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This paper provides a comprehensive analysis of recent advancements in additive manufacturing, a transformative approach to industrial production that allows for the layer-by-layer construction of complex parts directly from digital models. Focusing specifically on Directed Energy Deposition, it begins by clarifying the fundamental principles of metal additive manufacturing as defined by International Organization of Standardization and American Society for Testing and Materials standards, with an emphasis on laser- and powder-based methods that are pivotal to Directed Energy Deposition. It explores the critical process mechanisms that can lead to defect formation in the manufactured parts, offering in-depth insights into the factors that influence these outcomes. Additionally, the unique mechanisms of defect formation inherent to Directed Energy Deposition are examined in detail. The review also covers the current landscape of process evaluation and non-destructive testing methods essential for quality assurance, including both traditional and contemporary in situ monitoring techniques, with a particular focus given to advanced machine-vision-based methods for geometric analysis. Furthermore, the integration of process monitoring, multiphysics simulation models, and data analytics is discussed, charting a forward-looking roadmap for the development of Digital Twins in Laser–Powder-based Directed Energy Deposition. Finally, this review highlights critical research gaps and proposes directions for future research to enhance the accuracy and efficiency of Directed Energy Deposition systems. [ABSTRACT FROM AUTHOR]

Masonry domes have always represented one of the most fascinating and daring architectural structures, with great symbolic relevance. However, their structural behaviour under environmental actions, such as earthquakes, has not yet been thoroughly investigated. This paper defines an interdisciplinary methodological approach, based on advanced digital survey techniques to evaluate constructive, geometrical, and structural aspects of historical domes. The applied procedure comprises the study of archival documents and diagnostic reports, in-situ analyses and advanced three-dimensional surveys, the development of simplified discrete-element-based structural models, and the performing of pushover analyses. The procedure is applied to a case study represented by the dome of the 'Badia di Sant'Agata' church in Catania, an earthquake-prone town located in Eastern Sicily. This church was built after the devastating 1693 earthquake and its dome represents one of the most significant examples among hundreds of masonry domes built in Sicily. The historical and constructive investigation showed that the dome was built without supporting formworks, by laying 21 concentric courses of limestone voussoirs on a peculiar springing ring made of mortared rubble stones. The numerical simulations highlighted that this peculiar ring affects the dome's failure mechanism and limits its displacement capacity and the maximum shear transferred to the substructure. [ABSTRACT FROM AUTHOR]

We present a geometric approach to portfolio theory with a focus on risk-adjusted returns, in particular Jensen's alpha. We find that while the alpha/beta approach has severe limitations, especially in higher dimensions, only minor conceptual modifications (e.g., using orthogonal Sharpe ratios rather than risk-adjusted returns) are needed to identify the efficient trading strategies. We further show that, in a complete market, the so-called market price of risk vector is identical to the growth optimal Kelly vector, albeit expressed in coordinates of a different basis. This implies that a derivative, having an orthogonal Sharpe ratio of zero, has a price given by the minimal martingale measure. [ABSTRACT FROM AUTHOR]

Multi-packer completion strings have played a pivotal role in secure exploitation of oil and gas, increasing production and mitigating greenhouse gas emissions. While the integration of multi-packer structures has proven advantageous in oil and gas exploration, it has introduced complexities in load distribution, consequently giving rise to safety concerns. This study undertakes a thorough examination of the mechanical analysis pertaining to multi-packer completion strings. We present an analytical model for predicting the axial forces acting on the multi-packer string, utilizing the geometric constraint arising from the immobility of packers. It is demonstrated that the pressure differentials at the packers exhibit uniqueness in relation to both initial and boundary conditions, as well as the geometrical constraint. This paper provides an analytical solution for these pressure differentials. Novel concepts regarding the eigen-matrix of an N-packer completion string, influenced solely by Poisson's ratio, a virtual (N + 1) th packer and the eigen-depth and eigen-ratio of two adjacent packers, are proposed and their applications are discussed. Furthermore, this paper delves into a deeper examination of the multi-packer string's underlying mechanism. A consistent algorithm grounded in geometric analysis is developed based on the analytical model. Validation of our model is performed using three practical cases across various operation conditions, and the results demonstrate the efficacy of this methodology in accurately predicting failure occurrences. Sensitivity analysis results further substantiate the robustness of this method in practical applications. Additionally, it has been shown that strategically positioning the packers in areas where the string is highly prone to fractures significantly enhances the safety of the multi-packer string system. The findings presented in this paper offer a foundational framework for analyzing the mechanical behavior of constrained strings. Furthermore, there is potential for the development of the analytical model to incorporate additional factors, such as string system with packers of semi-free or free movement. The proposed method is also of fundamental significance for safety evaluation of string systems in carbon storage projects, which is obtaining increasing attention in the context of carbon neutralization. [ABSTRACT FROM AUTHOR]

This paper reports results on the existence and characterization of nonnegative stationary solutions (including the weak sense) of a spatial one-dimensional degenerate parabolic equations. The characterization of stationary solutions given in this paper refers to the enumeration of those that exist and the presentation of solution information such as the profile and asymptotic behavior of each of them. Due to the influence of terms derived from the degeneracy of the equations, it is not easy to classify and characterize the existence of stationary solutions of the equations considered in this paper. These results are obtained by using dynamical systems theory and geometric approaches (in particular, Poincaré compactification). In addition, an application of the results obtained in this paper is given. The results of the characterization of nonnegative weak stationary solutions of the spatial one-dimensional porous medium equation with special nonlinear terms are shown. These can be obtained by carefully using transformations known from previous studies. [ABSTRACT FROM AUTHOR]

Because they are safe and easy to use, collaborative robots are revolutionizing many sectors, including industry, medicine, and agriculture. Controlling their dynamics, movements, and postures are key points in this evolution. Inverse kinematics is then crucial for robot motion planning. In 6R serial robots, achieving a desired pose is possible with different joint combinations. In this paper, our focus lies in studying forward and, mainly, inverse kinematics of the FANUC CRX-10iA cobot, a 6R cobotic arm with a non-spherical wrist. Its specific structural parameters implies that no analytical solutions exist except for some particular situations. FANUC does not provide the complete set of inverse kinematic solutions, even when 16 solutions are possible, only 8 of them are provided in Roboguide software. Furthermore, the existing literature on joints-to-workspace mapping for CRX cobots is currently very limited. It either lacks or provides partial or inconsistent inverse kinematics analysis. We present and detail a novel fully geometric method for numerically solving inverse kinematics meeting the requirement of high precision and a fast response. This approach provides both the exact number of inverse kinematics solutions and the sets of joint angles even for singular configuration. Its effectiveness was verified through simulations using the Roboguide Software and experimentation on the actual CRX-10iA cobot. Several examples (8, 12, or 16 inverse kinematic solutions) have enabled us to validate and prove the robustness and reliability of this geometric approach. [ABSTRACT FROM AUTHOR]

