Based at Thornton Science Park, the new Faculty of Science and Engineering is located in a major research and innovation hub for the North West which is only a 20-minute bus trip from the main Chester Campus. The Faculty offers degrees in engineering and science disciplines using a strongly interdisciplinary teaching philosophy.

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  • Local-Partial Signal Combining Schemes for Cell-Free Large-Scale MU-MIMO Systems with Limited Fronthaul Capacity and Spatial Correlation Channels

    Alammari, Amr A.; Sharique, Mohd; Moinuddin, Athar Ali; Ansari, Mohammad Samar; Aligarh Muslim University; University of Chester (MDPI, 2022-09-01)
    Cell-free large-scale multi-user MIMO is a promising technology for the 5G-and-beyond mobile communication networks. Scalable signal processing is the key challenge in achieving the benefits of cell-free systems. This study examines a distributed approach for cell-free deployment with user-centric configuration and finite fronthaul capacity. Moreover, the impact of scaling the pilot length, the number of access points (APs), and the number of antennas per AP on the achievable average spectral efficiency are investigated. Using the dynamic cooperative clustering (DCC) technique and large-scale fading decoding process, we derive an approximation of the signal-tointerference-plus-noise ratio in the criteria of two local combining schemes: Local-Partial Regularized Zero Forcing (RZF) and Local Maximum Ratio (MR). The results indicate that distributed approaches in the cell-free system have the advantage of decreasing the fronthaul signaling and the computing complexity. The results also show that the Local-Partial RZF provides the highest average spectral efficiency among all the distributed combining schemes because the computational complexity of the Local-Partial RZF is independent of the UTs. Therefore, it does not grow as the number of user terminals (UTs) increases.
  • AFOM: Advanced Flow of Motion Detection Algorithm for Dynamic Camera Videos

    Aribilola, Ifeoluwapo; Asghar, Mamoona; Kanwal, Nadia; Ansari, Mohammad Samar; Lee, Brian; Technological University of the Shannon; National University of Ireland; University of Keele; University of Chester (IEEE, 2022-07-19)
    The surveillance videos taken from dynamic cam-eras are susceptible to multiple security threats like replay attacks, man-in-the-middle attacks, pixel correlation attacks etc. Using unsupervised learning, it is a challenge to detect objects in such surveillance videos, as fixed objects may appear to be in motion alongside the actual moving objects. But despite this challenge, the unsupervised learning techniques are efficient as they save object labelling and model training time, which is usually a case with supervised learning models. This paper proposes an effective computer vision-based object identification algorithm that can detect and separate stationary objects from moving objects in such videos. The proposed Advanced Flow Of Motion (AFOM) algorithm takes advantage of motion estimation between two consecutive frames and induces the estimated motion back to the frame to provide an improved detection on the dynamic camera videos. The comparative analysis demonstrates that the proposed AFOM outperforms a traditional dense optical flow (DOF) algorithm with an average increased difference of 56 % in accuracy, 61 % in precision, and 73 % in pixel space ratio (PSR), and with minimal higher object detection timing.
  • A single-layer asymmetric RNN with low hardware complexity for solving linear equations

    Ansari, Mohammad Samar; University of Chester (Elsevier, 2022-01-25)
    A single layer neural network for the solution of linear equations is presented. The proposed circuit is based on the standard Hopfield model albeit with the added flexibility that the interconnection weight matrix need not be symmetric. This results in an asymmetric Hopfield neural network capable of solving linear equations. PSPICE simulation results are given which verify the theoretical predictions. A simple technique to incorporate re-configurability into the circuit for setting the different weights of the interconnection is also included. Experimental results for circuits set up to solve small problems further confirm the operation of the proposed circuit.
  • FireNet-v2: Improved Lightweight Fire Detection Model for Real-Time IoT Applications

    Shees, Anwer; Ansari, Mohammad Samar; Varshney, Akshay; Asghar, Mamoona; Kanwal, Nadia; Aligarh Muslim University; University of Chester; Adobe India; University of Galway; Keele University
    Fire hazards cause huge ecological, social and economical losses in day to day life. Due to the rapid increase in the prevalence of fire accidents, it has become vital to equip the assets with fire prevention systems. There have been numerous researches to build a fire detection model in order to avert such accidents, with recent approaches leveraging the enormous improvements in computer vision deep learning models. However, most deep learning models have to compromise with their performance and accurate detection to maintain a reasonable inference time and parameter count. In this paper, we present a customized lightweight convolution neural network for early detection of fire. By virtue of low parameter count, the proposed model is amenable to embedded applications in real-time fire monitoring equipment, and even upcoming fire monitoring approaches such as unmanned aerial vehicles (drones). The fire detection results show marked improvement over the predecessor low-parameter-count models, while further reducing the number of trainable parameters. The overall accuracy of FireNet-v2, which has only 318,460 parameters, was found to be 98.43% when tested over Foggia's dataset.
  • Deep Learning based Wireless Channel Prediction: 5G Scenario

