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In quantum chemistry, Lowest Unoccupied Molecular Orbital (LUMO) is important for studying various chemical processes, including photochemical reactions, electron attached states, and electron excites states. Recently, an effective method has been introduced that involves the use of the Parametric Equation of Motion (PEM) in conjunction with the nuclear charge stabilization method for precise identification of true LUMO. However, the inclusion of extra diffuse functions in the basis set, which is necessary for describing electron-attached and electron-excited states, can cause issues due to the presence of the same symmetry states, leading to avoided crossing. Identifying the true LUMO among these avoided crossings is challenging due to the mixing of states and the exchange of their orbital character. This article introduces a modification of the PEM to identify the true LUMO by preventing the stabilization of specific states involved in avoided crossings. The present method is highly effective and requires minimal computational cost. [ABSTRACT FROM AUTHOR]

AbstractStatistical methods have been developed for regression modeling of the cumulative incidence function (CIF) given left-truncated right-censored competing risks data. Nevertheless, existing methods typically involve complicated weighted estimating equations or non parametric conditional likelihood function and often require a restrictive assumption that censoring and/or truncation times are independent of failure time. The pseudo-observation (PO) approach has been used in regression modeling of CIF for right-censored competing risks data under covariate-independent censoring or covariate-dependent censoring. We extend this approach to left-truncated right-censored competing risks data and propose to directly model the CIF based on POs, under general truncation and censoring mechanisms. We adjust for covariate-dependent truncation and/or covariate-dependent censoring by incorporating covariate-adjusted weights into the inverse probability weighted (IPW) estimator of the CIF. We derive large sample properties of the proposed estimators under reasonable model assumptions and regularity conditions and assess their finite sample performances by simulation studies under various scenarios. We apply the proposed method to a cohort study on pregnancy exposed to coumarin derivatives. [ABSTRACT FROM AUTHOR]

We study the problem of identifying the constant parameters involved in the right-hand sides of a linear non-autonomous system of differential equations with first-order ordinary derivatives. The specificity of the problem lies in the fact that additional conditions for identifying parameters, firstly, are non-local, and secondly, they include derivatives of an unknown function. We examine the conditions for the existence and uniqueness of a solution to the problem and propose two different approaches to the numerical solution of the problem. The results of computer experiments are presented. [ABSTRACT FROM AUTHOR]

Shear spinning is an incremental metal forming technique with several advantages when compared with traditional sheet metal forming technologies, including low force requirements, simpler tools, and the capability to produce complex geometries. In this process, a large deformation can be applied due to the localized compressive and shear stresses, which enhance the ductility more than conventional forming processes. Therefore, the final part improves considerably its mechanical properties due to the associated work hardening. Nevertheless, the shear spinning technology involves a complex material flow that is highly dependent on the process parameters. In consequence, it is essential to study the effect of the different parameters on the formability of any material in the shear spinning process, but more importantly, to develop a method to predict the forming limit in terms of such parameters to get most of the process. In this work, a novel approach to study the formability in terms of the radial force was introduced. In this sense, shear spinning experiments on mild steel (DC-04) and martensitic stainless steel (AISI 420) were carried out, to derive forming limit maps as a function of the mandrel wall angle (30–7°), the spindle speed (400–1000 rpm), the feed rate (0.01–1.2 mm/rev), and the roller attack angle (0–30°). It was found that the maximum thickness reduction that the materials can withstand was 84% and 80% for the DC-04 and AISI 420, respectively. The results showed that the spindle speed presents a minor effect while the feed rate and radial force greatly affect the formability; therefore, a parametric equation was proposed to describe the boundary between the safe and failure zones in terms of the maximum thickness reduction. Regarding the mechanical properties, the maximum hardness increment regarding the as-received materials was 73% and 42% for the DC-04 and AISI 420, respectively. [ABSTRACT FROM AUTHOR]

