep Prototype-Based Neural Networks: Can a 1 Look like a 0?
by Esteban García-Cuesta, Daniel Manrique and Radu Constantin Ionescu
Electronics 2025, 14(18), 3584; https://doi.org/10.3390/electronics14183584 - 10 Sep 2025
Viewed by 866
Abstract
Prototype-Based Networks (PBNs) are inherently interpretable architectures that facilitate understanding of model outputs by analyzing the activation of specific neurons—referred to as prototypes—during the forward pass. The learned prototypes serve as transformations of the input space into a latent representation that more effectively [...] Read more.
Prototype-Based Networks (PBNs) are inherently interpretable architectures that facilitate understanding of model outputs by analyzing the activation of specific neurons—referred to as prototypes—during the forward pass. The learned prototypes serve as transformations of the input space into a latent representation that more effectively encapsulates the main characteristics shared across data samples, thereby enhancing classification performance. Crucially, these prototypes can be decoded and projected back into the original input space, providing direct interpretability of the features learned by the network. While this characteristic marks a meaningful advancement toward the realization of fully interpretable artificial intelligence systems, our findings reveal that prototype representations can be deliberately or inadvertently manipulated without compromising the superficial appearance of explainability. In this study, we conduct a series of empirical investigations that demonstrate this phenomenon, framing it as a structural paradox potentially intrinsic to the architecture or its design, which may represent a significant robustness challenge for explainable AI methodologies. Full article
(This article belongs to the Special Issue Feature Papers in Artificial Intelligence)
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13 pages, 4460 KB  
Article
Interstitial Ag+ Engineering Enables Superior Resistive Switching in Quasi-2D Halide Perovskites
by Haiyang Qin, Zijia Wang, Qinrao Li, Jianxin Lin, Dongzhu Lu, Yicong Huang, Wenke Gao, Huachuan Wang and Chenghao Bi
Nanomaterials 2025, 15(16), 1267; https://doi.org/10.3390/nano15161267 - 16 Aug 2025
Viewed by 741
Abstract
Halide perovskite-based memristors are promising neuromorphic devices due to their unique ion migration and interface tunability, yet their conduction mechanisms remain unclear, causing stability and performance issues. Here, we engineer interstitial Ag+ ions within a quasi-two-dimensional (quasi-2D) halide perovskite ((C6H [...] Read more.
Halide perovskite-based memristors are promising neuromorphic devices due to their unique ion migration and interface tunability, yet their conduction mechanisms remain unclear, causing stability and performance issues. Here, we engineer interstitial Ag+ ions within a quasi-two-dimensional (quasi-2D) halide perovskite ((C6H5C2H4NH3)2Csn−1PbnI3n+1) to enhance device stability and controllability. The introduced Ag+ ions occupy organic interlayers, forming thermodynamically stable structures and introducing deep-level energy states without structural distortion, which do not act as non-radiative recombination centers, but instead serve as efficient charge trapping centers that stabilize intermediate resistance states and facilitate controlled filament evolution during resistive switching. This modification also leads to enhanced electron transparency near the Fermi level, contributing to improved charge transport dynamics and device performance. Under external electric fields, these Ag+ ions act as mobile ionic species, facilitating controlled filament formation and stable resistive switching. The resulting devices demonstrate exceptional performance, featuring an ultrahigh on/off ratio (∼108) and low operating voltages (∼0.31 V), surpassing existing benchmarks. Our findings highlight the dual role of Ag+ ions in structural stabilization and conduction modulation, providing a robust approach for high-performance perovskite memristor engineering. Full article
(This article belongs to the Special Issue Quantum Dot Materials and Their Optoelectronic Applications)
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15 pages, 5436 KB  
Article
Effect of Surface Passivation on the Quasi-Two-Dimensional Perovskite X2Cs(n−1) PbnI(3n+1)
by Min Li, Haoyan Zheng, Xianliang Ke, Dawei Zhang and Jie Huang
Condens. Matter 2025, 10(3), 44; https://doi.org/10.3390/condmat10030044 - 9 Aug 2025
Viewed by 627
Abstract
The two-dimensional (2D) Ruddlesden–Popper perovskite exhibits superior chemical stability but suffers from compromised photoelectric properties due to the van der Waals gap. This study presents a novel investigation of surface passivation effects on quasi-2D perovskite X2Csn−1PbnI3n+1 [...] Read more.
