Multiple Values Research Articles

Identifying and analyzing variables in musculoskeletal healthcare research

This review intends to help the students and researchers to organize, write, and analyze to make a quality research paper and to know the importance of variables in healthcare clinicians/researchers. Variables are fundamental components of the research question, serving as properties that distinguish members of groups or sets. They explain the different aspects of the sample that is under study. Variables are capable of having multiple values. Variables vary from subject to subject in the sample. A variable is a fundamental concept in research, representing a concept, image, or perception that can take different measurable values. Research design (RD) involves comparing independent groups or conditions, exploring predictive relationships, experimental studies for causal links, and comparative studies with multiple independent variables (IVs) or dependent variables (DVs). Variables play a crucial role in research and are broadly classified into causal relationships, study design, and units of measurement. IV is predictors that influence outcomes known as DV. Extraneous variables are unrelated but impactful. Moderating variables alter the relationship between IV and DV. Intervening variables or confounding variables (CVs) can complicate the cause-and-effect relationships. Active variables are manipulated, while attribute variables, inherent traits such as age, cannot be controlled. Even tests are based on variables. This review provides a comprehensive guide for students and young researchers to understand the role of variables and RD in conducting quality research papers, emphasizing the importance of operationalizing variables, controlling for CVs, and selecting appropriate RDs.

A Study on Non-Homogeneous Multiplicative Boundary Value Problems

The paper deals with the non-homogeneous boundary value problems established in the multiplicative calculus. For this problem, the multiplicative meanings of concepts such as adjoint operator, self-adjoint operator, Lagrange identity, Green's formula are obtained. Then, a criterion is given for the existence of solutions to non-homogeneous multiplicative boundary value problems.

Neuroprotection and mechanisms of ginsenosides in nervous system diseases: Progress and perspectives.

Ginsenosides are the primary component discernible from ginseng, including Rb1, Rb2, Rd, Rg1, Rg2, and compound K, and so forth. They have been shown to have multiple pharmacological activities. In recent years, more and more studies have been devoted to the neuroprotection of various ginsenosides against neurological diseases and their potential mechanisms. This paper comprehensively summarizes and reviews the neuroprotective effects of various ginsenosides on neurological diseases, especially acute and chronic neurodegenerative diseases, and their mechanisms, as well as their potential therapeutic applications to promote neuroprotection in disease prevention, treatment, and prognosis. Briefly, ginsenosides exert effective neuroprotective effects on neurological conditions, including stroke, Alzheimer's disease, Parkinson's disease, and brain/spinal cord injuries through a variety of molecular mechanisms, including anti-inflammatory, antioxidant, and anti-apoptotic. Among them, some signaling pathways play important roles in related processes, such as PI3K/Akt, TLR4/NF-κB, ROS/TXNIP/NLRP3, HO-1/Nrf2, Wnt/β-catenin, and Ca2+ pathway. In conclusion, the present study reviews the research progress on the neuroprotective effects of ginsenosides in the last decade, with the aim of furnishing essential theoretical underpinning and effective references for further research and exploration of the multiple medicinal values of Chinese herbal medicines and their small molecule compounds, including ginseng and panax ginseng. Because there is less evidence in the existing clinical studies, future research should be focused on clinical trials in order to truly reflect the clinical value of various ginsenosides for the benefit of patients.