In computer numerical control (CNC) operations, smooth motions generated using a parametric curve such as non-uniform rational B-splines (NURBS), Bezier curve, and B-spline are crucial for surface quality and machine control. A parametric curve is formed by micro-lines in CAD/CAM software; however, instead of micro-lines, polylines are used to define a parametric curve to prevent an increase in NC-file size. It is also possible to see situations where smooth motion is not required due to production requirements. In these cases, point-to-point motion is necessary instead of smooth motion. In this study, a polyline analysis algorithm is proposed to classify the normal line and polyline within the NC-file. A global tool path smoothing method is also applied, where the data points in a polyline group are converted into NURBS because CAD/CAM software commonly uses NURBS as a parametric curve for precision control of CNC machines. In addition, this study presents an optimal feedrate scheduling algorithm for the NURBS curve that generates an optimal feedrate profile for each segment by considering critical parameters such as segment length, initial and final velocity of each segment, maximum jerk, acceleration, and velocity. As the proposed algorithm is compatible with existing feedrate scheduling methods in which different feedrate values at the transition point of the NURBS segments are determined, this study provides an important contribution to the literature. The proposed methods in this study are verified by operating a spherical surface using industrial NC-files on a three-axis CNC-milling machine. The proposed methods demonstrate effective performance in experiments that require precision production operations. [ABSTRACT FROM AUTHOR]

This study examines the impact of slippage in hazard ratios (tending toward the null over subsequent datacuts) for overall survival for combination treatment with a PD-(L)-1 inhibitor and a tyrosine kinase inhibitor in advanced renal cell carcinoma. Four trials' Kaplan-Meier curves were digitized over several datacuts and fitted with standard parametric curves. Accuracy and consistency of early data projections were calculated versus observed restricted mean survival time and fitted lifetime survival from the longest follow-up datacut. The change in economically justifiable price (eJP) was calculated fitting the same curve to both arms, using an assumed average utility of 0.7 and willingness-to-pay threshold of £30 000 per quality-adjusted life-year. The eJP represents the lifetime justifiable price increment for the new treatment, including differences in drug-, administration-, and disease-related costs. Slippage in hazard ratios was observed in trials with longer follow-up, potentially influenced by subsequent PD-(L)-1 use after tyrosine kinase inhibitor monotherapy, early stoppage of PD-(L)-1, and development of resistance. Lognormal and log-logistic curves were more likely to overpredict the observed result; Gompertz and gamma underpredicted. Statistical measures of goodness of fit did not select the curves that resulted in the RMST closest to what was observed in the final data cut. Large differences in incremental mean life-years were observed between even the penultimate and final datacuts for most of the fitted curves, meaningfully affecting the eJP. This work demonstrates the challenge in predicting treatment benefits with novel therapies using immature data. Incorporating information on the impact of subsequent treatment is likely to play a key role in improving predictions. • Three trials of combination immuno-oncology and tyrosine kinase inhibitor treatment for first-line advanced renal cell carcinoma showed slippage in the hazard ratios (tending toward the null over subsequent datacuts) for overall survival. Predicted incremental mean life-years varied substantially between datacuts, suggesting earlier evaluations may overestimate treatment benefits. • The availability of subsequent treatment [PD-(L)-1 inhibitor therapy after tyrosine kinase inhibitor monotherapy] may be a driver of slippage in treatment effects. Duration of immuno-oncology treatment may also play a role given that slippage was not seen in Javelin Renal 101, although the longest follow-up in this trial was not as long as in the other studies. • Certain standard parametric curves showed trends in accuracy and consistency in predicting overall survival. Lognormal and log-logistic curves tended to overpredict, whereas Gompertz and gamma curves tended to underpredict. The Weibull curve most consistently predicted incremental life-years across datacuts, despite potential over- or underprediction for individual arms compared with other curves. Goodness of fit measures did not provide useful insights. [ABSTRACT FROM AUTHOR]

ISO 12135, a representative standard for the evaluation of fracture toughness, was revised in 2021. In the previous standard published in 2016, only yield strength was reflected for material property in fracture toughness calculation. In the revised standard, however, yield to tensile strength ratio is additionally reflected in consideration of strain hardening effect of the material. The current fracture toughness calculation formula evaluates fracture toughness reasonably considering the tensile characteristic. However, in some cases of cryogenic steels, fracture toughness is evaluated very conservatively than the previous procedure in 2016. In the case of STS 304L, there is no ductile-brittle transition phenomenon and it has excellent ductility and strength in cryogenic temperature. However, due to the secondary hardening phenomenon, yield to tensile strength ratio is relatively low, and as a result, fracture toughness is evaluated very low. The target of this study is to resolve the undervaluation of fracture toughness of cryogenic steel in the revised ISO 12135. For this purpose, the elastic and the plastic parameters in crack tip opening displacement (CTOD) equation in ISO 12135 are examined. The crack tip profile is derived by a series of finite element analysis and crack tip opening displacement is calculated for the derivation of new parametric equations for cryogenic materials. The calculated crack tip opening displacement is validated by experimental results. [ABSTRACT FROM AUTHOR]