Background: Trigonocephaly occurs due to the premature fusion of the metopic suture, leading to a triangular forehead and hypotelorism. This condition often requires surgical correction for morphological and functional indications. Metopic ridges also originate from premature metopic closure but are only associated with mid-frontal bulging; their surgical correction is rarely required. Differential diagnosis between these two conditions can be challenging, especially in minor trigonocephaly. Methods: Two hundred seven scans of patients with trigonocephaly (90), metopic rigdes (27), and controls (90) were collected. Geometric morphometrics were used to quantify skull and orbital morphology as well as the interfrontal angle and the cephalic index. An innovative method was developed to automatically compute the frontal curvature along the metopic suture. Different machine-learning algorithms were tested to assess the predictive power of morphological data in terms of classification. Results: We showed that control patients, trigonocephaly and metopic rigdes have distinctive skull and orbital shapes. The 3D frontal curvature enabled a clear discrimination between groups (sensitivity and specificity > 92%). Furthermore, we reached an accuracy of 100% in group discrimination when combining 6 univariate measures. Conclusion: Two diagnostic tools were proposed and demonstrated to be successful in assisting differential diagnosis for patients with trigonocephaly or metopic ridges. Further clinical assessments are required to validate the practical clinical relevance of these tools. [ABSTRACT FROM AUTHOR]

Graph theory plays a crucial role in various applications of mathematics and applied sciences. One specialised branch of graph theory is mathematical chemistry, which focuses on mathematical modelling and analysing chemical compounds and their properties. In this context, graphs are used to represent the structural and topological features of molecules, enabling chemists to gain insights into chemical reactions and make predictions about molecular properties. Recently, new versions of Sombor indices have been introduced using a geometric approach. This article specifically focuses on entropy-based variations of these Sombor indices, which includes SO, $ SO_{red} $ S O red , $ SO_{avg} $ S O avg , $ {}^mSO $ m SO , $ SO_3 $ S O 3 and $ SO_4 $ S O 4 , in the context of graphene sheet. Graphene has gained significant attention in the scientific and technological communities due to its exceptional properties. It finds widespread applications in diverse fields such as nanotechnology, electronics, energy storage, sensors, materials science and optoelectronics. Given the promising applications of graphene, it becomes essential to theoretically analyse its structure. Molecular descriptors play a crucial role, as they are strongly linked to various characteristics of chemical compounds. To better understand the Sombor indices, this article graphically represents their entropy measures. [ABSTRACT FROM AUTHOR]

Summary: Voronoi diagram is a versatile geometric structure used in solving many computational geometry problems in the field of image analysis, computer vision, engineering, medicine, molecular biology and various other disciplines. As a result it is a popular research domain. The literature boasts of many elegant and efficient methods for the construction of the Voronoi diagram in the real space. But these algorithms often fail to perform well when applied to the discrete domain. We present in this paper a novel algorithm for the construction of discrete Voronoi diagram using digital geometric concepts that takes care of the geometric perspective of the Voronoi diagram in the discrete domain. Although our technique is more suited for the CPU but by careful use of some cut‐off heuristics we are able to design a parallel programming approach for the construction of the Voronoi diagram making use of the GPU computational power. The speedup of our algorithm is compatible with the popular GPU technique for Voronoi diagram construction, namely, Jump flood algorithm. But since our algorithm uses the geometric perspective for the construction of Voronoi diagram it gives more accurate output and hence requires no correction step as is the case for the Jump flood algorithm. Furthermore, our approach focuses on extensibility, allowing for easy amenability for construction of higher dimensional Voronoi diagram. This ensures that the Voronoi diagram generation process can be adapted and expanded upon to meet specific requirements or incorporate future enhancements. We present an implementation of the proposed algorithm on a GPU using TAICHI language and describe its performance benefits on multiple input sites. [ABSTRACT FROM AUTHOR]

Chemical graph theory facilitates different tools that have significant applications in drug design and development. Recently Gutman et al., introduced a novel vertex‐degree‐based topological index called elliptic Sombor index and its application through geometric approach. This article focuses on chemical applicability of elliptic Sombor index through regression models using 22 benzenoid hydrocarbons. In this process, the chemical applicability of elliptic Sombor index with various physicochemical properties of benzenoid hydrocarbons through curvilinear regression models is carried out. The data of the statistical analysis shows that elliptic Sombor index is potential predictive index for the property polarizability. [ABSTRACT FROM AUTHOR]

This paper considers the problem of finding maximum volume (axis-aligned) inscribed boxes in a compact convex set, defined by a finite number of convex inequalities, and presents optimization and geometric approaches for solving them. Several optimization models are developed that can be easily generalized to find other inscribed geometric shapes such as triangles, rhombi, and squares. To find the largest volume axis-aligned inscribed rectangles in the higher dimensions, an interior-point method algorithm is presented and analyzed. For two-dimensional space, a parametrized optimization approach is developed to find the largest area (axis-aligned) inscribed rectangles in convex sets. The optimization approach provides a uniform framework for solving a wide variety of relevant problems. Finally, two computational geometric (1−ε) –approximation algorithms with sublinear running times are presented that improve the previous results. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms—Continuous. Funding: The author is grateful for the support of Northeastern University for this research. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information (https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0239) as well as from the IJOC GitHub software repository (https://github.com/INFORMSJoC/2022.0239). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/. [ABSTRACT FROM AUTHOR]

The selection of the optimal methodology for the 3D geometric documentation of cultural heritage is a subject of high concern in contemporary scientific research. As a matter of fact, it requires a multi-source data acquisition process and the fusion of datasets from different sensors. This paper aims to demonstrate the workflow for the proper implementation and integration of geodetic, photogrammetric and laser scanning techniques so that high-quality photorealistic 3D models and other documentation products can be generated for a complicated, large-dimensional architectural monument and its surroundings. As a case study, we present the monitoring of the Mehmet Bey Mosque, which is a landmark in the city of Serres and a significant remaining sample of the Ottoman architecture in Greece. The surveying campaign was conducted in the context of the 2022–2023 annual workshop of the Interdepartmental Program of Postgraduate Studies "Protection Conservation and Restoration of Cultural Monuments" of the Aristotle University of Thessaloniki, and it served as a geometric background for interdisciplinary cooperation and decision-making on the monument restoration process. The results of our study encourage the fusion of terrestrial laser scanning and photogrammetric datasets for the 3D modeling of the mosque, as they supplement each other as regards geometry and texture. [ABSTRACT FROM AUTHOR]

