Linear Representation Research Articles

Preview control for linear parameter‐varying discrete‐time systems using a linear fractional representation

AbstractIn order to create a mixed tracking controller for discrete‐time linear parameter‐varying (LPV) systems, this work introduces a unique control strategy employing a linear fractional representation (LFR). Under the premise that the reference signal can be previewable, a design strategy for a controller with preview compensations is suggested. First, a preview tracking control issue is changed into a regulator problem by adding two additional linked auxiliary parameters into the initial system state/input, which further creates more enhanced error systems incorporating future information of reference signals. The suggested conditions then depend on utilizing slack variables with decision matrices related to the LFR approach in order to give innovative preview controller designs. We also construct reduced conservative synthesis situations by taking into account parameter‐dependent Lyapunov functions and full‐block multipliers. In order to create state‐feedback and output‐feedback preview tracking controllers, design requirements are characterized as linear matrix inequalities. Two simulation studies are completed as the last step to highlight how successful the suggested control strategy is.

A New Marshall-Olkin Generalized-k Family of Distributions with Applications

Abstract In this article, we describe a new Marshall-Olkin family of distributions based on the Marshall-Olkin family of distributions by employing a different generator. We have derived several statistical properties of the new Marshall-Olkin family, including survival function, hazard rate function, quantile function, moments, and linear representation. We have taken the Weibull distribution as a baseline distribution and found that the new Marshall-Olkin Weibull distribution exhibits a variety of shapes for its hazard rates and densities. We have tested the performance of various estimation methods for the new Marshall-Olkin family of distributions, including eight different estimation methods. Additionally, we conducted extensive simulations to assess the robustness of the estimation methods across various scenarios and determine which methods are most effective for varying sample sizes. To further validate the new Marshall-Olkin Weibull distribution, we have used two different data sets to compare the fitted new Marshall-Olkin Weibull distribution with the well-known comparative models. By testing the performance of different estimation methods and using multiple data sets, we can demonstrate the versatility and usefulness of the new Marshall-Olkin family of distributions for modeling a wide range of phenomena.

A procedure for addressing the soil-abutment interaction problem in the safety assessment of bridges

The paper proposes a methodology for assessing the seismic soil-foundation-abutment response in the spirit of the sub-structure approach. The methodology is suitable for conventional bridge abutments characterized by massive structural components and accommodates for various structural and soil configurations, from both stratigraphic and topographic perspectives. The abutment is assumed to be rigid, which is a generally acceptable hypothesis given the typical geometry of conventional bridge abutments with walls, orthogonal wingwalls, and a concrete deep or strip foundation. The methodology aligns with codes, which foresees a linear behaviour for the abutments and suggest a low behaviour factor to account for dissipative capabilities due to the backfills and the radiation phenomena in the soil. The soil is assumed to perform elastically, but the overall soil nonlinearity induced by seismic wave propagation can be incorporated into the method using a linear equivalent representation of soil properties. The methodology is applied to a real case study in order to demonstrate its effectiveness in addressing the soil-structure interaction problem and to provide practical suggestions for the method implementation, including possible simplifications.

Cartan projections of fiber products and non‐quasi‐isometric embeddings

AbstractLet be a finitely generated group and be a normal subgroup of . The fiber product of with respect to is the subgroup of the direct product . For every representation , where is a local field, we establish upper bounds for the norm of the Cartan projection of in terms of a fixed word length function on . As an application, we exhibit examples of finitely generated and finitely presented fiber products , where is linear and Gromov hyperbolic, such that does not admit linear representations that are quasi‐isometric embeddings.