This article proposed a novel method to categorize the best student in all progressive studies by using the single valued Neutrosophic soft set-in variable sense. An ambivalence set of multi-observer data which is related to analyse the students, taken as input for categorizing the best student identification. Neutrosophic soft set is an immense application to find out the choice-making problem in the Neutrosophic area. The creation of an analogous table has shaped the classification investigation. It helps to put up things, people into groups according to their quality, ability, performance etc., in Neutrosophic environment [ABSTRACT FROM AUTHOR]

In this paper, we establish the well‐posedness of the Cauchy problem for incompressible magneto‐hydrodynamic equations by using parametric Meyer wavelets. The velocity field and magnetic field are expanded according to parametric wavelets given by a discrete set of thresholds. The evolution of these two types of fields is described by the relationship of parameterized flag microlocal quantities over binary time intervals. The corresponding iteration space is defined by the decay of the single norm, and the single norm is defined by the parametric flag microlocal quantities at a set of binary time interval. [ABSTRACT FROM AUTHOR]

In this work, a subspace method is proposed for efficient solution of parametric linear systems with a symmetric and positive definite coefficient matrix of the form I − A (σ). The motivation is to use the method for solution of linear systems appearing when solving parameter dependent elliptic PDEs using the finite element method (FEM). In the proposed method, one first computes a method subspace and then uses it to approximately solve the linear system for any parameter vector. The method subspace is designed in such a way that it contains the j + 1 -term truncated Neumann series approximation of the solution to desired accuracy for any admissible parameter vector. This allows us to use the best approximation property of subspace methods to show that the subspace solution is at least as accurate as the truncated Neumann series approximation. The performance of the method is demonstrated by numerical examples with the parametric diffusion equation. In these examples, the method yields much smaller errors than anticipated by the Neumann series based error analysis. We study this phenomenon in some special cases. [ABSTRACT FROM AUTHOR]

We introduce a novel formulation for the evolution of parametric curves by anisotropic curve shortening flow in |${{\mathbb{R}}}^{d}$|⁠ , |$d\geq 2$|⁠. The reformulation hinges on a suitable manipulation of the parameterization's tangential velocity, leading to a strictly parabolic differential equation. Moreover, the derived equation is in divergence form, giving rise to a natural variational numerical method. For a fully discrete finite element approximation based on piecewise linear elements we prove optimal error estimates. Numerical simulations confirm the theoretical results and demonstrate the practicality of the method. [ABSTRACT FROM AUTHOR]

The ferrofluid flow between torsionally oscillating disks spans a spectrum of applications including enhanced damping, controlled energy dissipation, magnetic actuators, cooling systems, and advanced aerospace engineering. The current study focuses on exploring ferrofluid flow between torsionally oscillating disks under two selected cases: Case I, where the lower disk torsionally oscillates, while the upper disk remains stationary, and Case II, where both the upper and lower disks torsionally oscillate, with the upper disk 180 ° out of phase relative to the lower disk. The novel influence of magnetic field-dependent (MFD) viscosity and variable thermal conductivity are incorporated into the dealing equations of the ferrofluid flow to identify the real character of ferrofluids. The parametric form of these equations is obtained after utilizing the suitable similarity function approach. The patterns of the velocity field, streamlines, and isotherms along with velocity and temperature variations are portrayed pictorially after a series of experimental simulations conducted in the MATLAB environment by using Bvp5c. The variations in the coefficients of skin friction and Nusselt numbers at the respective upper and lower disks are displayed through graphs and tables with the emerging parameters. It is identified from the comparative analysis of above-mentioned two cases that the ferrofluid flow is a decreasing function of the MFD parameter due to the reduced magnitude of velocity, whereas a direct relationship between the ferrofluid temperature and the MFD parameter is noted. It is also observed that ferrofluid heat is enhanced by the variable thermal conductivity owing to the increased value of the ferrofluid temperature. This study may be helpful in understanding magnetically damped oscillations. [ABSTRACT FROM AUTHOR]