    Varshney, Rajat; Gangal, Chirag; Sharique, Mohd; Ansari, Mohammad Samar; Aligarh Muslim University; Aligarh Muslim University; Aligarh Muslim University; University of Chester
    In the area of wireless communication, channel estimation is a challenging problem due to the need for real-time implementation as well as system dependence on the estimation accuracy. This work presents a Long-Short Term Memory (LSTM) based deep learning (DL) approach for the prediction of channel response in real-time and real-world non-stationary channels. The model uses the pre-defined history of channel impulse response (CIR) data along with two other features viz. transmitter-receiver update distance and root-mean-square delay spread values which are also changing in time with the channel impulse response. The objective is to obtain an approximate estimate of CIRs using prediction through the DL model as compared to conventional methods. For training the model, a sample dataset is generated through the open-source channel simulation software NYUSIM which realizes samples of CIRs for measurement-based channel models based on various multipath channel parameters. From the model test results, it is found that the proposed DL approach provides a viable lightweight solution for channel prediction in wireless communication.
  • DNA codes from skew dihedral group ring

    Dougherty, Steven; Korban, Adrian; Şahinkaya, Serap; Ustun, Deniz (American Institute of Mathematical Sciences (AIMS), 2022)
    <p style='text-indent:20px;'>In this work, we present a matrix construction for reversible codes derived from skew dihedral group rings. By employing this matrix construction, the ring <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{F}_{j, k} $\end{document}</tex-math></inline-formula> and its associated Gray maps, we show how one can construct reversible codes of length <inline-formula><tex-math id="M2">\begin{document}$ n2^{j+k} $\end{document}</tex-math></inline-formula> over the finite field <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{F}_4. $\end{document}</tex-math></inline-formula> As an application, we construct a number of DNA codes that satisfy the Hamming distance, the reverse, the reverse-complement, and the GC-content constraints with better parameters than some good DNA codes in the literature.</p>
  • Cholesterol transport in blood, lipoproteins, and cholesterol metabolism.

    Mc Auley, Mark T.; Morgan, Amy; University of Chester (Elsevier, 2022-04-26)
    The aim of this chapter is to critically discuss recent work which has focused on the dynamics of cholesterol transport and its intersection with health. Firstly, we provide an overview of the main lipoproteins, and their role in whole-body cholesterol metabolism. We then focus on low-density lipoprotein cholesterol (LDL-C) and high-density lipoprotein cholesterol (HDL-C), paying particular attention to a diverse array of evidence which associates perturbations to these lipoproteins with cardiovascular disease (CVD). Next, we explain how aging and obesity disrupt the biological mechanisms that regulate cholesterol metabolism. Crucially, we reveal the parallels between aging and obesity, underscoring that obesity superimposed on the aging process has the potential to exacerbate the age-related dysregulation of cholesterol metabolism. Following this, we unveil how mathematical modeling can be used to deepen our understanding of cholesterol metabolism. We conclude the chapter by discussing the future of this area; in doing so, we reveal how recent experimental findings could open the way for novel therapeutic approaches which could help maintain optimal blood lipoprotein levels and thus increase health span.
  • Watchdog Monitoring for Detecting and Handling of Control Flow Hijack on RISC-V-based Binaries