Euler-type sums involving odd harmonic numbers and binomial coefficients

In this paper, by iterated integral expression of multiple t-polylogarithm function, we establish some expressions of series involving multiple t-harmonic sums in terms of coloured multiple zeta values. Using these expressions and the stuffle relations, we discuss the evaluations of some Euler-type sums involving odd harmonic numbers and binomial coefficients, such as T odd , k ( q ; r ) := ∑ n = 1 ∞ h n − 1 ( k 1 ) ⋯ h n − 1 ( k p ) ( 2 n − 1 ) q ( n + r r ) , T evev , k ( q ; r ) := ∑ n = 1 ∞ h n ( k 1 ) ⋯ h n ( k p ) n q ( n + r r ) , T odd , k ( q ; r ) := ∑ n = 1 ∞ h n − 1 ( k ) ( 2 n − 1 ) q ∏ i = 1 b ( n + r i r i ) , T even , k ( q ; r ) := ∑ n = 1 ∞ h n ( k ) n q ∏ i = 1 b ( n + r i r i ) , T odd , k 1 , k 2 ( q ; r ) := ∑ n = 1 ∞ h n − 1 ( k 1 ) h n − 1 ( k 2 ) ( 2 n − 1 ) q ∏ i = 1 b ( n + r i r i ) , T even , k 1 , k 2 ( q ; r ) := ∑ n = 1 ∞ h n ( k 1 ) h n ( k 2 ) n q ∏ i = 1 b ( n + r i r i ) , and some other forms. We present some explicit evaluations as examples. It can be found that this work gives a unified approach to such sums, and generalizes many known results in the literature.

Validation of morphological ear classification devised by principal component analysis using three-dimensional images for human identification.

This study attempts to classify ear morphology for human identification in forensic investigations by distinguishing between the upper auricle and lobule areas. A three-dimensional homologous model of the ear was created using 414 ear images of males aged 17-93 years reconstructed from computed tomography scans of forensic autopsy cases. Morphological changes were visualized using principal component analysis and areas of significant individual differences within the entire ear were identified. The classification criterion images for the upper auricle (ten images) and lobule (12 images) were developed by combining multiple principal component values: components 1-5 for the upper auricle and 1-6 for the lobule. Three-dimensional ear images of the upper auricle and lobule areas from 414 subjects were categorized using a measurement method based on the minimum distance between 5,507 corresponding points. The results indicate the applicability of the criterion images for the morphological classification of ears in this study.

Extreme learning machine based on BDE feature selection to detect fault scenarios in grid-connected PV systems under MPPT mode

This paper considers real-time data-driven adaptive fault detection (FD) in grid-connected PV (GPV) systems under maximum power point tracking (MPPT) modes during large variations. Faults under MPPT modes remain undetected for longer periods, introducing new protection challenges and threats to the system. An intelligent FD algorithm is developed through real-time multi-sensor measurements and virtual Micro Phasor Measurement Unit (Micro-PMU) estimations. The high-dimensional and high-frequency multivariate features vary over time, and computational efficiency becomes crucial to realizing online adaptive FD. The goal of this study is to present an artificial intelligence (AI) technique for detecting seven faults: inverter fault, feedback sensor fault, grid anomaly, nonhomogeneous partial shading, open circuit in PV array, MPPT controller fault, and boost converter controller fault. In this work, it was found that the application of Extreme Learning Machine (ELM) plays an important role in fault detection and localization. Nine (9) statistical features and eight (8) wavelet packet parameters are extracted from the data based on multiple default values. These features were used as an input vector to train and test the ELM and determine whether the system is operating under normal conditions or is faulty. The BDE feature selection algorithm is adopted to optimize the seven-fault classification procedure to reduce the number of features. The results showed that the Extreme Learning Machine (ELM), based on statistical parameters followed by BDE, can detect faults with high accuracy (98.3%) compared to a case without optimization.

Some Symmetry and Duality Theorems on Multiple Zeta(-Star) Values

In this paper, we provide a symmetric formula and a duality formula relating multiple zeta values and zeta-star values. We find that the summation ∑a+b=r−1(−1)aζ★(a+2,{2}p−1)ζ★({1}b+1,{2}q) equals ζ★({2}p,{1}r,{2}q)+(−1)r+1ζ★({2}q,r+2,{2}p−1). With the help of this equation and Zagier’s ζ★({2}p,3,{2}q) formula, we can easily determine ζ★({2}p,1,{2}q) and several interesting expressions.