The two-dimensional (2D) Ruddlesden–Popper perovskite exhibits superior chemical stability but suffers from compromised photoelectric properties due to the van der Waals gap. This study presents a novel investigation of surface passivation effects on quasi-2D perovskite X2Csn−1PbnI3n+1 (n = 1–6; X = MA, FA, PEA) using DFT methods, revealing three key advances: First, we demonstrate that organic cation passivation (MA+, FA+, PEA+) enables exceptional stability improvements, with FA-passivated structures showing optimal stability—a crucial finding for materials design. Second, we identify a critical thickness effect (n > 3) where bandgaps converge to <1.6 eV (approaching bulk values) while maintaining strong absorption, establishing the minimum layer requirement for optimal performance. Third, we reveal that effective masses balance and absorption strengthens significantly when n > 3. These fundamental insights provide a transformative strategy to simultaneously enhance both stability and optoelectronic properties in quasi-2D perovskites. Full article
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15 pages, 2190 KB  
Article
Synthesis and Characterization of Covalent Triazine Frameworks Based on 4,4′-(Phenazine-5,10-diyl)dibenzonitrile and Its Application in CO2/CH4 Separation
by Hanibal Othman, Robert Oestreich, Vivian Küll, Marcus N. A. Fetzer and Christoph Janiak
Molecules 2025, 30(15), 3110; https://doi.org/10.3390/molecules30153110 - 24 Jul 2025
Viewed by 579
Abstract
Covalent triazine frameworks (CTFs) have gained recognition as stable porous organic polymers, for example, for CO2 separation. From the monomer 4,4′-(phenazine-5,10-diyl)dibenzonitrile (pBN), new pBN-CTFs were synthesized using the ionothermal method with a variation in temperature (400 and 550 °C) and the ZnCl [...] Read more.
Covalent triazine frameworks (CTFs) have gained recognition as stable porous organic polymers, for example, for CO2 separation. From the monomer 4,4′-(phenazine-5,10-diyl)dibenzonitrile (pBN), new pBN-CTFs were synthesized using the ionothermal method with a variation in temperature (400 and 550 °C) and the ZnCl2-to-monomer ratio (10 and 20). N2 adsorption yielded BET surface areas up to 1460 m2g −1. The pBN-CTFs are promising CO2 adsorbents and are comparable to other benchmark CTFs such as CTF-1 with a CO2 uptake of pBN-CTF-10-550 at 293 K of up to 54 cm3 g−1 or 96 mg g−1, with a CO2/CH4 IAST selectivity of 22 for a 50% mixture of CO2/CH4. pBN-CTF-10-400 has a very high heat of adsorption of 79 kJ mol−1 for CO2 near zero coverage in comparison to other CTFs, and it also stays well above the liquefaction heat of CO2 due to its high microporosity of 50% of the total pore volume. Full article
(This article belongs to the Section Materials Chemistry)
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15 pages, 1866 KB  
Article
Electron Spin Resonance Spectroscopy Suitability for Investigating the Oxidative Stability of Non-Alcoholic Beers
by Maria Cristina Porcu and Daniele Sanna
Oxygen 2025, 5(3), 14; https://doi.org/10.3390/oxygen5030014 - 16 Jul 2025
Viewed by 412
Abstract
Seven lager beers and seven non-alcoholic counterparts, marketed by the same producers, were analyzed for their total phenolic content (TPC), radical scavenging activity (RSA) towards the DPPH radical and ThioBarbituric Index (TBI). All beers were also subjected to spin trapping experiments at 60 [...] Read more.