We investigate normal magnetic curves in (2 n + s) -dimensional homothetic s-th Sasakian manifolds as a generalization of S-manifolds. We show that a curve γ is a normal magnetic curve in a homothetic s-th Sasakian manifold if and only if its osculating order satisfies r ≤ 3 and it belongs to a family of θ i -slant helices. Additionally, we construct a homothetic s-th Sasakian manifold using generalized D-homothetic transformations and present the parametric equations of normal magnetic curves in this manifold. [ABSTRACT FROM AUTHOR]

The paper studied the finite-time semi-global stabilisation problems for a class of nonlinear systems with a novel polynomial growth condition. By using some key properties of a class of parametric Lyapunov equations and constructing time-varying Lyapunov-like functions, two time-varying high-gain controllers, where scalar time-varying functions are generated by differential equations, are designed. Both state feedback and observer-based output feedback are considered. It is shown that the proposed methods can achieve finite-time semi-global stabilisation for closed-loop systems and the controls are bound. Two numerical examples verify the effectiveness of the established results. [ABSTRACT FROM AUTHOR]

Background: The long-term health-economic consequences of acute stroke are typically extrapolated from short-term outcomes observed in different studies, using models based on assumptions about longer-term morbidity and mortality. Inconsistency in these assumptions and the methods of extrapolation can create difficulties when comparing estimates of lifetime cost-effectiveness of stroke care interventions. Aims: To develop a long-term model consisting of a set of equations to estimate the lifetime effects of stroke care interventions to promote consistency in extrapolation of short-term outcomes. Methods: Data about further admissions and mortality were provided for acute stroke patients discharged between 2013 and 2014 from a large English service. This was combined with data from UK life tables to create a set of parametric equations in a model that use age, sex, and modified Rankin Scores to predict the lifetime risk of mortality and secondary care resource utilization including ED attendances, non-elective admissions, and elective admissions. A cohort of 1509 (male 51%; mean age 74) stroke patients had median follow-up of 7 years and represented 7111 post-discharge patient years. A logistic model estimated mortality within 12 months of discharge, and a Gompertz model was used over the remainder of the lifetime. Hospital attendances were modeled using a Weibull distribution. Non-elective and elective bed days were both modeled using a log-logistic distribution. Results: Mortality risk increased with age, dependency, and male sex. Although the overall pattern was similar for resource utilization, there were different variations according to dependency and gender for ED attendances and non-elective/elective admissions. For example, 65-year-old women with a mRS at discharge of 1 would gain an extra 6.75 life years compared to 65-year-old women with a mRS at discharge of 3. Over their lifetime, 65-year-old women with an mRS at discharge of 1 would experience 0.09 less ED attendances, 2.12 less non-elective bed days, and 1.28 additional elective bed days than 65-year-old women with an mRS at discharge of 3. Conclusions: Using long-term follow-up publicly available data from a large clinical cohort, this new model promotes standardized extrapolation of key outcomes over the life course and potentially can improve the real-world accuracy and comparison of long-term cost-effectiveness estimates for stroke care interventions. Data assess statement: Data are available upon reasonable request from third parties. [ABSTRACT FROM AUTHOR]

Teak [Tectona grandis L.f.] has a wide distribution range in tropical countries and is Nepal's second most planted commercial tree species. This study aimed to develop a robust and reliable taper equation for Teak species in Nepal. To achieve this, 15 parametric taper equations were fitted and evaluated using the diameter and height data of 100 trees sampled from two stands of the Sagarnath Plantation projects, Nepal. The data set was split into training (90%) and testing (10%) sets based on the trees' ID, and model fitting was conducted in two phases. In the first phase, nonlinear models were fitted to the training data using 10-fold cross-validation, and the performance was evaluated based on fit and validation statistics. The top five models were further analyzed in the second phase using a mixed effects framework to account for variance and correlation structures. The modified Bi model performed best under a fixed effects modeling framework (R2 = 0.96, RMSE = 1.83 cm). However, the Sharma and Zhang model performed the best under a mixed-effects modeling framework (R2 = 0.97, RMSE = 1.54 cm). Therefore, we suggest using the modified Bi under fixed effects and variable exponent equation of Sharma and Zhang under mixed-effects modeling as a taper equation for Teak. The Sharma and Zhang's equation is recommended for its high accuracy and better performance over previously recommended variable exponents equations. [ABSTRACT FROM AUTHOR]