    Oyinloye, Toyosi; Speakman, Lee; Eze, Thaddeus; O'Mahony, Lucas; University of Chester; University of Salford (SAI Organization, 2022-08-31)
    Abstract—Control flow hijacking has been a major challenge in software security. Several means of protections have been developed but insecurities persist. This is because existing protections have sometimes been circumvented while some resilient protections do not cover all applications. Studies have revealed that a holistic way of tackling software insecurity could involve watchdog monitoring and detection via Control Flow Integrity (CFI). The CFI concept has shown a good measure of reliability to mitigate control flow hijacking. However, sophisticated attack techniques in the form of Return Oriented Programming (ROP) have persisted. A flexible protection is desirable, which not only covers as many architecture structures as possible but also mitigates known resilient attacks like ROP. The solution proffered here is a hybrid of CFI and watchdog timing via inter-process signaling (IP-CFI). It is a software-based protection that involves recompilation of the target program. The implementation here is on vulnerable RISC-V-based process but is flexible and could be adapted on other architectures. We present a proof of concept in IP-CFI which when applied to a vulnerable program, ROP is mitigated. The target program incurs a run-time overhead of 1.5%. The code is available.
  • A Mathematical Model which Examines Age-Related Stochastic Fluctuations in DNA Maintenance Methylation

    Zagkos, Loukas; Roberts, Jason; Mc Auley, Mark T.; University of Chester; Imperial College London (Elsevier, 2021-11-11)
    Due to its complexity and its ubiquitous nature the ageing process remains an enduring biological puzzle. Many molecular mechanisms and biochemical process have become synonymous with ageing. However, recent findings have pinpointed epigenetics as having a key role in ageing and healthspan. In particular age related changes to DNA methylation offer the possibility of monitoring the trajectory of biological ageing and could even be used to predict the onset of diseases such as cancer, Alzheimer's disease and cardiovascular disease. At the molecular level emerging evidence strongly suggests the regulatory processes which govern DNA methylation are subject to intracellular stochasticity. It is challenging to fully understand the impact of stochasticity on DNA methylation levels at the molecular level experimentally. An ideal solution is to use mathematical models to capture the essence of the stochasticity and its outcomes. In this paper we present a novel stochastic model which accounts for specific methylation levels within a gene promoter. Uncertainty of the eventual site-specific methylation levels for different values of methylation age, depending on the initial methylation levels were analysed. Our model predicts the observed bistable levels in CpG islands. In addition, simulations with various levels of noise indicate that uncertainty predominantly spreads through the hypermethylated region of stability, especially for large values of input noise. A key outcome of the model is that CpG islands with high to intermediate methylation levels tend to be more susceptible to dramatic DNA methylation changes due to increasing methylation age.
  • DNA Methylation in Genes Associated with the Evolution of Ageing and Disease: A Critical Review

    Mc Auley, Mark T.; University of Chester (Elsevier, 2021-10-15)
    Ageing is characterised by a physical decline in biological functioning which results in a progressive risk of mortality with time. As a biological phenomenon, it is underpinned by the dysregulation of a myriad of complex processes. Recently, however, ever-increasing evidence has associated epigenetic mechanisms, such as DNA methylation (DNAm) with age-onset pathologies, including cancer, cardiovascular disease, and Alzheimer’s disease. These diseases compromise healthspan. Consequently, there is a medical imperative to understand the link between epigenetic ageing, and healthspan. Evolutionary theory provides a unique way to gain new insights into epigenetic ageing and health. This review will: (1) provide a brief overview of the main evolutionary theories of ageing; (2) discuss recent genetic evidence which has revealed alleles that have pleiotropic effects on fitness at different ages in humans; (3) consider the effects of DNAm on pleiotropic alleles, which are associated with age related disease; (4) discuss how age related DNAm changes resonate with the mutation accumulation, disposable soma and programmed theories of ageing; (5) discuss how DNAm changes associated with caloric restriction intersect with the evolution of ageing; and (6) conclude by discussing how evolutionary theory can be used to inform investigations which quantify age-related DNAm changes which are linked to age onset pathology.
  • Modelling Cholesterol Metabolism and Atherosclerosis

    Mc Auley, Mark T.; University of Chester (Wiley, 2021-12-20)
    Atherosclerotic cardiovascular disease (ASCVD) is the leading cause of morbidity and mortality among Western populations. Many risk factors have been identified for ASCVD; however, elevated low-density lipoprotein cholesterol (LDL-C) remains the gold standard. Cholesterol metabolism at the cellular and whole-body level is maintained by an array of interacting components. These regulatory mechanisms have complex behavior. Likewise, the mechanisms which underpin atherogenesis are nontrivial and multifaceted. To help overcome the challenge of investigating these processes mathematical modeling, which is a core constituent of the systems biology paradigm has played a pivotal role in deciphering their dynamics. In so doing models have revealed new insights about the key drivers of ASCVD. The aim of this review is fourfold; to provide an overview of cholesterol metabolism and atherosclerosis, to briefly introduce mathematical approaches used in this field, to critically discuss models of cholesterol metabolism and atherosclerosis, and to highlight areas where mathematical modeling could help to investigate in the future.
  • The interdependency and co-regulation of the vitamin D and cholesterol metabolism.