An experimental design of the blockchain business model using a soft system dynamics modeling approach

Purpose The growing discussion on blockchain and business models often falls short of demonstrating and evaluating systems consistently exposed to settings of dynamic complexity. Therefore, in practicing systems thinking, this study aims to provide a depiction of dynamic complexity in blockchain business models and develop policy-based scenarios to enhance blockchain-based systems behavior. Design/methodology/approach This study integrated the soft system dynamic (SD) methodology approach, which focuses on a situation analysis and SDs in policy design. This single case study chose a firm engaged in the content industry, where the adoption of blockchains is a solution to tackle the industry’s significant challenges. Data were collected using a qualitative approach and then adapted into a simulation model. Findings The study pinpointed key parameters significantly affecting the system through a sensitivity analysis. Then, this experimental study found that all improvement initiatives delivered better system performance. At the same time, the study also identified counterintuitive findings, where the interventions using multiple value subsystems had insignificant effects on the system compared to a single advent. Originality/value This study illuminates the growing field of blockchain and business models through system modeling and experimentation, using an integrative approach like soft system dynamics methodology. It also identifies and demonstrates value distribution and the dynamic complexity inherent in the blockchain business model.

A Novel Defect Quantification Method Utilizing Multi-Sensor Magnetic Flux Leakage Signal Fusion

In the assessment of pipeline integrity using magnetic flux leakage (MFL) detection, it is crucial to quantify defects accurately and efficiently using MFL signals. However, in complex detection environments, traditional defect inversion methods exhibit low quantification accuracy and efficiency due to the complexity of their algorithms or excessive reliance on a priori knowledge and expert experience. To address these issues, this study presents a novel defect quantification method based on multi-sensor signal fusion (MSSF). The method employs a multi-sensor probe to fuse the MFL signals under multiple lift-off values, enhancing the diversity of defect information. This enables defect-opening profile recognition using the characteristic approximation approach (CAA). Subsequently, the MSSF method is based on a 3D magnetic dipole model and integrates the structural features of multi-sensor probes to develop an algorithm. This algorithm iteratively determines the defect depth at multiple data acquisition points within the defect region to obtain the maximum defect depth. The feasibility of the MSSF quantification method is validated through finite element simulation and physical experiments. The results demonstrate that the proposed method achieves accurate defect quantification while enhancing efficiency, with the number of iterations for each defect depth calculation point consistently requiring fewer than 15 iterations. For rectangular metal loss, perforation, and conical defects, quantification errors are less than 10%, meeting practical inspection requirements.

Exploring the impact of multiple stressors on cultural and ecological values: A case study from Te Rerenga Parāoa (Whangārei Harbour), Aotearoa New Zealand

Estuarine systems often have degraded ecology and functioning due to the multiple interacting stressors that they face. While conventional management might address such a situation with narrow focus, ecosystem based management provides a more holistic and adaptive option allowing for multiple objectives and values. Such an approach has some commonality with that of the Indigenous Māori of New Zealand, who have holistic, interconnected and intergenerational aspirations. In this study we utilised Te-Rerenga-Parāoa, Whangārei Harbour, Aotearoa New Zealand as a case study to explore how ecological and Māori knowledge systems and approaches may align to address complex multi-stressor and multi-value scenarios common to estuarine systems. To achieve this we co-developed a Bayesian network heuristic modelling tool which provided a probabilistic framework to consider the relationship between stressors and the outcome states for ecological function and cultural values. The process of model development itself was a rewarding exercise of joint learning. For example, this process illustrated that while western ecological values can focus on endpoints such as abundance, cultural values were more diverse, experiential in nature, with value often centred on a cultural practise being enabled, not just the ecological component the value was connected to. Once the model was developed we considered three specific scenarios relating to harbour dredging, environmental degradation from land use changes, and fishery extraction. Model predictions for these scenarios generally matched expectations, but were generic in nature and could therefore benefit from more specificity relating to aspects such as spatial scale and context around cultural interpretation. Overall, the model demonstrated generic utility as an interactive educational tool for resource managers considering the broader impacts (on ecology and cultural values) of major societal challenges.