Background: Current therapies for complex Crohn's perianal fistulas (CPF) have a limited ability to achieve long-term healing. Darvadstrocel (DVS) is an expanded allogeneic adipose-derived mesenchymal stem cell therapy that has demonstrated efficacy in treating complex CPF in clinical trials. There are, however, limited long-term comparative data with standard of care (SoC). The aim of this study was to combine clinical trial data and real-world evidence using statistical methodologies to predict long-term effectiveness of DVS versus SoC in patients with CPF. Methods: Data were pooled from a clinical trial (ADMIRE-CD) and two retrospective chart review studies (INSPECT and PREFACE). Predictive statistical models extrapolated clinical outcomes beyond observed follow-up using parametric curves, which were implemented into a semi-Markov model to obtain the number of patients in remission. The setting was multinational and multicenter. ADMIRE-CD was conducted in 49 hospitals in 7 European countries and Israel. INSPECT used data from the ADMIRE study. PREFACE involved patients from Belgium, France, Germany, Italy, and Spain. The participants were patients with complex CPF treated with DVS or SoC. Times to remission and relapse (clinical, and clinical plus patient-centric remission) were analyzed. Additionally, the proportion of patients in clinical and patient-centric remission was examined. Results: In total, 513 patients were included in the analysis (ADMIRE-CD [N = 200] and PREFACE [N = 313]). Patients in ADMIRE-CD and PREFACE were similar in age (median [interquartile range, IQR], 36 [20.0] versus 36 [22.0] years, respectively) and gender (males, 54% and 52%, respectively). The median (IQR) duration of Crohn's disease was 9.4 [11.3] years for patients in ADMIRE-CD and 6.5 [12.9] years for patients in PREFACE. The estimated time to remission was shorter for patients treated with DVS versus SoC. The estimated time to relapse was longer for patients treated with DVS versus SoC. A higher estimated proportion of patients treated with DVS versus SoC had clinical and patient-centric remission at 24 months (48% and 35%, respectively) and 48 months (49% and 32%, respectively). Conclusion: This novel approach enabled pooled data from a clinical trial and real-world settings to predict long-term effectiveness of DVS versus SoC in patients with complex CPF. [ABSTRACT FROM AUTHOR]

Using the STEAM framework brings disciplines together by exploiting the synergy between modelling processes, mathematics and engineering. Nowadays, educational robotics is a powerful and flexible educational tool with a great motivational factor at all levels of education, including higher education. Cycloids (prolate, cusped, curtate) have been used in many engineering and physics-related problems over the centuries, have several interesting and unique properties, and are still widely used today by engineers and designers in new fields. In this paper, we present a new STEAM-based methodology for visualisation and multiple representations of cycloid curves using educational LEGO robots in the classroom environment. The first step is to draw the curves with light, using the classical definition of cycloids and a LEGO robot of our own design, and then follow the same principle to draw them with a marker pen, where the drawing head is moved by a robot. To strengthen visualisation, we have created another LEGO robot structure with a more complex working principle, which is also capable of drawing all three types of cycloids. The paper shows the details of the construction and engineering aspects of both types of drawing robots, together with the mathematical background. [ABSTRACT FROM AUTHOR]

In this work, we investigate the synthesis problem of slant spacelike ruled surfaces and associated Bertrand offsets (B O) in E 1 3 (Minkowsk 3-space). We provide the parametric equation for a non-developable spacelike ruled surface (S L R S) by using the Blaschke frame (B F). This results in the amplitude to control a family of curvature functions defining the domestic form of this S L R S. Therefore, we found the appropriate S L R S criteria to be slant S L R S. Thus, several new Bertrand offsets (B O) for slant S L R S are investigated and constructed. [ABSTRACT FROM AUTHOR]

In 2011 Hefez and Hernandes completed Zariski's analytic classification of plane branches belonging to a given equisingularity class by creating "very short" parameterizations over the complex numbers. Their results were used by Mehmood and Pfister to classify unimodal plane branches in characteristic 0 by giving lists of normal forms. The aim of this paper is to give a complete classification of unimodal plane branches over an algebraically closed field of positive characteristic. Since the methods of Hefez and Hernandes cannot be used in positive characteristic, we use a different approach and, for some sporadic singularities in small characteristic, computations with Singular. Our methods are characteristic-independent and provide a different proof of the classification in characteristic 0 showing at the same time that this classification holds also in large characteristic. The main theoretical ingredients are the semicontinuity of the semigroup and of the modality, which we prove and which may be of independent interest. [ABSTRACT FROM AUTHOR]