Seven lager beers and seven non-alcoholic counterparts, marketed by the same producers, were analyzed for their total phenolic content (TPC), radical scavenging activity (RSA) towards the DPPH radical and ThioBarbituric Index (TBI). All beers were also subjected to spin trapping experiments at 60 °C in the presence of PBN. To our knowledge, this is the first time that non-alcoholic beers (NABs) have been subjected to spin trapping experiments coupled with Electron Spin Resonance (ESR) spectroscopy. The evolution of the intensity of the PBN radical adducts during the first 150 min was represented graphically and the intensity at 150 min (I150) and the area under the curve (AUC), were measured. The I150 and the AUC of lagers and NABs are significantly different, whereas the TPC, the EC50 of the DPPH assay, and the TBI of the two groups are superimposed. A relationship, previously proposed by us, to correlate ESR spectroscopy parameters with others obtained from UV-Vis spectrophotometry, was also applied, demonstrating its practicability. Multivariate analysis shows that clustering in two separate groups occurs only if I150 and AUC are included in the model. Based on these results, ESR spectroscopy can be applied to study the oxidative stability of NABs. Full article
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16 pages, 2229 KB  
Article
Investigation of the Effect of Molecules Containing Sulfonamide Moiety Adsorbed on the FAPbI3 Perovskite Surface: A First-Principles Study
by Shiyan Yang, Yu Zhuang, Youbo Dou, Jianjun Wang, Hongwen Zhang, Wenjing Lu, Qiuli Zhang, Xihua Zhang, Yuan Wu and Xianfeng Jiang
Molecules 2025, 30(11), 2463; https://doi.org/10.3390/molecules30112463 - 4 Jun 2025
Viewed by 736
Abstract
First-principles calculations were conducted to examine the impact of three sulfonamide-containing molecules (H4N2O2S, CH8N4O3S, and C2H2N6O4S) adsorbed on the FAPbI3(001) perovskite [...] Read more.
First-principles calculations were conducted to examine the impact of three sulfonamide-containing molecules (H4N2O2S, CH8N4O3S, and C2H2N6O4S) adsorbed on the FAPbI3(001) perovskite surface, aiming to establish a significant positive correlation between the molecular structures and their regulatory effects on the perovskite surface. A systematic comparison was conducted to evaluate the adsorption stability of the three molecules on the two distinct surface terminations. The results show that all three molecules exhibit strong adsorption on the FAPbI3(001) surface, with C2H12N6O4S demonstrating the most favorable binding stability due to its extended frameworks and multiple electron-donating/withdrawing groups. Simpler molecules lacking carbon skeletons exhibit weaker adsorption and less dependence on surface termination. Ab initio molecular dynamics simulations (AIMD) further corroborated the thermal stability of the stable adsorption configurations at elevated temperatures. Electronic structure analysis reveals that molecular adsorption significantly reconstructs the density of states (DOS) on the PbI2-terminated surface, inducing shifts in band-edge states and enhancing energy-level coupling between molecular orbitals and surface states. In contrast, the FAI-terminated surface shows weaker interactions. Charge density difference (CDD) analysis indicates that the molecules form multiple coordination bonds (e.g., Pb–O, Pb–S, and Pb–N) with uncoordinated Pb atoms, facilitated by –SO2–NH2 groups. Bader charge and work function analyses indicate that the PbI2-terminated surface exhibits more pronounced electronic coupling and interfacial charge transfer. The C2H12N6O4S adsorption system demonstrates the most substantial reduction in work function. Optical property calculations show a distinct red-shift in the absorption edge along both the XX and YY directions for all adsorption systems, accompanied by enhanced absorption intensity and broadened spectral range. These findings suggest that sulfonamide-containing molecules, particularly C2H12N6O4S with extended carbon skeletons, can effectively stabilize the perovskite interface, optimize charge transport pathways, and enhance light-harvesting performance. Full article
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10 pages, 2024 KB  
Article
Bifunctional 4,5-Diiodoimidazole Interfacial Engineering Enables Simultaneous Defect Passivation and Crystallization Control for High-Efficiency Inverted Perovskite Solar Cells
by Huaxi Gao, Yu Zhang, Ihtesham Ghani, Min Xin, Danish Khan, Junyu Wang, Di Lu, Tao Cao, Wei Chen, Xin Yang and Zeguo Tang
Nanomaterials 2025, 15(10), 766; https://doi.org/10.3390/nano15100766 - 20 May 2025
Viewed by 691
Abstract
Despite the rapid efficiency advancement of perovskite solar cells (PSCs), non-radiative recombination at the buried interface between self-assembled monolayers (SAMs) and perovskite remains a critical bottleneck, primarily due to interfacial defects and energy level mismatch. In this study, we demonstrate a bifunctional interlayer [...] Read more.