We prove a strong localized gluing result for the general relativistic constraint equations (with or without cosmological constant) in n ≥ 3 spatial dimensions. We glue an ϵ -rescaling of an asymptotically flat data set (γ ^ , k ^) into the neighborhood of a point p ∈ X inside of another initial data set (X , γ , k) , under a local genericity condition (non-existence of KIDs) near p . As the scaling parameter ϵ tends to 0, the rescalings x ϵ of normal coordinates x on X around p become asymptotically flat coordinates on the asymptotically flat data set; outside of any neighborhood of p on the other hand, the glued initial data converge back to (γ , k) . The initial data we construct enjoy polyhomogeneous regularity jointly in ϵ and the (rescaled) spatial coordinates. Applying our construction to unit mass black hole data sets (X , γ , k) and appropriate boosted Kerr initial data sets (γ ^ , k ^) produces initial data which conjecturally evolve into the extreme mass ratio inspiral of a unit mass and a mass ϵ black hole. The proof combines a variant of the gluing method introduced by Corvino and Schoen with geometric singular analysis techniques originating in Melrose's work. On a technical level, we present a fully geometric microlocal treatment of the solvability theory for the linearized constraints map. [ABSTRACT FROM AUTHOR]

The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The research underscores the effectiveness of this geometric approach in the linearization of a class of Newtonian systems that cannot be linearized through symmetry analysis. [ABSTRACT FROM AUTHOR]

The scanner is an important unit of the Coherent Doppler wind Lidar (CDWL) for long range detection. In the compact telescope system, a prism wedge rotating around the optical axis is often added to realize conical scanning of the laser beam. For the focused CDWL, the prism wedge can produce astigmatism in the measuring light, which will change the energy distribution in the detection space. Based on the geometric optics technique, the relation between the wedge parameters and the beam propagation status is analyzed in this paper. A modified range weighting function is also constructed by using the collection efficiency mode. A CDWL with 100 m focused distance was employed in the wind measurement experiments. Compared the measurement data from the CDWL with the anemometer at 50 m and 100 m height at the wind tower, the result indicates that the CDWL detection data is closer to the wind field at 50 m height, which is consistent with theoretical analysis. The optimized wedge parameters and the modified range weighting function are helpful for the further development of the miniaturized and lightweight CDWL. [ABSTRACT FROM AUTHOR]

For a wide class of curvature energy functionals defined for planar curves under the fixed-length constraint, we obtain optimal necessary conditions for global and local minimizers. Our results extend Maddocks' and Sachkov's rigidity principles for Euler's elastica by a new, unified and geometric approach. This in particular leads to complete classification of stable closed p-elasticae for all p ∈ (1 , ∞) and of stable pinned p-elasticae for p ∈ (1 , 2 ] . Our proof is based on a simple but robust "cut-and-paste" trick without computing the energy nor its second variation, which works well for planar periodic curves but also extends to some non-periodic or non-planar cases. An analytically remarkable point is that our method is directly valid for the highly singular regime p ∈ (1 , 3 2 ] in which the second variation may not exist even for smooth variations. [ABSTRACT FROM AUTHOR]

A geometric approach to the integrability and reduction of dynamical systems, both when dealing with systems of differential equations and in classical physics, is developed from a modern perspective. The main ingredients of this analysis are infinitesimal symmetries and tensor fields that are invariant under the given dynamics. A particular emphasis is placed on the existence of alternative invariant volume forms and the associated Jacobi multiplier theory, and then the Hojman symmetry theory is developed as a complement to the Noether theorem and non-Noether constants of motion. We also recall the geometric approach to Sundman infinitesimal time-reparametrisation for autonomous systems of first-order differential equations and some of its applications to integrability, and an analysis of how to define Sundman transformations for autonomous systems of second-order differential equations is proposed, which shows the necessity of considering alternative tangent bundle structures. A short description of alternative tangent structures is provided, and an application to integrability, namely, the linearisability of scalar second-order differential equations under generalised Sundman transformations, is developed. [ABSTRACT FROM AUTHOR]

We study the space of open string effective field theories by combining the constraint of unitarity and monodromy relations for the four-point amplitude. The latter is a reflection of an underlying disk correlator with singularities at the boundary. By assuming maximal susy the resulting bootstrap isolates Wilson coefficients to at least 10−4 of the Type-I superstring. Furthermore, utilizing our geometric approach, we obtain the critical dimension of 10 from the low energy coefficients alone. Remarkably, relaxing SUSY but requiring the massless states to carry four-dimensional helicities, the Wilson coefficients are again constrained to superstring values within 10−4. Thus we conclude that type-I string theory is the unique solution to the monodromy bootstrap with either maximal susy or vector external states. We also introduce Tachyons to the bootstrap and demonstrate for the scattering of external vectors, the bosonic and superstring span the allowed region. Allowed regions for closed string effective field theories are obtained by implementing the KLT relations. [ABSTRACT FROM AUTHOR]