    Warren, Tara; McAliister, Roisin; Morgan, Amy; Singh Rai, Taranjit; McGilligan, Victoria; Ennis, Matthew; Page, Christopher; Kelly, Catriona; Peace, Aaron; Corfe, Bernard; et al. (MDPI, 2021-08-06)
    Vitamin D and cholesterol metabolism overlap significantly in the pathways that contribute to their biosynthesis. However, our understanding of their independent and co-regulation is limited. Cardiovascular disease is the leading cause of death globally and atherosclerosis, the pathology associated with elevated cholesterol, is the leading cause of cardiovascular disease. It is therefore important to understand vitamin D metabolism as a contributory factor. From the literature, we compile evidence of how these systems interact, relating the understanding of the molecular mechanisms involved to the results from observational studies. We also present the first systems biology pathway map of the joint cholesterol and vitamin D metabolisms made available using the Systems Biology Graphical Notation (SBGN) Markup Language (SBGNML). It is shown that the relationship between vitamin D supplementation, total cholesterol, and LDL-C status, and between latitude, vitamin D, and cholesterol status are consistent with our knowledge of molecular mechanisms. We also highlight the results that cannot be explained with our current knowledge of molecular mechanisms: (i) vitamin D supplementation mitigates the side-effects of statin therapy; (ii) statin therapy does not impact upon vitamin D status; and critically (iii) vitamin D supplementation does not improve cardiovascular outcomes, despite improving cardiovascular risk factors. For (iii), we present a hypothesis, based on observations in the literature, that describes how vitamin D regulates the balance between cellular and plasma cholesterol. Answering these questions will create significant opportunities for advancement in our understanding of cardiovascular health
  • Miyamoto groups of code algebras

    Castillo-Ramirez, Alonso; McInroy, Justin; Universidad de Guadalajara; University of Bristol; Heilbronn Institute for Mathematical Research
    A code algebra A_C is a nonassociative commutative algebra defined via a binary linear code C. In a previous paper, we classified when code algebras are Z_2-graded axial (decomposition) algebras generated by small idempotents. In this paper, for each algebra in our classification, we obtain the Miyamoto group associated to the grading. We also show that the code algebra structure can be recovered from the axial decomposition algebra structure.
  • A unique ternary Ce(III)-quercetin-phenanthroline assembly with antioxidant and anti-inflammatory properties

    Halevas, E; Matsia, S; Hatzidimitriou, A; Geromichalou, E; Papadopoulos, Theodoros; Katsipis, G; Pantazaki, A; Litsardakis, G; Salifoglou, A; Aristotle Univeristy of Thessaloniki; National and Kapodistrian University of Athens; University of Chester
    Quercetin is one of the most bioactive and common dietary flavonoids, with a significant repertoire of biological and pharmacological properties. The biological activity of quercetin, however, is influenced by its limited solubility and bioavailability. Driven by the need to enhance quercetin bioavailability and bioactivity through metal ion complexation, synthetic efforts led to a unique ternary Ce(III)-quercetin-(1,10-phenanthroline) (1) compound. Physicochemical characterization (elemental analysis, FT-IR, Thermogravimetric analysis (TGA), UV–Visible, NMR, Electron Spray Ionization-Mass Spectrometry (ESI-MS), Fluorescence, X-rays) revealed its solid-state and solution properties, with significant information emanating from the coordination sphere composition of Ce(III). The experimental data justified further entry of 1 in biological studies involving toxicity, (Reactive Oxygen Species, ROS)-suppressing potential, cell metabolism inhibition in Saccharomyces cerevisiae (S. cerevisiae) cultures, and plasmid DNA degradation. DFT calculations revealed its electronic structure profile, with in silico studies showing binding to DNA, DNA gyrase, and glutathione S-transferase, thus providing useful complementary insight into the elucidation of the mechanism of action of 1 at the molecular level and interpretation of its bio-activity. The collective work projects the importance of physicochemically supported bio-activity profile of well-defined Ce(III)-flavonoid compounds, thereby justifying focused pursuit of new hybrid metal-organic materials, effectively enhancing the role of naturally-occurring flavonoids in physiology and disease.
  • Multifunctional cellular sandwich structures with optimised core topologies for improved mechanical properties and energy harvesting performance