Despite the rapid efficiency advancement of perovskite solar cells (PSCs), non-radiative recombination at the buried interface between self-assembled monolayers (SAMs) and perovskite remains a critical bottleneck, primarily due to interfacial defects and energy level mismatch. In this study, we demonstrate a bifunctional interlayer engineering strategy by introducing 4,5-diiodoimidazole (4,5-Di-I) at the Me-4PACz/perovskite interface. This approach uniquely addresses two fundamental limitations of SAM-based interfaces: the insufficient defect passivation capability of conventional Me-4PACz due to steric hindrance effects and the poor perovskite wettability on hydrophobic SAM surfaces that exacerbates interfacial voids. The imidazole derivatives not only form strong Pb–N coordination bonds with undercoordinated Pb2+ but also modulate the surface energy of Me-4PACz, enabling the growth of pinhole-free perovskite films with preferential crystal orientation. The champion device with 4,5-Di-I modification achieves a power conversion efficiency (PCE) of 24.10%, with a VOC enhancement from 1.12 V to 1.14 V, while maintaining 91% of initial PCE after 1300 h in N₂ atmosphere (25 °C), demonstrating superior stability under ISOS-L-2 protocols. This work establishes a universal strategy for interfacial multifunctionality design, proving that simultaneous defect suppression and crystallization control can break the long-standing trade-off between efficiency and stability in solution-processed photovoltaics. Full article
(This article belongs to the Special Issue Advanced Nanoscale Materials and (Flexible) Devices)
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19 pages, 4494 KB  
Article
Spacer Loss upon 2D Ruddlesden–Popper Halide Perovskite Annealing Raises Film Properties and Solar Cell Performances
by Tao Zhu, Min Liu, Marie Cresp, Daming Zheng, Karol Vegso, Peter Siffalovic and Thierry Pauporté
Nanomaterials 2025, 15(10), 750; https://doi.org/10.3390/nano15100750 - 16 May 2025
Cited by 1 | Viewed by 1306
Abstract
Using reduced-dimensional halide perovskites is emerging as a promising strategy for enhancing the stability of optoelectronic devices such as solar cells, even if their performances remain a step below those of the 3D halide perovskites. Two-dimensional Ruddlesden–Popper (2D-RP) structures are characterized by the [...] Read more.
Using reduced-dimensional halide perovskites is emerging as a promising strategy for enhancing the stability of optoelectronic devices such as solar cells, even if their performances remain a step below those of the 3D halide perovskites. Two-dimensional Ruddlesden–Popper (2D-RP) structures are characterized by the n parameter that represents the number of PbI6 layers in the spacer-separated perovskite slabs. The present study focuses on formamidinium (FA)-based 2D-RP type perovskites denoted as PMA2FAn−1PbnI3n+1 (PMA = Phenylmethylammonium or benzylammonium). We investigate the effect of n on the one step growth mechanism and the film morphology, microstructure, phase purity, and optoelectronic properties. Our findings demonstrate that the average n is not only determined by the initial spacer content in the precursor solution but also by the thermal annealing process that leads to a partial spacer loss. Depending on n, perovskite solar cells achieving a power conversion efficiency up to 21%, coupled with enhanced film stability compared to 3D perovskites have been prepared. By using MACl additive and an excess of PbI2 in the perovskite precursor solution, we have been able to achieve high efficiency and to stabilize the n = 5 perovskite solar cells. This research represents a significant stride in comprehending the formation of FA-based layered perovskites through one-step sequential deposition, enabling control over their phase distribution, composition, and orientation. Full article
(This article belongs to the Special Issue Advances in Nanomaterials for Optoelectronics: Second Edition)
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11 pages, 1449 KB  
Article
A Learning Probabilistic Boolean Network Model of a Manufacturing Process with Applications in System Asset Maintenance
by Pedro Juan Rivera Torres, Chen Chen, Sara Rodríguez González and Orestes Llanes Santiago
Entropy 2025, 27(5), 463; https://doi.org/10.3390/e27050463 - 25 Apr 2025
Viewed by 900
Abstract
Probabilistic Boolean Networks (PBN) can model the dynamics of complex biological systems, as well as other non-biological systems like manufacturing systems and smart grids. In this proof-of-concept paper, we propose a PBN architecture with a learning process that significantly enhances fault and failure [...] Read more.