Restricted selection is used to control genetic changes in one or more characters. Three main selection indices are adopted for this purpose. First, Kempthorne's index is used to maximize aggregate breeding value (BV) with changes in some traits restricted to zero; second, Harville's index is used to maximize aggregate BV with proportional changes for some traits; and third, Yamada's index is mathematically used to achieve the relative desired changes for all traits. Kempthorne's index is equivalent to Harville's index. However, the relationship between Kempthorne's and Yamada's indices has not been clarified. In addition, the characteristics of restricted selection indices and the relationship between BV and restricted BV (RBV) are also unknown. The aim of this study was to clarify the characteristics of restricted selection indices and describe the relationship between BV and RBV by using linear algebra and geometric techniques, respectively. First, I proved that Yamada's index is part of Kempthorne's index. Second, I investigated the relationship between BVs that were estimated using an ordinary selection index (EBVs) and RBVs estimated using a restricted selection index (ERBVs) and proved that the ERBVs of the restricted traits are proportional to the relative desired changes. Third, I proved that RBV is represented by a linear function of BV and geometrically represented the relationship between BV and RBV. In this study, new findings on restricted selection indices and RBV were obtained. This useful clarification of the relationship between BV and RBV will make it possible to evaluate the response to selection using not only a restricted selection index, but also a restricted BLUP in computer simulation studies. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the single server queueing model where arrival of units is based on Markovian arrival process and service provided by server is in phase type distribution (MAP/PH/1) with working vacation and working breakdown. The study of working vacation and working breakdown is very important and applicable in real life. The absence of a server during vacation or server breakdown due to any reason may cause service interruption in queueing systems. We consider the repair process in two phases to recover the server from breakdown to working state with improved service rate phase-wise. Arrival process is based on Neuts' versatile point process that follows MAP and service times follow phase type distributions. The arrival rate, the service rate, working vacation rate and working breakdown are mutually independent. Matrix geometric solution method is used to obtain stationary probability vectors. Long run probabilities for all the states are derived in the results. Also with the different combination of arrival and service process the flow of average expected length with several parameters have been presented graphically. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a new profit maximization inventory model in which demand is a power function of selling price, marketing expenditure, and service expenditure, and unit cost is a power function of order quantity. In reality, the exponents of selling price, marketing expenditure, and service expenditure in a demand function and the exponent of order quantity in a unit cost function may not be exact. So, we consider the model in an uncertain-based framework with the assumption that the exponents of selling price, marketing expenditure, and service expenditure in a demand function and the exponent of order quantity in a unit cost function are uncertain. Based on uncertainty theory, we develop a method to find the equivalent crisp model. In addition, we use the geometric programming method to solve the crisp model. Furthermore, we give a suitable numerical example to show how the proposed methodology can be used to solve the model for different uncertainty distributions at different uncertainty levels. In the numerical example, we use five different uncertainty distributions and figure out the selling price, marketing cost, service cost, order quantity, and profit with respect to the uncertainty level for each uncertainty distribution. Finally, we perform a sensitivity analysis for a better understanding of the proposed model. [ABSTRACT FROM AUTHOR]

This study investigates the optimal distribution and pattern of ground control points (GCPs) in aerial photogrammetric projects. Aerial triangulation (AT), also known as bundle adjustment, is the fundamental step in refining 3D reconstruction models and camera positions, thereby minimizing reprojection errors. The study utilizes data from a national project in Romania, employing high-resolution aerial images acquisition using photogrammetric sensors. The project has rigorous requirements of ground control points (GCP) placement and field measurements using GNSS and geometric leveling techniques. The study employs various scenarios, manipulating the number and distribution of GCPs, to assess their influence on planimetric and altimetric accuracy. Results indicate that the configuration and number of GCPs significantly affect the accuracy of photogrammetric products, such as dense image point clouds, digital surface models, and orthophotos. Moreover, the study underscores the importance of precise GCP determination methods, especially in regions lacking a precise gravimetric geoid model. In scenarios with inadequate GCP coverage the outcomes have inferior quality, emphasizing the critical role of GCPs in ensuring the quality of photogrammetric products. Overall, the research gives a clear view on the best placement patterns of GCPs and their influence on AT process evaluation performed in check points (CHKs). [ABSTRACT FROM AUTHOR]

The precision of workpiece machining is critically influenced by the geometric errors in the kinematics of grind robots, which directly affect their absolute positioning accuracy. To tackle this challenge, this paper introduces a logistic-tent chaotic mapping Levenberg Marquardt algorithm designed to accurately identify and compensate for this geometric error. the approach begins with the construction of a forward kinematic model and an error model specific to the robot. Then the algorithm is adopted to identify and compensate for the geometric error. The method establishes a mapping interval around the initial candidate solutions derived from iterative applications of the Levenberg Marquardt algorithm. Within this interval, the logistic-tent chaotic mapping method generates a diverse set of candidate solutions. These candidates are evaluated based on their fitness values, with the optimal solution selected for subsequent iterations. Empirical compensation experiments have validated the proposed method's precision and effectiveness, demonstrating a 6% increase in compensation accuracy and a 47.68% improvement in efficiency compared to existing state-of-the-art approaches. This process not only minimizes the truncation error inherent in the Levenberg Marquardt algorithm but also significantly enhances solution efficiency. Moreover, simulation experiments on grind processes further validate the method's ability to significantly improve the quality of workpiece machining. [ABSTRACT FROM AUTHOR]

Background: Heart rate variability (HRV) is believed to possess the potential for disease detection. However, early identification of heart disease remains challenging, as HRV analysis in dogs primarily reflects the advanced stages of the disease. Hypothesis/objective: The aim of this study is to compare 24-h HRV with sleep HRV to assess the potential utility of sleep HRV analysis. Animals: Thirty healthy dogs with no echocardiographic abnormalities were included in the study, comprising 23 females and 7 males ranging in age from 2 months to 8 years (mean [standard deviation], 1.4 [1.6]). Methods: This study employed a cross-sectional study. 24-h HRV and sleep HRV were measured from 48-h Holter recordings. Both linear analysis, a traditional method of heart rate variability analysis, and nonlinear analysis, a novel approach, were conducted. Additionally, circadian rhythm parameters were assessed. Results: In frequency analysis of linear analysis, the parasympathetic index nHF was significantly higher during sleep compared to the mean 24-h period (mean sleep HRV [standard deviation] vs. mean 24 h [standard deviation], 95% confidence interval, p value, r-family: 0.24 [0.057] vs. 0.23 [0.045], 0.006-0.031, p = 0.005, r = 0.49). Regarding time domain analysis, the parasympathetic indices SDNN and RMSSD were also significantly higher during sleep (SDNN: 179.7 [66.9] vs. 156.6 [53.2], 14.5-31.7, p < 0.001, r = 0.71 RMSSD: 187.0 [74.0] vs. 165.4 [62.2], 13.2-30.0, p < 0.001, r = 0.70). In a geometric method of nonlinear analysis, the parasympathetic indices SD1 and SD2 showed significantly higher values during sleep (SD1: 132.4 [52.4] vs. 117.1 [44.0], 9.3-21.1, p < 0.001, r = 0.70 SD2: 215.0 [80.5] vs. 185.9 [62.0], 17.6-40.6, p < 0.001, r = 0.69). Furthermore, the circadian rhythm items of the parasympathetic indices SDNN, RMSSD, SD1, and SD2 exhibited positive peaks during sleep. Conclusion: The findings suggest that focusing on HRV during sleep can provide a more accurate representation of parasympathetic activity, as it captures the peak circadian rhythm items. [ABSTRACT FROM AUTHOR]