Efficient Estimation of Synthetic Indicators for the Assessment of Nonlinear Systems Quality

The availability of synthetic indicators of the degree and type of nonlinearity in systems is used in various fields to assess system quality or to highlight possible malfunctions. Different distortion or damage indexes are synthetic measures designed (and standardized) to evaluate the frequency trend of specific aspects resulting from the nonlinear behavior of the system under consideration. The different measures of deviation from linear behavior quantitatively consider the system and its nonlinearity characteristics; they were defined according to practically feasible measurement methodologies and the various aspects of the system’s nonlinearity that needed to be highlighted. In parallel, techniques for representing and modeling nonlinear systems have been defined, capable of describing the system in a more general way, attempting to capture its input–output characteristics by varying the level of stress to which the system is subjected. Numerous modeling techniques have been proposed, aimed at representing the nonlinear behavior of physical devices. In this paper, after an extensive description of the Hammerstein model identification technique based on swept sinusoidal signals, we show how the nonlinear model of the system can be used to obtain accurate estimates of the parameter aimed at describing the nonlinearity characteristics of the system. This extensive description makes it possible to point out that the same Hammerstein model can be obtained not only from a single type of excitation, but it is shown that the identification technique can be extended to input signals of different types. The description of the method also makes clear the motivation behind the introduction of the proposed original technique for estimating, from a single measurement, the model parameters of the nonlinear system—and from these the synthetic estimators—relative to multiple values of the input signal amplitude, thus enabling a considerable increase in the estimation efficiency of these parameters. The proposed technique is verified with both synthetic and laboratory experiments, demonstrating the effectiveness of the method in evaluating nonlinear system parameters, distortion estimates, and parameters defined for an early detection of defects of the structure.

What hinders the transition towards a bio-based construction sector? A global innovation system perspective on its value chain

The construction sector is heavily polluting and an actual threat to the natural environment, therefore its transition towards becoming bio-based is imperative. This transition is currently unfolding and it is driven in particular by innovation activities taking place along the sector’s value chain. In the upstream segment, bio-based materials are being improved, while novel building techniques in the core segment enable the use of these materials. In this paper, we utilize the global innovation systems (GIS) framework to examine these innovation activities and their valuation dynamics. In particular, we investigate how the GIS of the bio-based construction sector is organized along its value chain, providing insights into the barriers to the sector's sustainability transition. Our empirical analysis, based on a rich set of expert interviews, demonstrates that the GIS configuration changes along the value chain, driven by profound differences in the innovation mode. This situation creates a bottleneck that hinders the sector's transition, where knowledge about bio-based materials developed upstream fails to translate down the value chain. However, we also find that several niche firms cover and integrate multiple value chain segments and overcome this knowledge gap, suggesting that the transition towards a bio-based construction sector could accelerate with further innovation system reconfigurations.

Non-holomorphic modular forms from zeta generators

We study non-holomorphic modular forms built from iterated integrals of holomorphic modular forms for SL(2, ℤ) known as equivariant iterated Eisenstein integrals. A special subclass of them furnishes an equivalent description of the modular graph forms appearing in the low-energy expansion of string amplitudes at genus one. Notably the Fourier expansion of modular graph forms contains single-valued multiple zeta values. We deduce the appearance of products and higher-depth instances of multiple zeta values in equivariant iterated Eisenstein integrals, and ultimately modular graph forms, from the appearance of simpler odd Riemann zeta values. This analysis relies on so-called zeta generators which act on certain non-commutative variables in the generating series of the iterated integrals. From an extension of these non-commutative variables we incorporate iterated integrals involving holomorphic cusp forms into our setup and use them to construct the modular completion of triple Eisenstein integrals. Our work represents a fully explicit realisation of the modular graph forms within Brown’s framework of equivariant iterated Eisenstein integrals and reveals structural analogies between single-valued period functions appearing in genus zero and one string amplitudes.