Direct evidence for logarithmic magnitude representation in the central nervous system

Fechner’s law proposes a logarithmic relationship between the physical intensity and perceived magnitude of a stimulus. The principle of logarithmic magnitude representation has been extensively utilized in various theoretical frameworks. Although the neural correlates of Weber’s law have been considered as possible evidence for Fechner’s law, there is still a lack of direct evidence for a logarithmic representation in the central nervous system. In our study, participants were asked to reproduce the time intervals between two circles and ignore their spatial distances while electroencephalogram (EEG) signals were recorded synchronously. Behavioral results showed that a Bayesian model, which assumes a logarithmic representation of spatiotemporal information, was better at predicting production times than a model relying on a linear representation. The EEG results revealed that P2 and P3b amplitudes increased linearly with the logarithmic transformation of spatiotemporal information, and these event-related potentials were localized in the parietal cortex. Our study provides direct evidence supporting logarithmic magnitude representation in the central nervous system.

From human explanations to explainable AI: Insights from constrained optimization

Many complex decision-making scenarios encountered in the real-world, including energy systems and infrastructure planning, can be formulated as constrained optimization problems. Solutions for these problems are often obtained using white-box solvers based on linear program representations. Even though these algorithms are well understood and the optimality of the solution is guaranteed, explanations for the solutions are still necessary to build trust and ensure the implementation of policies. Solution algorithms represent the problem in a high-dimensional abstract space, which does not translate well to intuitive explanations for lay people. Here, we report three studies in which we pose constrained optimization problems in the form of a computer game to participants. In the game, called Furniture Factory, participants manage a company that produces furniture. In two qualitative studies, we first elicit representations and heuristics with concurrent explanations and validate their use in post-hoc explanations. We analyze the complexity of the explanations given by participants to gain a deeper understanding of how complex cognitively adequate explanations should be. Based on insights from the analysis of the two qualitative studies, we formalize strategies that in combination can act as descriptors for participants’ behavior and optimal solutions. We match the strategies to decisions in a large behavioral dataset (>150 participants) gathered in a third study, and compare the complexity of strategy combinations to the complexity featured in participants’ explanations. Based on the analyses from these three studies, we discuss how these insights can inform the automatic generation of cognitively adequate explanations in future AI systems.

ReLU surrogates in mixed-integer MPC for irrigation scheduling

Efficient water management in agriculture is important for mitigating the growing freshwater scarcity crisis. Mixed-integer Model Predictive Control (MPC) has emerged as an effective approach for addressing the complex scheduling problems in agricultural irrigation. However, the computational complexity of mixed-integer MPC still poses a significant challenge, particularly in large-scale applications. This study proposes an approach to enhance the computational efficiency of mixed-integer MPC-based irrigation schedulers by employing Rectified Linear Unit (ReLU) surrogate models to describe the soil moisture dynamics of the agricultural field. By leveraging the mixed-integer linear representation of the ReLU operator, the proposed approach transforms the mixed-integer MPC-based scheduler with a quadratic cost function into a mixed-integer quadratic program, which is the simplest class of mixed-integer nonlinear programming problems that can be efficiently solved using global optimization solvers. The effectiveness of this approach is demonstrated through comparative studies conducted on a large-scale agricultural field across two growing seasons, involving other machine learning surrogate models, specifically Long Short-Term Memory (LSTM) networks, and the triggered irrigation scheduling method. The ReLU-based approach significantly reduces solution times—by up to 99.5%—while achieving comparable performance to the LSTM approach in terms of water savings and Irrigation Water Use Efficiency (IWUE). Moreover, the ReLU-based approach achieves enhanced performance in terms of irrigation water savings and IWUE compared to the triggered approach.

Modeling and analysis of a flexible spinning Euler-Bernoulli beam with centrifugal stiffening and softening: A linear fractional representation approach with application to spinning spacecraft

The derivation of a linear fractional representation (LFR) model for a flexible, spinning and uniform Euler-Bernoulli beam is accomplished using the Lagrange technique, fully capturing the centrifugal force generated by the spinning motion and accounting for its dependence on the angular velocity. This six degrees of freedom (DOF) model accounts for the behavior of deflection in the moving body frame, encompassing the bending, traction and torsion dynamics. The model is also designed to be compliant with the Two-Input-Two-Output Port (TITOP) approach, which offers the possibility to model complex multibody mechanical systems, while keeping the uncertain nature of the plant and condensing all the possible mechanical configurations in a single LFR. To evaluate the effectiveness of the model, various scenarios are considered and their results are tabulated. These scenarios include uniform beams with fixed root boundary conditions for different values of tip mass, root offset and angular velocity. The results from the analysis of the uniform cantilever beam are compared with solutions found in the literature and obtained from a commercial finite element software. Ultimately, this paper presents a multibody model for a spinning spacecraft mission scenario. A comprehensive analysis of the system dynamics is conducted, providing insights into the behavior of the spacecraft under spinning conditions.