Compared with large enterprises, the development scale and organizational structure of small and medium-sized enterprises are insufficient, which brings certain limitations to the development of small and medium-sized enterprises in China. In order to promote the long-term development of small and medium-sized enterprises in the new era, it is necessary to require enterprise leaders to innovate marketing plans, strengthen risk management of enterprises, and enhance their strength in market competition. The market risk evaluation of small and medium sized enterprises (SMSEs) in the new era is a multiple-attribute decision-making (MADM). The IVIFSs are employed as the tool for portraying uncertain information during the market risk evaluation of SMSEs in the new era. In this paper, the interval-valued intuitionistic fuzzy (IVIF) Hamacher interactive power geometric (IVIFHIPG) technique is addressed based on IVIF Hamacher interactive weighted geometric (IVIFHIWG) technique and power geometric (PG) technique. Some properties of IVIFHIPG technique were addressed. Then, the IVIFHIPG technique is employed to manage MADM under IVIFSs. Finally, an example for market risk evaluation of SMSEs in the new era is employed to verify the IVIFHIPG technique. Thus, the main contributions of this paper are addressed: (1) the IVIFHIPG technique is addressed based on IVIFHIWG technique and PG technique; (2) the IVIFHIPG technique is came up with to manage the MADM under IVIFSs; (3) a numerical example for market risk evaluation of SMSEs in the new era has been came up with to show the IVIFHIPG technique; and (4) some comparative analysis is addressed to verify the I IVIFHIPG technique. [ABSTRACT FROM AUTHOR]

Short-span bridges are one of the most frequent infrastructures along railway tracks where railway track stiffness suddenly changes. The sudden variation in the vertical stiffness of railway tracks increases dynamic loads and causes numerous defects in ballasted railroads. Therefore, improving the dynamic performance of railway tracks can be conducted by constructing countermeasures along the transition zone. In this regard, the approach slab is a practical technique used in railway-bridge transition zones. As the dimensional shape of the approach slab plays a significant role in the dynamic response of transition zones, this study evaluated the effects of its main geometric parameters. A three-dimensional model of the railway portal bridge, including the approach slab, was built using the finite element method and analyzed by imposing moving wheel loads as a series of acting force points along rail elements. The model was validated with field-obtained results acquired through a laser/camera-based measuring technique. Then, some sensitivity analyses were performed to find optimized geometric dimensions of approach slabs to improve the dynamic behavior of railway-bridge transition zones. Obtained results of the geometrical sensitivity analysis show that when the geometrically optimized approach slab is used along the railway-bridge transition zone, the displacements of rail and ballast are decreased by 24% and 18%, respectively. [ABSTRACT FROM AUTHOR]

Large machine tools are critically affected by ambient temperature fluctuations, impacting their performance and the quality of machined products. Addressing the challenge of accurately measuring thermal effects on machine structures, this study introduces the Machine Tool Integrated Inverse Multilateration method. This method offers a precise approach for assessing geometric error parameters throughout a machine's working volume, featuring a low level of uncertainty and high speed suitable for effective temperature change monitoring. A significant innovation is found in the capability to automatically realise the volumetric error characterisation of medium- to large-sized machine tools at intervals of 40–60 min with a measurement uncertainty of 10 µm. This enables the detailed study of thermal errors which are generated due to variations in ambient temperature over extended periods. To validate the method, an extensive experimental campaign was conducted on a ZAYER Arion G™ large machine tool using a LEICA AT960™ laser tracker with four wide-angle retro-reflectors under natural workshop conditions. This research identified two key thermal scenarios, quasi-stationary and changing environments, providing valuable insights into how temperature variations influence machine behaviour. This novel method facilitates the optimization of machine tool operations and the improvement of product quality in industrial environments, marking a significant advancement in manufacturing metrology. [ABSTRACT FROM AUTHOR]

The application of contemporary artificial intelligence techniques to address geometric problems and automated deductive proofs has always been a grand challenge to the interdisciplinary field of mathematics and artificial intelligence. This is the fourth article in a series of our works, in our previous work, we established a geometric formalized system known as FormalGeo. Moreover, we annotated approximately 7000 geometric problems, forming the FormalGeo7k dataset. Despite the fact that FGPS (Formal Geometry Problem Solver) can achieve interpretable algebraic equation solving and human-like deductive reasoning, it often experiences timeouts due to the complexity of the search strategy. In this paper, we introduced FGeo-TP (theorem predictor), which utilizes the language model to predict the theorem sequences for solving geometry problems. The encoder and decoder components in the transformer architecture naturally establish a mapping between the sequences and embedding vectors, exhibiting inherent symmetry. We compare the effectiveness of various transformer architectures, such as BART or T5, in theorem prediction, and implement pruning in the search process of FGPS, thereby improving its performance when solving geometry problems. Our results demonstrate a significant increase in the problem-solving rate of the language model-enhanced FGeo-TP on the FormalGeo7k dataset, rising from 39.7% to 80.86%. Furthermore, FGeo-TP exhibits notable reductions in solution times and search steps across problems of varying difficulty levels. [ABSTRACT FROM AUTHOR]

This paper introduces an efficient numerical method for approximating solutions to geometric boundary value problems. We propose the multiplicative sinc–Galerkin method, tailored specifically for solving multiplicative differential equations. The method utilizes the geometric Whittaker cardinal function to approximate functions and their geometric derivatives. By reducing the geometric differential equation to a system of algebraic equations, we achieve computational efficiency. The method not only proves to be computationally efficient but also showcases a valuable symmetric property, aligning with inherent patterns in geometric structures. This symmetry enhances the method's compatibility with the often-present symmetries in geometric boundary value problems, offering both computational advantages and a deeper understanding of geometric calculus. To demonstrate the reliability and efficiency of the proposed method, we present several examples with both homogeneous and non-homogeneous boundary conditions. These examples serve to validate the method's performance in practice. [ABSTRACT FROM AUTHOR]