Research on defect image inversion method based on multiple lift-off values magnetic memory signals

Metal magnetic memory (MMM) technique can detect macroscopic defects and stress concentration zones of ferromagnets, and it can identify the approximate positions of these flaws, while a little further information about defects characteristics can be provided. To promote study in this area, defects were modeled as located magnetic dipoles whose strength and locations should be determined, and the technique of truncated generalized inverse and reduced space inversion were used to image the dipole strength. Through the simulation experiment, we found that the inversion image quality was influenced by the lift-off value. When the magnetic field data for inversion was measured with a small lift-off value, the inner dipoles contributions were easily buried and the inversion image can only shown the surface dipole well. In contrast, if the magnetic field data was measured with a greater lift-off value, it was possible to inverse the deep dipoles while the image was somewhat out of focus. To overcome this limitation, we composed the magnetic field data measured under different lift-off values together for inversion. We used the same model to test this approach and applied it on real crack defects magnetic field data. It demonstrates that the new inversion approach shows better accuracy than the single lift-off value and it was possible to infer the location and size of defect by the evaluation of preset dipoles.

Value Set Hub: Software for developing and curating high-quality value sets.

Terminology value sets are central to research studies using real-world patient data. We describe the considerable challenges that arise when researchers develop value sets for their studies or reuse those made by others. We present Value Set Hub (VS-Hub), software to overcome these challenges, describing its design, implementation, features, use in the field, lessons learned, and future directions. Over a five-month period, VS-Hub has been used by over 200 users and has been used in the development and curation of 95 recommended value sets for commonly studied conditions, treatments, and lab tests. Particular innovations include the presentation of multiple value sets on the same screen for easy comparison, the display of compared value sets in the context of vocabulary hierarchies, the integration of these analytic features and value set authoring, and value set browsing features that encourage users to review existing value sets that may be relevant to their needs.

Identification and Spatial Characterization of suburban areas in Chengdu

As transitional territories where urban and rural functions interpenetrate, suburban areas have multiple values such as recreation, ecology, agriculture and landscape. Promoting their benign development is the key to realizing global sustainable development of urban and rural areas. Planning control is based on precisely defining the spatial extent of suburban areas, and understanding suburban area differentiation and its driving forces scientifically is a key component in raising the bar for focused planning. However, there are currently few research on fine-scale quantitative identification of suburban regions because of their spatial complexity, ambiguous boundaries, and structural dynamic features. Based on this, this paper develops a multi-dimensional identification index system for suburban areas by using multi-source big data and remote sensing information technology, employs the random forest model and the K-Medoids clustering algorithm, identifies the distribution of suburban areas and their subdivided types, analyzes spatial differentiation characteristics, and conducts empirical research using Chengdu City as an example. The findings demonstrate that: (1) The majority of suburban areas in Chengdu City are concentrated in groups or belts surrounding the urban centers of various districts and counties, and the higher the functional class of a district or county, the denser the distribution of suburban areas. (2)The distribution pattern of suburban areas in Chengdu basically conforms to its planned urban hierarchy system.(3)Urban-oriented, suburban-interacted and rural-oriented suburban areas have their own characteristics in land use level, economic development and regional population characteristics.(4)Regional openness, government behavior, social development and geographical location are the core driving factors of spatial differentiation of suburban areas in Chengdu, and the interaction between urban and rural systems, the flow of social resources and the transformation of development models are the spatial mapping dimensions that affect its differentiation.(5)Chengdu can draw up detailed regulatory planning of suburban units to standardize and guide the overall development of suburban areas. In support of the global sustainable development goals, this study offers a methodology for precisely and impartially defining suburban areas, assisting in the implementation of integrated urban-rural development globally.