Identification of a Triangular Random Coefficient Model using a Correction Function

Abstract Previously, identification of triangular random coefficient models required a restriction on the dimension of the first stage heterogeneity or independence assumptions across the different sources of the heterogeneity. This note proposes a new identification strategy that does not rely on either of these restrictions but rather assumes conditional means have a conditional linear projection representation in order to construct “correction functions” to address endogeneity and gain identification of the average partial effect. This identification strategy allows for both continuous and discrete instruments. Finally, the proposed identification method is illustrated in estimating the returns to education.

Collision dynamics between breather and lump-type localized waves in the (3+1)-dimensional shallow water wave equation

In this paper, the collision dynamics of breather and lump-type localized waves in the (3+1)-dimensional shallow water wave equation are investigated in detail. Firstly, the auto-Bäcklund transformation and the linear representation of the (3+1)-dimensional shallow water wave equation are derived in virtue of the truncated Painlevé expansion method, which provide convenience in solving the (3+1)-dimensional shallow water wave equation. Secondly, based on the linear representation and the principle of linear superposition, the rational solutions in the exponential and polynomial forms are constructed. Tuning the free parameters of the rational solutions, localized waves of various patterns are obtained such as breather, lump-type localized waves and their hybrid structure. The anomalous inelastic interaction phenomenons of breather and lump-type localized waves are exhibited. Thirdly, combining the large-time behaviors of solution with the velocity relationship of localized waves, the dynamical properties and the classification of localized wave solutions are discussed in detail. Finally, we discuss the bound state of breather and lump-type localized waves under the velocity resonance condition, three different types of lump-breather molecules are displayed. The obtained results further enrich the structures and dynamical behaviors of localized waves. It is expected that the interaction phenomena taking place in the (3+1)-dimensional shallow water wave equation will be helpful in predicting or controlling some related shallow water wave phenomena.

On irreducible representation of cover of Lie algebras

Batten introduced the concept of a cover for Lie algebras in [3]. In this article, we describe the irreducible representation of cover of Lie algebras by establishing a bijective correspondence between the set of irreducible linear representations of cover of Lie algebras and the set of projective representations of Lie algebras.

Predictive Models for Aggregate Available Capacity Prediction in Vehicle-to-Grid Applications

The integration of vehicle-to-grid (V2G) technology into smart energy management systems represents a significant advancement in the field of energy suppliers for Industry 4.0. V2G systems enable a bidirectional flow of energy between electric vehicles and the power grid and can provide ancillary services to the grid, such as peak shaving, load balancing, and emergency power supply during power outages, grid faults, or periods of high demand. In this context, reliable prediction of the availability of V2G as an energy source in the grid is fundamental in order to optimize both grid stability and economic returns. This requires both an accurate modeling framework that includes the integration and pre-processing of readily accessible data and a prediction phase over different time horizons for the provision of different time-scale ancillary services. In this research, we propose and compare two data-driven predictive modeling approaches to demonstrate their suitability for dealing with quasi-periodic time series, including those dealing with mobility data, meteorological and calendrical information, and renewable energy generation. These approaches utilize publicly available vehicle tracking data within the floating car data paradigm, information about meteorological conditions, and fuzzy weekend and holiday information to predict the available aggregate capacity with high precision over different time horizons. Two data-driven predictive modeling approaches are then applied to the selected data, and the performance is compared. The first approach is Hankel dynamic mode decomposition with control (HDMDc), a linear state-space representation technique, and the second is long short-term memory (LSTM), a deep learning method based on recurrent nonlinear neural networks. In particular, HDMDc performs well on predictions up to a time horizon of 4 h, demonstrating its effectiveness in capturing global dynamics over an entire year of data, including weekends, holidays, and different meteorological conditions. This capability, along with its state-space representation, enables the extraction of relationships among exogenous inputs and target variables. Consequently, HDMDc is applicable to V2G integration in complex environments such as smart grids, which include various energy suppliers, renewable energy sources, buildings, and mobility data.