Probabilistic Boolean Networks (PBN) can model the dynamics of complex biological systems, as well as other non-biological systems like manufacturing systems and smart grids. In this proof-of-concept paper, we propose a PBN architecture with a learning process that significantly enhances fault and failure prediction in manufacturing systems. This concept was tested using a PBN model of an ultrasound welding process and its machines. Through various experiments, the model successfully learned to maintain a normal operating state. Leveraging the complex properties of PBNs, we utilize them as an adaptive learning tool with positive feedback, demonstrating that these networks may have broader applications than previously recognized. This multi-layered PBN architecture offers substantial improvements in fault detection performance within a positive feedback network structure that shows greater noise tolerance than other methods. Full article
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31 pages, 12346 KB  
Review
Phosphorylated Nitrones—Synthesis and Applications
by Iwona Rozpara, José Marco-Contelles, Dorota G. Piotrowska and Iwona E. Głowacka
Molecules 2025, 30(6), 1333; https://doi.org/10.3390/molecules30061333 - 16 Mar 2025
Cited by 2 | Viewed by 1958
Abstract
Phosphorylated nitrones belong to an important class of compounds with several applications, such as their therapeutic potency to reduce oxidative stress or as spin-trapping agents. This review covers available synthetic methods for the preparation of both non-cyclic and cyclic phosphorylated nitrones, including the [...] Read more.
Phosphorylated nitrones belong to an important class of compounds with several applications, such as their therapeutic potency to reduce oxidative stress or as spin-trapping agents. This review covers available synthetic methods for the preparation of both non-cyclic and cyclic phosphorylated nitrones, including the possibilities of the modification of structures with selected functional groups, as well as examples of their application. As reported, the incorporation of diethoxyphosphoryl function into the structure of PBN and DMPO resulted in obtaining their phosphorylated analogs, i.e., N-benzylidene-1-diethoxyphosphoryl-1-methylethylamine N-oxide (PPN) and 5-(diethoxyphosphoryl)-5-methyl-1-pyrroline-N-oxide (DEPMPO), respectively, both forming spin adducts of improved stability in comparison to the reference non-phosphorus nitrones. Moreover, antioxidant and neuroprotective activity observed in the group of phosphorylated nitrones makes them promising candidates for therapeutics. Full article
(This article belongs to the Special Issue Design, Synthesis, and Analysis of Potential Drugs, 3rd Edition)
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21 pages, 4632 KB  
Article
A Learning Probabilistic Boolean Network Model of a Smart Grid with Applications in System Maintenance
by Pedro Juan Rivera Torres, Chen Chen, Jaime Macías-Aguayo, Sara Rodríguez González, Javier Prieto Tejedor, Orestes Llanes Santiago, Carlos Gershenson García and Samir Kanaan Izquierdo
Energies 2024, 17(24), 6399; https://doi.org/10.3390/en17246399 - 19 Dec 2024
Cited by 2 | Viewed by 1559
Abstract
Probabilistic Boolean Networks can capture the dynamics of complex biological systems as well as other non-biological systems, such as manufacturing systems and smart grids. In this proof-of-concept manuscript, we propose a Probabilistic Boolean Network architecture with a learning process that significantly improves the [...] Read more.