    Chen, Boyue; Narita, Fumio; Wang, Congsi; Jia, Yu; Shi, Yu; Univerisity of Chester; Tohuku University; Xidian University; Aston University (Elsevier, 2022-04-13)
    This paper developed a multifunctional composite sandwich structure with optimised design on topological cores. As the main concern, full composite sandwich structures were manufactured with carbon fibre reinforced polymer (CFRP) facesheets and designed cores. Three-point bending tests have been performed to assess the mechanical performance of designed cellular sandwich structures. To evaluate the energy harvesting performance, the piezoelectric transducer was integrated at the interface between the upper facesheet and core, with both sinusoidal base excitation input and acceleration measured from real cruising aircraft and vehicle. It has been found that the sandwich with conventional honeycomb core has demonstrated the best mechanical performance, assessed under the bending tests. In terms of energy harvesting performance, sandwich with re-entrant honeycomb manifested approximately 20% higher RMS voltage output than sandwiches with conventional honeycomb and chiral structure core, evaluated both numerically and experimentally. The resistance sweep tests further suggested that the power output from sandwich with re-entrant honeycomb core was twice as large as that from sandwiches with conventional honeycomb and chiral structure cores, under optimal external resistance and sinusoidal base excitation.
  • Thermal Induced Interface Mechanical Response Analysis of SMT Lead-Free Solder Joint and Its Adaptive Optimization

    Liu, Shaoyi; Yan, Yuefei; Zhou, Yijiang; Han, Baoqing; Wang, Benben; Zhang, Daxing; Xue, Song; Wang, Zhihai; Yu, Kunpeng; Shi, Yu; et al. (MDPI, 2022-06-08)
    Surface mount technology (SMT) plays an important role in integrated circuits, but due to thermal stress alternation caused by temperature cycling, it tends to have thermo-mechanical reliability problems. At the same time, considering the environmental and health problems of lead (Pb)-based solders, the electronics industry has turned to lead-free solders, such as ternary alloy Sn-3Ag-0.5Cu (SAC305). As lead-free solders exhibit visco-plastic mechanical properties significantly affected by temperature, their thermo-mechanical reliability has received considerable attention. In this study, the interface delamination of an SMT solder joint using a SAC305 alloy under temperature cycling has been analyzed by the nonlinear finite element method. The results indicate that the highest contact pressure at the four corners of the termination/solder horizontal interface means that delamination is most likely to occur, followed by the y-direction side region of the solder/land interface and the top arc region of the termination/solder vertical interface. It should be noted that in order to keep the shape of the solder joint in the finite element model consistent with the actual situation after the reflow process, a minimum energy-based morphology evolution method has been incorporated into the established finite element model. Eventually, an Improved Efficient Global Optimization (IEGO) method was used to optimize the geometry of the SMT solder joint in order to reduce the contact pressure at critical points and critical regions. The optimization result shows that the contact pressure at the critical points and at the critical regions decreases significantly, which also means that the probability of thermal-induced delamination decreases.
  • Electromechanical characterization and kinetic energy harvesting of piezoelectric nanocomposites reinforced with glass fibers

    Maruyama, Kohei; Kawakami, Yoshihiro; Mori, Kotaro; Kurita, Hiroki; Shi, Yu; Jia, Yu; Narita, Fumio; Tohoku University; Research Institute for Electromagnetic Materials; Ibaraki University; University of Chester; Aston University (Elsevier, 2022-03-21)
    Piezoelectric composites are a significant research field because of their excellent mechanical flexibility and sufficient stress-induced voltage. Furthermore, due to the widespread use of the Internet of Things (IoT) in recent years, small-sized piezoelectric composites have attracted a lot of attention. Also, there is an urgent need to develop evaluation methods for these composites. This paper evaluates the piezoelectric and mechanical properties of potassium sodium niobate (KNN)-epoxy and KNN-glass fiber-reinforced polymer (GFRP) composites using a modified small punch (MSP) and nanoindentation tests in addition to d33 measurements. An analytical solution for the piezoelectric composite thin plate under bending was obtained for the determination of the bending properties. Due to the glass fiber inclusion, the bending strength increased by about four times, and Young's modulus in the length direction increased by approximately two times (more than that of the KNN-epoxy); however, in the thickness direction, Young's modulus decreased by less than half. An impact energy harvesting test was then performed on the KNN-epoxy and KNN-GFRP composites. As a result, the output voltage of KNN-GFRP was larger than that of KNN-epoxy. Also, the output voltage was about 2.4 V with a compressive stress of 0.2 MPa, although the presence of the glass fibers decreased the piezoelectric constants. Finally, damped flexural vibration energy harvesting test was carried out on the KNN-epoxy and KNN-GFRP composites. The KNN-epoxy was broken during the test, however KNN-GFRP composite with a load resistance of 10 generated 35 nJ of energy. Overall, through this work, we succeeded in developing piezoelectric energy harvesting composite materials that can withstand impact and bending vibration using glass fibers and also established a method for evaluating the electromechanical properties with small test specimen.
  • Numerical prediction of the chip formation and damage response in CFRP cutting with a novel strain rate based material model