Whole-class discussions in mathematics are envisioned as spaces for the sharing of ideas and making connections among them. We pursue how Palestinian/Arab Israeli teachers consider making use of multiple solutions to a task: which three they indicate that they would select for a whole-class discussion and in what sequence. Under the hypothetical assumption that each solution has been authored by boys and girls (with below-average and with above-average grades), how do teachers consider distributing opportunities to present the designated solutions? Participants most commonly selected a direct model, an error, or an inductive approach supported by a geometric representation, typically in that sequence (or error, direct model, inductive). Participants tended to indicate that they would invite a girl with below-average grades to present the direct model; a student with above-average grades to present the error; and a boy with above-average grades to present the inductive/geometric solution. Our analysis extends to participants' explanations of their choices and illuminates their intentions: making use of existing status arrangements, leveraging existing hierarchies to benefit others in the class, or modifying the indicated student's status by making space for them to participate on the public floor. Our analysis highlights implications of these patterns, especially in terms of classroom opportunities to learn. [ABSTRACT FROM AUTHOR]

Compressed Sensing (CS) encompasses a broad array of theoretical and applied techniques for recovering signals, given partial knowledge of their coefficients, cf. Candés (C. R. Acad. Sci. Paris, Ser. I 346, 589–592 (2008)), Candés et al. (IEEE Trans. Inf. Theo (2006)), Donoho (IEEE Trans. Inf. Theo. 52(4), (2006)), Donoho et al. (IEEE Trans. Inf. Theo. 52(1), (2006)). Its applications span various fields, including mathematics, physics, engineering, and several medical sciences, cf. Adcock and Hansen (Compressive Imaging: Structure, Sampling, Learning, p. 2021), Berk et al. (2019 13th International conference on Sampling Theory and Applications (SampTA) pp. 1-5. IEEE (2019)), Brady et al. (Opt. Express 17(15), 13040–13049 (2009)), Chan (Terahertz imaging with compressive sensing. Rice University, USA (2010)), Correa et al. (2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) pp. 7789–7793 (2014, May) IEEE), Gao et al. (Nature 516(7529), 74–77 (2014)), Liu and Kang (Opt. Express 18(21), 22010–22019 (2010)), McEwen and Wiaux (Mon. Notices Royal Astron. Soc. 413(2), 1318–1332 (2011)), Marim et al. (Opt. Lett. 35(6), 871–873 (2010)), Yu and Wang (Phys. Med. Biol. 54(9), 2791 (2009)), Yu and Wang (Phys. Med. Biol. 54(9), 2791 (2009)). Motivated by our interest in the mathematics behind Magnetic Resonance Imaging (MRI) and CS, we employ convex analysis techniques to analytically determine equivalents of Lagrange multipliers for optimization problems with inequality constraints, specifically a weighted LASSO with voxel-wise weighting. We investigate this problem under assumptions on the fidelity term A x - b 2 2 , either concerning the sign of its gradient or orthogonality-like conditions of its matrix. To be more precise, we either require the sign of each coordinate of 2 (A x - b) T A to be fixed within a rectangular neighborhood of the origin, with the side lengths of the rectangle dependent on the constraints, or we assume A T A to be diagonal. The objective of this work is to explore the relationship between Lagrange multipliers and the constraints of a weighted variant of LASSO, specifically in the mentioned cases where this relationship can be computed explicitly. As they scale the regularization terms of the weighted LASSO, Lagrange multipliers serve as tuning parameters for the weighted LASSO, prompting the question of their potential effective use as tuning parameters in applications like MR image reconstruction and denoising. This work represents an initial step in this direction. [ABSTRACT FROM AUTHOR]

Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields, including computer vision, medical imaging, and machine learning. The dominant geometric paradigm for such applications has consisted of a few Riemannian geometries associated with spectral computations that are costly at high scale and in high dimensions. We present a route to a scalable geometric framework for the analysis and processing of SPD-valued data based on the efficient computation of extreme generalized eigenvalues through the Hilbert and Thompson geometries of the semidefinite cone. We explore a particular geodesic space structure based on Thompson geometry in detail and establish several properties associated with this structure. Furthermore, we define a novel inductive mean of SPD matrices based on this geometry and prove its existence and uniqueness for a given finite collection of points. Finally, we state and prove a number of desirable properties that are satisfied by this mean. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive review of mitigation approaches, with a specific focus on geometric modifications, to mitigate shaft voltage and bearing current in electric drives. While existing literature extensively covers inverter-based and coupling path-based methods, the potential of geometric modifications has been relatively underexplored. The geometric modification approach offers a cost-effective and weight-saving solution that can be incorporated during the early design phase of the machine. The review begins by outlining various mitigation techniques applied at the victim machine, including bearing insulation, ceramic ball bearings, shaft grounding, and more. Subsequently, it delves into the machine geometric modification approach, examining methods related to winding design, stator and rotor iron geometry, rotor surface insulation, and stator slot wedge modifications. The paper highlights the practical relevance of geometric modifications, emphasizing their advantages over conventional methods. The review paper aims to shed new light on the potential of geometric modifications in mitigating shaft voltage and bearing current in electric drives. By providing valuable insights into this approach, researchers and engineers can develop robust and efficient electric drive systems, advancing the field of electrical machine design and operation. [ABSTRACT FROM AUTHOR]

This study introduces a mathematical model for unemployment that incorporates nonlinear functions in the matching process and job creation. Derived from observed data and constraints on job creation, the nonlinearity enhances the model's realism. Through a geometric approach, sufficient conditions are identified to ensure a successful application of global stability analysis. The dynamics of the general system, including thresholds and global stability of the nontrivial equilibrium, are fully determined. Existence and stability of both trivial and non-trivial equilibria are rigorously proven using Lyapunov functions, Jacobian matrices, and the Lozinski measure. Numerical analysis illustrates the theoretical findings, emphasizing the global asymptotic stability of the nontrivial equilibrium under specific conditions. This work offers valuable insights into unemployment dynamics, bridging complex mathematics with practical economic models. [ABSTRACT FROM AUTHOR]

The attribute alignment approach to team composition allows researchers to assess variation in team member attributes, which occurs simultaneously within and across individual team members. This approach facilitates the development of theory testing the proposition that individual members are themselves complex systems comprised of multiple attributes and that the configuration of those attributes affects team-level processes and outcomes. Here, we expand this approach, originally developed for two attributes, by describing three ways researchers may capture the alignment of three or more team member attributes: (a) a geometric approach, (b) a physical approach accentuating ideal alignment, and (c) an algebraic approach accentuating the direction (as opposed to magnitude) of alignment. We also provide examples of the research questions each could answer and compare the methods empirically using a synthetic dataset assessing 100 teams of three to seven members across four attributes. Then, we provide a practical guide to selecting an appropriate method when considering team-member attribute patterns by answering several common questions regarding applying attribute alignment. Finally, we provide code (https://github.com/kjem514/Attribute-Alignment-Code) and apply this approach to a field data set in our appendices. [ABSTRACT FROM AUTHOR]