Probabilistic Boolean Networks can capture the dynamics of complex biological systems as well as other non-biological systems, such as manufacturing systems and smart grids. In this proof-of-concept manuscript, we propose a Probabilistic Boolean Network architecture with a learning process that significantly improves the prediction of the occurrence of faults and failures in smart-grid systems. This idea was tested in a Probabilistic Boolean Network model of the WSCC nine-bus system that incorporates Intelligent Power Routers on every bus. The model learned the equality and negation functions in the different experiments performed. We take advantage of the complex properties of Probabilistic Boolean Networks to use them as a positive feedback adaptive learning tool and to illustrate that these networks could have a more general use than previously thought. This multi-layered PBN architecture provides a significant improvement in terms of performance for fault detection, within a positive-feedback network structure that is more tolerant of noise than other techniques. Full article
(This article belongs to the Section A1: Smart Grids and Microgrids)
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18 pages, 2087 KB  
Article
Longitudinal Study of SARS-CoV-2 Vaccinations and Infections in Patients with Gastrointestinal Cancer: Stabilizing Immune Responses and Neutralizing Emerging Variants with Variant-Adapted Antigen Exposures
by Maria A. Gonzalez-Carmona, Alina M. Schmitz, Moritz Berger, Leona I. Baier, Jens G. Gorny, Farsaneh Sadeghlar, Thomas Anhalt, Xin Zhou, Taotao Zhou, Robert Mahn, Christian Möhring, Thomas Linnemann, Matthias Schmid, Christian P. Strassburg, Christoph Boesecke, Jürgen K. Rockstroh, Anna-Maria Eis-Hübinger and Malte B. Monin
Int. J. Mol. Sci. 2024, 25(24), 13613; https://doi.org/10.3390/ijms252413613 - 19 Dec 2024
Viewed by 1240
Abstract
This longitudinal study examined how active gastrointestinal (GI) cancer types affect immune responses to SARS-CoV-2, focusing on the ability to neutralize the Omicron variants. Patients with GI cancer (n = 168) were categorized into those with hepatocellular carcinoma, hepatic metastatic GI cancer, [...] Read more.
This longitudinal study examined how active gastrointestinal (GI) cancer types affect immune responses to SARS-CoV-2, focusing on the ability to neutralize the Omicron variants. Patients with GI cancer (n = 168) were categorized into those with hepatocellular carcinoma, hepatic metastatic GI cancer, non-hepatic metastatic GI cancer, and two control groups of patients with and without underlying liver diseases. Humoral and cellular immune responses were evaluated before and after Omicron antigen exposures. In the pre-Omicron era, humoral SARS-CoV-2 immunity decreased after three antigen contacts without further antigen exposure. While Omicron neutralization was significantly lower than wildtype neutralization (p < 0.01), Omicron infections were yet mild to moderate. Additional Omicron exposures improved IgG levels (p < 0.01) and Omicron neutralization (p < 0.01). However, this effect was significantly less intense in patients with active GI cancer, particularly in patients with pancreaticobiliary neoplasms (PBN; p = 0.04), with underlying immunodeficiency (p = 0.05), and/or under conventional chemotherapy (p = 0.05). Pre-Omicron SARS-CoV-2 immunity prevented severe clinical courses of infections with Omicron variants in patients with GI cancer. However, in patients with PBN, with underlying immunodeficiency, and/or under conventional chemotherapy initial contacts with Omicron antigens triggered only reduced immune responses. Thus, subgroups could be identified for whom booster vaccinations are of special clinical significance. Full article
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15 pages, 9260 KB  
Article
Multiple Posterior Insula Projections to the Brainstem Descending Pain Modulatory System
by Despoina Liang and Charalampos Labrakakis
Int. J. Mol. Sci. 2024, 25(17), 9185; https://doi.org/10.3390/ijms25179185 - 24 Aug 2024
Cited by 5 | Viewed by 3048
Abstract
The insular cortex is an important hub for sensory and emotional integration. It is one of the areas consistently found activated during pain. While the insular’s connections to the limbic system might play a role in the aversive and emotional component of pain, [...] Read more.