Modelling the complementarity and flexibility between different shared modes available in smart electric mobility hubs (eHUBS)

eHUBS are physical locations that integrate two or more electric shared mobility modes. As they provide transport users easier access to a wide range of transport modes, multimodal behaviour is expected to be more common. However, this issue has not been addressed in previous stated preference studies on mode choices involving innovative transport modes. In this study, multimodal behaviour is explicitly addressed both in measurement and in modelling by adopting the multiple discrete–continuous (MDC) modelling framework in contrast to discrete choice models. Instead of asking transport users to indicate the most preferred alternative, they were allowed to choose more than one alternative by allocating trips between several modes. This study aims to answer two questions: 1) whether there is complementarity between the multiple shared modes offered in eHUBS and 2) how would transport users adapt when one of the shared modes that they plan to use becomes unavailable. Using stated mode choice data of non-commuting trips from transport users whose current mode is driving a private car in Manchester, UK, several models under the MDC framework were estimated including Multiple Discrete-Continuous Extreme Value (MDCEV) model, mixed MDCEV model, and the extended Multiple Discrete Continuous (eMDC) model. The results show that there is complementarity between shared electric vehicle (EV) and electric bike (e-bike) offered in the eHUBS. In addition, the research show that the flexibility between those two shared modes is stronger than assumed in the MDCEV model, and common preference heterogeneity cannot fully account for this phenomenon.

Examining the combined benefits of photobiomodulation and apigenin for the treatment of asthenozoospermia: An innovative therapeutic strategy.

Individuals suffering from asthenospermia, an infertility disorder, have reduced sperm motility. This study's goal was to identify the impacts of diverse photobiomodulation procedures on the motility of sperm in vitro in patients with asthenospermia, either in isolation or in combination with Apigenin. At 633 nm and 808 nm, the lasers are used with multiple dose values (0.6, 1.2, and 2.4) J/cm2 and altering Apigenin concentrations (5, 10, 25, and 50 μM). All of the photobiomodulation procedures were assessed. Assessing factors were the DNA fragmentation index, sperm viability, as well as progressive sperm motility. The progressive sperm motility results for 633 nm and 808 nm show a significant increase over 633 nm + 808 nm after 60 min after irradiation. Sperm motility increased more quickly under the 808 nm procedure than under the other procedures (p < 0.02). The observation of progressive sperm motility indicated that a 10 μM concentration of Apigenin created higher results than other concentrations (p < 0.01). Apigenin with 808 nm at 1.2 J/cm2 resulted in better sperm motility (p < 0.01) and decreased DNA fragmentation index.There was a notable increase (p < 0.05) in the DNA fragmentation index with the 633 nm + 808 nm procedure. At a 10 μM concentration of Apigenin, the DNA fragmentation index was lower than at a 50 μM concentration (p < 0.02). Neither Apigenin nor photobiomodulation significantly decreased sperm viability. The study suggests that asthenozoospermia patients may benefit from apigenin utilized alongside photobiomodulation, while further investigation is required.

6938 An Unusual Case of New-onset autoimmune insulin dependent Diabetes Mellitus Complicating Gestational Diabetes Mellitus in A Pregnant Female Diagnosed Early Postpartum - The Enigma of Autoimmunity and Diabetes During Pregnancy