Landmark-based geometric morphometrics (GM) was used to examine, for the first time, spontaneous hybridization between Alnus incana (L.) Moench and Alnus rohlenae Vít, Douda and Mandák, and to assess inter- and intrapopulation variability in leaf shape, leaf size and venation in natural populations in Serbia (Western Balkans). Two geographically distant (30 km) and two close (1.2 km) populations were selected to examine hybridization. The variability in leaf shapes was assessed by canonical variate analysis and linear discriminant analysis performed on the symmetric component of variation. Covariation between the symmetric component of shape variation and the number of pairs of secondary leaf veins was investigated with partial least squares analysis. Static allometry was examined for the first time in the genus Alnus Mill. A higher proportion of A. incana leaves was classified as A. rohlenae in geographically close populations, which is in accordance with the hypothesis about spontaneous hybridization. No single leaf of A. rohlenae was classified as A. incana, indicating that putative hybrids can only be found in grey alder populations. This study demonstrates that GM is a powerful tool for species delimitation and hybrid detection in the genus Alnus and it can be used for preliminary screening in hybrid zones. [ABSTRACT FROM AUTHOR]

In the context of this investigation, we introduce an innovative mathematical model designed to elucidate the intricate dynamics underlying the transmission of Anthroponotic Cutaneous Leishmania. This model offers a comprehensive exploration of the qualitative characteristics associated with the transmission process. Employing the next-generation method, we deduce the threshold value $ R_0 $ for this model. We rigorously explore both local and global stability conditions in the disease-free scenario, contingent upon $ R_0 $ being less than unity. Furthermore, we elucidate the global stability at the disease-free equilibrium point by leveraging the Castillo-Chavez method. In contrast, at the endemic equilibrium point, we establish conditions for local and global stability, when $ R_0 $ exceeds unity. To achieve global stability at the endemic equilibrium, we employ a geometric approach, a Lyapunov theory extension, incorporating a secondary additive compound matrix. Additionally, we conduct sensitivity analysis to assess the impact of various parameters on the threshold number. Employing center manifold theory, we delve into bifurcation analysis. Estimation of parameter values is carried out using least squares curve fitting techniques. Finally, we present a comprehensive discussion with graphical representation of key parameters in the concluding section of the paper.

In this paper, we consider the problem of quantifying parametric uncertainty in classical empirical interatomic potentials (IPs) using both Bayesian (Markov Chain Monte Carlo) and frequentist (profile likelihood) methods. We interface these tools with the Open Knowledgebase of Interatomic Models and study three models based on the Lennard-Jones, Morse, and Stillinger–Weber potentials. We confirm that IPs are typically sloppy, i.e., insensitive to coordinated changes in some parameter combinations. Because the inverse problem in such models is ill-conditioned, parameters are unidentifiable. This presents challenges for traditional statistical methods, as we demonstrate and interpret within both Bayesian and frequentist frameworks. We use information geometry to illuminate the underlying cause of this phenomenon and show that IPs have global properties similar to those of sloppy models from fields, such as systems biology, power systems, and critical phenomena. IPs correspond to bounded manifolds with a hierarchy of widths, leading to low effective dimensionality in the model. We show how information geometry can motivate new, natural parameterizations that improve the stability and interpretation of uncertainty quantification analysis and further suggest simplified, less-sloppy models. [ABSTRACT FROM AUTHOR]

The Art Gallery Problem (AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon P and an integer k, the goal is to decide if there exists a setG of k guards within P such that every point p ∈ P is seen by at least one guard д ∈ G. Each guard corresponds to a point in the polygon P, and we say that a guard д sees a point p if the line segment pд is contained in P. We prove that the AGP is ∃R-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) the AGP is not in the complexity class NP unless NP = ∃R. As a corollary of our construction, we prove that for any real algebraic number α, there is an instance of the AGP where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. As an illustration of our techniques, we show that for every compact semi-algebraic set S ⊆ [0, 1]2, there exists a polygon with corners at rational coordinates such that for every p ∈ [0, 1]2, there is a set of guards of minimum cardinality containing p if and only if p ∈ S. In the ∃R-hardness proof for the AGP, we introduce a new ∃R-complete problem ETR-INV. We believe that this problem is of independent interest, as it has already been used to obtain ∃R-hardness proofs for other problems. [ABSTRACT FROM AUTHOR]

Environmental toxicants overwhelmingly occur together as mixtures. The variety of possible chemical interactions makes it difficult to predict the danger of the mixture. In this work, we propose the novel Reflected Generalized Concentration Addition (RGCA), a piece-wise, geometric technique for sigmoidal dose-responsed inverse functions that extends the use of generalized concentration addition (GCA) for 3+ parameter models. Since experimental tests of all relevant mixtures is costly and intractable, we rely only on the individual chemical dose responses. Additionally, RGCA enhances the classical two-step model for the cumulative effects of mixtures, which assumes a combination of GCA and independent action (IA). We explore how various clustering methods can dramatically improve predictions. We compare our technique to the IA, CA, and GCA models and show in a simulation study that the two-step approach performs well under a variety of true models. We then apply our method to a challenging data set of individual chemical and mixture responses where the target is an androgen receptor (Tox21 AR-luc). Our results show significantly improved predictions for larger mixtures. Our work complements ongoing efforts to predict environmental exposure to various chemicals and offers a starting point for combining different exposure predictions to quantify a total risk to health. [ABSTRACT FROM AUTHOR]

The connection between contextuality and graph theory has paved the way for numerous advancements in the field. One notable development is the realization that sets of probability distributions in many contextuality scenarios can be effectively described using well-established convex sets from graph theory. This geometric approach allows for a beautiful characterization of these sets. The application of geometry is not limited to the description of contextuality sets alone; it also plays a crucial role in defining contextuality quantifiers based on geometric distances. These quantifiers are particularly significant in the context of the resource theory of contextuality, which emerged following the recognition of contextuality as a valuable resource for quantum computation. In this paper, we provide a comprehensive review of the geometric aspects of contextuality. Additionally, we use this geometry to define several quantifiers, offering the advantage of applicability to other approaches to contextuality where previously defined quantifiers may not be suitable. This article is part of the theme issue 'Quantum contextuality, causality and freedom of choice'. [ABSTRACT FROM AUTHOR]