    Wang, Xiaonan; Wang, Fuji; Jin, Xinghai; Fu, Rao; Shi, Yu; Dalian University of Technology; University of Chester (Elsevier, 2022-05-15)
    Carbon fibre reinforced plastics (CFRPs) are susceptible to various cutting damages. An accurate model that could efficiently predict the material removal and chip formation mechanisms will thus help to reduce the damages during cutting and further improved machining quality can be pursued. In previous studies, macro numerical models have been proposed to predict the orthogonal cutting of the CFRP laminates with subsurface damages under quasi-static loading conditions. However, the strain rate effect on the material behaviours has rarely been considered in the material modelling process, which would lead to the inaccurate prediction of the cutting process and damage extent, especially at high cutting speed. To address this issue, a novel material failure model is developed in this work by incorporating the strain rate effect across the damage initiation (combined Hashin and Puck laws) and evolution criteria. The variation in material properties with the strain rate is considered for the characterization of the stress-strain relationships under different loading speeds. With this material model, a three-dimensional macro numerical model is established to simulate the orthogonal cutting of CFRPs under four typical fibre cutting angles. The machining process and cutting force simulated by the proposed model are well agreed with the results of the CFRP orthogonal cutting experiments, and the prediction accuracy has been improved compared with the model without considering the strain rate effect. In addition, the effects of processing conditions on the subsurface damage in machining CFRPs under 135° are assessed. The subsurface damage is found to decrease with the cutting speed increases to 100 mm/s, afterwards, it tends to be stable when the cutting speed is over 100 mm/s. The increased severity of the subsurface damage is predicted with the higher cutting depths.
  • Split spin factor algebras

    McInroy, Justin; Shpectorov, Sergey; University of Bristol; Heilbronn Institute for Mathematical Research, Bristol; University of Birmingham; University of Chester (Elsevier, 2021-12-22)
    Motivated by Yabe's classification of symmetric $2$-generated axial algebras of Monster type \cite{yabe}, we introduce a large class of algebras of Monster type $(\alpha, \frac{1}{2})$, generalising Yabe's $\mathrm{III}(\alpha,\frac{1}{2}, \delta)$ family. Our algebras bear a striking similarity with Jordan spin factor algebras with the difference being that we asymmetrically split the identity as a sum of two idempotents. We investigate the properties of these algebras, including the existence of a Frobenius form and ideals. In the $2$-generated case, where our algebra is isomorphic to one of Yabe's examples, we use our new viewpoint to identify the axet, that is, the closure of the two generating axes.
  • Enumerating 3-generated axial algebras of Monster type

    McInroy, Justin; Shpectorov, Sergey; Khasraw, Sanhan; Salahaddin University-Erbil; University of Bristol; Heilbronn Institute for Mathematical Research, Bristol; University of Birmingham; University of Chester (Elsevier, 2021-06-17)
    An axial algebra is a commutative non-associative algebra generated by axes, that is, primitive, semisimple idempotents whose eigenvectors multiply according to a certain fusion law. The Griess algebra, whose automorphism group is the Monster, is an example of an axial algebra. We say an axial algebra is of Monster type if it has the same fusion law as the Griess algebra. The 2-generated axial algebras of Monster type, called Norton-Sakuma algebras, have been fully classified and are one of nine isomorphism types. In this paper, we enumerate a subclass of 3-generated axial algebras of Monster type in terms of their groups and shapes. It turns out that the vast majority of the possible shapes for such algebras collapse; that is they do not lead to non-trivial examples. This is in sharp contrast to previous thinking. Accordingly, we develop a method of minimal forbidden configurations, to allow us to efficiently recognise and eliminate collapsing shapes.

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