Abstract Disclosure: R. Abdelmasih: None. W. Pan: None. Introduction: Gestation diabetes is the most common metabolic disorder in pregnancy affecting 25% of pregnancies. T1DM accounts for 0.2-0.5% of all pregnancies in the United States yearly. New-onset autoimmune insulin-dependent diabetes mellitus during pregnancy is very uncommon and it might be overlooked. Diabetes-related autoantibodies have been studied in women with GDM with prevalence of 0-10%. There is controversial data regarding the clinical significance of checking diabetes antibodies in pregnancy. Case Presentation: A 22-year-old G1P1 female with history of Hashimoto hypothyroidism presented during early second trimester for evaluation of GDM with serum glucose of 300s mg/dl in a random blood sample with multiple subsequent high values, hemoglobin A1C of 9.4% (prepartum value was 5.3%). Patient was started on insulin during pregnancy. Pregnancy was complicated with cesarian section and macrosomia. 1 week postpartum, A1c was 5.4%, Insulin was discontinued, and patient was continued on Metformin. 2 months Postpartum, patient presented with persistent high blood glucose values of 300-400 mg/dl with no signs of DKA. Glipizide and Glargine 15 units daily were added though patient continued to have hyperglycemia. Laboratory workup were remarkable for high blood glucose of 342 mg/dl, low C peptide of 0.6 ng/dl, and positive glutamic acid decarboxylase antibodies (GAD Ab) 38.8 U/ml. oral antidiabetics were discontinued and prandial insulin was added. Discussion: GDM is a risk factor for DMII after pregnancy given high insulin resistance postpartum, Data regarding new-onset DMI and diabetes autoimmunity during pregnancy is lacking. About 0-10% of women with gestational diabetes develop DM autoantibodies. GAD Ab exhibits the highest sensitivity of 63% for type 1 diabetes prediction compared to Islet cell antibodies (48%), and tyrosine phosphatase-related islet antigen 2 (34%). Pathophysiology of autoimmune DM during pregnancy is not quite understood; It is thought that when autoimmune tendency is present, GDM causes β-cells to deteriorate due to excessive insulin production in setting of insulin resistance, and the burden of third-trimester gestation may cause the insulin deficiency to be expressed sooner than non-pregnant female. So far, Literature about the predictive value or specific clinical features of checking diabetes autoantibodies is discordant and inconclusive. It is suggested to screen for autoantibodies only when array of clinical hallmarks suggestive of an autoimmune DM form of GDM present as young age, low BMI, need for insulin therapy early in pregnancy, presence of ketones, concurrent other autoimmune conditions (e.g., thyroiditis). This case is to shed light on autoimmunity in pregnancy. More data is needed with longer follow-up to better identify women at risk, their characteristics, indications for autoantibodies testing, follow and prognosis. Presentation: 6/2/2024

Crossed Product Interpretation of the Double Shuffle Lie Algebra Attached to a Finite Abelian Group

Racinet studied a scheme associated with the double shuffle and regularization relations between multiple polylogarithm values at N th roots of unity and constructed a group scheme attached to the situation; he also showed it to be the specialization for G=\mu_{N} of a group scheme \operatorname{\mathsf{DMR}}_{0}^{G} attached to a finite abelian group G . Then Enriquez and Furusho proved that \operatorname{\mathsf{DMR}}_{0}^{G} can be essentially identified with the stabilizer of a coproduct element arising in Racinet’s theory with respect to the action of a group of automorphisms of a free Lie algebra attached to G . We reformulate Racinet’s construction in terms of crossed products. Racinet’s coproduct can then be identified with a coproduct \widehat{\Delta}^{\mathcal{M}}_{G} defined on a module \widehat{\mathcal{M}}_{G} over an algebra \widehat{\mathcal{W}}_{G} , which is equipped with its own coproduct \widehat{\Delta}^{\mathcal{W}}_{G} , and the group action on \widehat{\mathcal{M}}_{G} extends to a compatible action of \widehat{\mathcal{W}}_{G} . We then show that the stabilizer of \widehat{\Delta}^{\mathcal{M}}_{G} , hence \operatorname{\mathsf{DMR}}_{0}^{G} , is contained in the stabilizer of \widehat{\Delta}^{\mathcal{W}}_{G} thus generalizing a result of Enriquez and Furusho [Selecta Math. (N.S.) 29 (2023), article no. 3]. This yields an explicit group scheme containing \operatorname{\mathsf{DMR}}_{0}^{G} , which we also express in the Racinet formalism.