The insular cortex is an important hub for sensory and emotional integration. It is one of the areas consistently found activated during pain. While the insular’s connections to the limbic system might play a role in the aversive and emotional component of pain, its connections to the descending pain system might be involved in pain intensity coding. Here, we used anterograde tracing with viral expression of mCherry fluorescent protein, to examine the connectivity of insular axons to different brainstem nuclei involved in the descending modulation of pain in detail. We found extensive connections to the main areas of descending pain control, namely, the periaqueductal gray (PAG) and the raphe magnus (RMg). In addition, we also identified an extensive insular connection to the parabrachial nucleus (PBN). Although not as extensive, we found a consistent axonal input from the insula to different noradrenergic nuclei, the locus coeruleus (LC), the subcoereuleus (SubCD) and the A5 nucleus. These connections emphasize a prominent relation of the insula with the descending pain modulatory system, which reveals an important role of the insula in pain processing through descending pathways. Full article
(This article belongs to the Section Molecular Neurobiology)
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16 pages, 337 KB  
Article
Probability Bracket Notation for Probability Modeling
by Xing M. Wang and Tony C. Scott
Axioms 2024, 13(8), 564; https://doi.org/10.3390/axioms13080564 - 20 Aug 2024
Viewed by 1655
Abstract
Following Dirac’s notation in Quantum Mechanics (QM), we propose the Probability Bracket Notation (PBN), by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc. Using the PBN, many formulae, such as normalizations and expectations in systems of one or more random variables, can now [...] Read more.
Following Dirac’s notation in Quantum Mechanics (QM), we propose the Probability Bracket Notation (PBN), by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc. Using the PBN, many formulae, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions, which are easy to expand by inserting a proper P-identity. The time evolution of homogeneous Markov processes can also be formatted in such a way. Our system P-kets are identified with probability vectors and our P-bra system is comparable with Doi’s state function or Peliti’s standard bra. In the Heisenberg picture of the PBN, a random variable becomes a stochastic process, and the Chapman–Kolmogorov equations are obtained by inserting a time-dependent P-identity. Also, some QM expressions in Dirac notation are naturally transformed to probability expressions in PBN by a special Wick rotation. Potential applications show the usefulness of the PBN beyond the constrained domain and range of Hermitian operators on Hilbert Spaces in QM all the way to IT. Full article
(This article belongs to the Special Issue Stochastic Processes in Quantum Mechanics and Classical Physics)
12 pages, 3298 KB  
Article
Structural Evolution of Small-Sized Phosphorus-Doped Boron Clusters: A Half-Sandwich-Structured PB15 Cluster
by Danyu Wang, Yueju Yang, Shixiong Li and Deliang Chen
Molecules 2024, 29(14), 3384; https://doi.org/10.3390/molecules29143384 - 18 Jul 2024
Viewed by 1525
Abstract
The present study is a theoretical investigation into the structural evolution, electronic properties, and photoelectron spectra of phosphorus-doped boron clusters PBn0/− (n = 3–17). The results of this study revealed that the lowest energy structures of PBn (n = [...] Read more.
The present study is a theoretical investigation into the structural evolution, electronic properties, and photoelectron spectra of phosphorus-doped boron clusters PBn0/− (n = 3–17). The results of this study revealed that the lowest energy structures of PBn (n = 3–17) clusters, except for PB17, exhibit planar or quasi-planar structures. The lowest energy structures of PBn (n = 3–17), with the exceptions of PB7, PB9, and PB15, are planar or quasi-planar. The ground state of PB7 has an umbrella-shaped structure, with C6V symmetry. Interestingly, the neutral cluster PB15 has a half-sandwich-like structure, in which the P atom is attached to three B atoms at one end of the sandwich, exhibiting excellent relative and chemical stability due to its higher second-order energy difference and larger HOMO–LUMO energy gap of 4.31 eV. Subsequently, adaptive natural density partitioning (AdNDP) and electron localization function (ELF) analyses demonstrate the bonding characteristics of PB7 and PB15, providing support for the validity of their stability. The calculated photoelectron spectra show distinct characteristic peaks of PBn (n = 3–17) clusters, thus providing theoretical evidence for the future identification of doped boron clusters. In summary, our work has significant implications for understanding the structural evolution of doped boron clusters PBn0/− (n = 3–17), motivating further experiments regarding doped boron clusters. Full article
(This article belongs to the Special Issue Boron Chemistry and Applications)
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