In this paper, we study an SSvEIQR model with nonlinear contact rate, isolation rate and vaccination rate driven by media coverage. First, the basic reproduction number R0 is derived. Then, the threshold dynamics of the disease are obtained in terms of R0: when R0 ≤ 1, the global stability of the disease-free equilibrium is obtained by constructing an appropriate Lyapunov function; when R0 > 1, the sufficient conditions to prove the globally stability of endemic equilibrium are obtained by applying the geometric method into the four-dimensional system, which needs to estimate the Lozinskiǐ measure of a 6 × 6 matrix. Further, we conduct some numerical simulations to validate our theoretical results, and analyze the impact of media coverage on disease transmission, the results show that media coverage could effectively suppress the spread of the disease and reduce the number of infected individuals. Finally, through the sensitivity analysis of R0, we obtain some measures to control the spread of the disease, such as reducing contact, strengthening isolation and vaccination. [ABSTRACT FROM AUTHOR]

We perform a Hamiltonian analysis of unimodular gravity in its first-order formulation, specifically a modification of the Holst action. In order to simplify the analysis, prior studies on this theory have introduced (for several reasons) additional elements, such as parametrization, complex fields, or considering the Barbero–Immirzi parameter as imaginary. We show that, by using a geometric implementation of the Dirac algorithm, a comprehensive analysis of the theory can be conducted without relying on these additional ingredients. The resulting theory reproduces the behavior of metric unimodular gravity. [ABSTRACT FROM AUTHOR]

The United Nations' Human Development Index remains a widely used and accepted measure of human development. Although it has been revised over the years to address various critiques, a remaining concern is the way the three dimensions are aggregated into the single index. A deterioration in one dimension can be compensated for by an improvement in another. Since compensability is inextricably linked with trade-offs and intensity of preferences, a non-compensatory (i.e., Condorcet) approach to aggregation is employed in this paper. Although non-compensatory approaches have been employed previously, this paper adds to the literature by undertaking an application of the Condorcet approach to the entire HDI. This approach, which does not use intensities of preferences, ensures that the degree of compensability connected with the aggregation model is at the minimum possible level. To achieve this, country level rankings are then compared to those for the 2020 Human Development Index which aggregates dimensions using a geometric mean. The findings demonstrated substantial changes in rank-order between the HDI and Condorcet approach. This outcome provides empirical evidence which demonstrates that the non-compensatory Condorcet approach can mitigate issues of compensation present within the geometric aggregation technique currently employed by the HDI. These findings have potential implications in aiding the identification and employment of potential policy priorities—specifically, the notion that policy should emphasise the development of a country as opposed to economic growth alone. [ABSTRACT FROM AUTHOR]

The geometric renormalization technique for complex networks has successfully revealed the multiscale self-similarity of real network topologies and can be applied to generate replicas at different length scales. Here, we extend the geometric renormalization framework to weighted networks, where the intensities of the interactions play a crucial role in their structural organization and function. Our findings demonstrate that the weighted organization of real networks exhibits multiscale self-similarity under a renormalization protocol that selects the connections with the maximum weight across increasingly longer length scales. We present a theory that elucidates this symmetry, and that sustains the selection of the maximum weight as a meaningful procedure. Based on our results, scaled-down replicas of weighted networks can be straightforwardly derived, facilitating the investigation of various size-dependent phenomena in downstream applications. Geometric renormalization reveals hidden network symmetries by scaling them down while retaining key features. Extended to weighted networks, in which link intensities matter, here the authors present empirical evidence and theory to justify selecting links with maximum weights across increasingly longer length scales to reduce resolution, enabling self-similar replicas and study of size-dependent phenomena. [ABSTRACT FROM AUTHOR]

This paper analyses a general qualitative depiction of changes in the configuration of the non-linear model of a tractor-semitrailer with excessive turnability under stationary circular motions within a broad range of control parameters, such as the longitudinal velocity of motion and angle of rotation of the wheels of the controlled unit. It allows us to foresee the jackknifing effect and to estimate the conditions necessary for safe travel. The proposed analysis combines the transparent geometric approach to analyse the configuration of a tractor-semitrailer, which was developed earlier by J.P. Pauwelussen, and the ability to investigate real bifurcations of the non-linear equation of handling and the non-linear equation of folding angle tractor-semitrailer analytically. Based on this approach, we obtain new analytical dependencies that determine the boundaries of divergent loss of stability of the entire manifold of stationary states of a tractor-semitrailer model and estimates of the maximum values of the angle of folding under different values of the vehicle's longitudinal velocity. This is achieved without the prior determination of the manifold of stationary states itself. [ABSTRACT FROM AUTHOR]

The geometric approach to the study of the Herglotz problem developed in Massa and Pagani [J. Math. Phys. 64, 102902 (2023)] is extended to the case in which the evolution of the system is subject to a set of non-holonomic constraints. The original setup is suitably adapted to the case in study. Various aspects of the problem are considered: the direct derivation of the evolution equations; the super-lagrangian approach; the resulting super-Hamiltonian and its relation with Pontryagin's maximum principle; the abnormality index of the extremals; the invariance properties of the theory and the consequent existence of Herglotz Lagrangians gauge equivalent to ordinary ones. [ABSTRACT FROM AUTHOR]

In this article, the ( 1 + 1 )-dimensional Pochhammer–Chree (PC) model, which illustrates a nonlinear scheme of longitudinal wave propagation of elastic bars, is investigated to determine its analytical solutions using Lie infinitesimal generators. The solutions obtained using the symmetries of the PC model allow one to investigate the propagation of longitudinal deformation waves within an elastic rod. The infinitesimal generators are obtained by implementing the Lie symmetry technique and the geometric method. In the geometric method, the Estabrook and extended Harrison differential forms are used for establishing the infinitesimal generators. Using Olver’s conventional method, a system of optimal subalgebras has been built because there are an infinite number of possible linear combinations of infinitesimal generators. Furthermore, it is demonstrated by the use of formal Lagrangian that the aforementioned model satisfies the quasi-self-adjointness criteria. Additionally, the conservation laws associated with the symmetries of the PC model are derived using the quasi-self-adjointness condition and Ibragimov’s ‘new conservation theorem’. Finally, the three-dimensional surfaces of the acquired solutions have been plotted with the corresponding density and contour plots. The newly developed results show the effectiveness, dependability and validity of the symmetry techniques for obtaining invariant solutions to this nonlinear governing model. The novelty of symmetry analysis is that the group of transformations under which the differential equations remain invariant can be used to simplify the given model. [ABSTRACT FROM AUTHOR]