A new extension of the Gumbel distribution with biomedical data analysis

In the field of biomedical research, data characteristics often exhibit significant variability, challenging the applicability of classical Gumbel distribution for biomedical data modeling. To address this, this paper introduces a novel extension of the Gumbel model known as the odd beta prime Gumbel (OBP-Gum) model. Derived from the odd beta prime family, the new distribution exhibits greater kurtosis compared to the traditional Gumbel distribution. Importantly, the proposed distribution is designed to capture right-skewed, left-skewed, and nearly symmetric density functions, as well as increasing, decreasing, constant, and upside-down bathtub shapes for its hazard rate function, providing excellent curvature features for creating flexible statistical models for biomedical research. We derive the fundamental features of the OBP-Gum model, such as the quantile function, linear representations, moment generating function, moments, skewness, kurtosis, incomplete moments, and Rényi and Tsallis entropies. Parameter estimation for this new model is conducted using the maximum likelihood estimation method. A simulation study demonstrates the performance of the model parameters. The empirical findings, based on applications to two biomedical datasets, suggest that the OBP-Gum distribution outperforms existing models, particularly in handling extreme observations. Instead of relying on conventional models for decision-making, this research provides relevant stakeholders with an improved statistical distribution for more accurate biomedical data modeling.

Probing the capacity of a spatiotemporal deep learning model for short-term PM2.5 forecasts in a coastal urban area

Accurate forecast of fine particulate matter (PM2.5) is crucial for city air pollution control, yet remains challenging due to the complex urban atmospheric chemical and physical processes. Recently deep learning has been routinely applied for better urban PM2.5 forecasts. However, their capacity to represent the spatiotemporal urban atmospheric processes remains underexplored, especially compared with traditional approaches such as chemistry-transport models (CTMs) and shallow statistical methods other than deep learning. Here we probe such urban-scale representation capacity of a spatiotemporal deep learning (STDL) model for 24-hour short-term PM2.5 forecasts at six urban stations in Rizhao, a coastal city in China. Compared with two operational CTMs and three statistical models, the STDL model shows its superiority with improvements in all five evaluation metrics, notably in root mean square error (RMSE) for forecasts at lead times within 12 h with reductions of 49.8 % and 47.8 % respectively. This demonstrates the STDL model's capacity to represent nonlinear small-scale phenomena such as street-level emissions and urban meteorology that are in general not well represented in either CTMs or shallow statistical models. This gain of small-scale representation in forecast performance decreases at increasing lead times, leading to similar RMSEs to the statistical methods (linear shallow representations) at about 12 h and to the CTMs (mesoscale representations) at 24 h. The STDL model performs especially well in winter, when complex urban physical and chemical processes dominate the frequent severe air pollution, and in moisture conditions fostering hygroscopic growth of particles. The DL-based PM2.5 forecasts align with observed trends under various humidity and wind conditions. Such investigation into the potential and limitations of deep learning representation for urban PM2.5 forecasting could hopefully inspire further fusion of distinct representations from CTMs and deep networks to break the conventional limits of short-term PM2.5 forecasts.

Effect of Model Fidelity on High-Speed Aeroelastic Behavior of a Cantilever Plate

Turbulence, flow separation, and shock dynamics challenge the modeling and analysis of high-speed aeroelastic behavior. Motivated by this, the importance of modeling the fidelity of the flow is explored in the aeroelastic response of a cantilever plate in an Ma=2.0 separating turbulent flow using unsteady Reynolds-averaged Navier–Stokes (URANS) and URANS-enriched local piston theory (LPT). Structural modeling assumptions are also evaluated using both linear and nonlinear representations. Close agreement in the predicted aeroelastic steady state is observed. However, large discrepancies in the dynamic aeroelastic response predictions are found and ultimately linked to the neglect of deformation-induced cavity pressure fluctuations and dynamic flow separation in the LPT model. Interestingly, the dynamic flow separation induces a fluid-driven limit cycle oscillation in the postflutter regime. Furthermore, structural nonlinearity is not found to have a strong impact on the conditions and configurations considered.

Persistent Extreme Surface Solar Radiation and Its Implications on Solar Photovoltaics

AbstractClimatic extreme events are important because they can strongly impact humans, infrastructure, and biodiversity and will be affected by a changing climate. Surface Solar Radiation (SSR) is the primary energy source for solar photovoltaics (PV), which will be indispensable in future zero‐emissions energy systems. Despite their pivotal role, extreme events in SSR remain under‐documented. We provide a starting point in extreme SSR analysis by focusing on events caused by internal variability alone and therefore building a baseline for future extreme SSR research. We analyze extreme SSR events using daily‐mean data from the pre‐industrial control simulations (piControl) of the Coupled Model Intercomparison Project—Phase 6. We investigate their role in PV energy generation using the Global Solar Energy Estimator with the intent of strengthening the energy system's resilience. Our results show a pronounced asymmetry between consecutive days with extremely high and extremely low solar radiation over land, the former occurring more frequently than the latter. Moreover, our results call for detailed PV generation modeling that includes panel geometry. Simple models based on linear SSR representations prove insufficient due to pronounced seasonal variations and strong non‐linear SSR dependency of high extremes. Our results demonstrate how climate model results can be leveraged to understand persistent radiation extremes that are relevant for future energy systems.

POSTMODERNIST VIEWS OF TIME AND SPACE IN LITERATURE

Rejecting canonical narrative forms and ideas, postmodern literature presents a fresh perspective on the passage of time and geographical location. In contrast to the linear and cohesive representations of time and space expected in modernist literature, postmodernist works often embrace dislocation, fragmentation, and distorted timelines. Through metafiction, intertextuality, and non-linear tales, this study investigates how postmodern writers question traditional ideas of space and time. These methods show time as cyclical, irrelevant, or fluid, which disrupts the reader's perception of a steady, chronological development. Similarly, postmodern literature frequently portrays space as hyperreal, fragmented, or disjointed, reflecting a world where the lines between the tangible and the abstract are increasingly blurred. This study analyses seminal postmodern works to see how these changed views of space and time fit into a more extensive critique of postmodern reality, identity, and meaning stability. According to the results, postmodern literature does more than make us rethink the passage of time; it also prompts us to wonder what it means to exist.

On The Representations and Characters of Quaternions Group Q8

We are interested in studying the linear representations of the Quaternions group Q8 by determining it’s charater table and irreducible representations which allows us to construct some representations of degre 6 and 8 of Q8.

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Valerij G. Bardakov and P. Bellingeri introduced a new linear representation ρ¯F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{\\rho }_F$$\\end{document} of degree n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+1$$\\end{document} of the braid group Bn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_n$$\\end{document}. We study the irreducibility of this representation. We prove that ρ¯F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{\\rho }_F$$\\end{document} is reducible to the degree n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-1$$\\end{document}. Moreover, we give necessary and sufficient conditions for the irreducibility of the complex specialization of its n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-1$$\\end{document} degree composition factor ϕ¯F\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\bar{\\phi }_F$$\\end{document}.

Controllability of delayed generalized asynchronous Boolean control networks under disturbances

For delayed generalized asynchronous Boolean networks with random disturbances, it is still uncertain about their robust controllability. This paper uses the semi-tensor product to transform these Boolean networks into linear representations to derive the dependence matrix of the network transition on the control mode. After that, three different control modes are utilized to define controllability. Each control mode provides both reachable trajectories set under a specific initial trajectory and an initial trajectory set under a specific target trajectory. It is also calculated for the likelihood of a transition between two states. Lastly, two concrete examples validate the theoretical findings under various control modes.