Korteweg–de Vries (KdV) equation with the influence of Landau damping of electrons is developed to explore the weakly nonlinear and weakly dispersive ion-acoustic waves in a non-collisional unmagnetized plasma system whose constituents are adiabatic warm ions, negative-charged immobile dust grains and combined Kappa–Cairns distributed electrons. It is found that the nonlinear coefficient of the KdV equation vanishes along different parametric curves in the parametric space. In this situation, a modified KdV equation with the Landau damping effect is obtained to demonstrate the nonlinear propagation of the ion-acoustic waves. By employing multiple time scale method, the solitary wave solutions of these equations are derived. The solitary wave's amplitudes have been noticed to decay gradually over time. [ABSTRACT FROM AUTHOR]

We propose CurveML, a benchmark for evaluating and comparing methods for the classification and identification of plane curves represented as point sets. The dataset is composed of 520k curves, of which 280k are generated from specific families characterised by distinctive shapes, and 240k are obtained from Bézier or composite Bézier curves. The dataset was generated starting from the parametric equations of the selected curves making it easily extensible. It is split into training, validation, and test sets to make it usable by learning-based methods, and it contains curves perturbed with different kinds of point set artefacts. To evaluate the detection of curves in point sets, our benchmark includes various metrics with particular care on what concerns the classification and approximation accuracy. Finally, we provide a comprehensive set of accompanying demonstrations, showcasing curve classification, and parameter regression tasks using both ResNet-based and PointNet-based networks. These demonstrations encompass 14 experiments, with each network type comprising 7 runs: 1 for classification and 6 for regression of the 6 defining parameters of plane curves. The corresponding Jupyter notebooks with training procedures, evaluations, and pre-trained models are also included for a thorough understanding of the methodologies employed. [ABSTRACT FROM AUTHOR]

After reviewing important historical and present day ideas about the concept of time, we develop some instances of mathematical examples where, from the interaction of concepts that model interactions of things in the observable world, time flow emerges in an intuitive and local interpretation. We present several instances of emergence of time flow in mathematical contexts, to wit, by specific parametrisation of deterministic and stochastic curves or of geodesics in Riemann manifolds. [ABSTRACT FROM AUTHOR]

Cotton topping is a crucial aspect of cotton production, inhibiting apical dominance in cotton plants. Existing cotton topping machinery often results in over-topping. To address this challenge, the characteristics of manual topping operations were emulated by incorporating bionic principles to analyze the motions involved. Studying the artificial topping action and the trajectory of hand movements led to the design of a bionic topping manipulator and a trajectory-generating mechanism, serving as the core component of the cotton topping device. A flat-bottomed follower disc cam mechanism was used to facilitate the automatic opening and closing of the manipulator. The cam's working area was divided, its contour curve selected, and the manipulator's pulling spring's action point and length determined. Subsequently, parametric equations for the motion trajectory of the bionic topping manipulator were established. Building on the topping mechanism's working principle, a mechanical model was developed to analyze the swing of cotton plants. The model demonstrates that the displacement at the free end of the stalk was primarily influenced by its length. A lifter was then designed to reduce plant swing amplitude and orderly distribute its top position. The designed prototype of a single-row cotton bionic topping device was tested and verified through orthogonal tests, using operating speed, rotational speed, and topping depth as test factors. The topping rate and over-topping rate served as the indices for testing. The results indicated an average topping rate of 78.67% and an over-topping rate of 8%. This was achieved at a 0.3 m/s operating speed, a 40 r/min rotational speed, and a 110 mm topping depth. Cotton topping devices demonstrated greater effectiveness in minimizing damage to cotton plants, and future research should focus on enhancing topping rates even further. This study provides a theoretical foundation and test data to support the design of cotton topping machinery, guiding future mechanical improvements and agricultural practices. [ABSTRACT FROM AUTHOR]

In recent years, deep learning has gained increasing popularity in the fields of Partial Differential Equations (PDEs) and Reduced Order Modeling (ROM), providing domain practitioners with new powerful data-driven techniques such as Physics-Informed Neural Networks (PINNs), Neural Operators, Deep Operator Networks (DeepONets) and Deep-Learning based ROMs (DL-ROMs). In this context, deep autoencoders based on Convolutional Neural Networks (CNNs) have proven extremely effective, outperforming established techniques, such as the reduced basis method, when dealing with complex nonlinear problems. However, despite the empirical success of CNN-based autoencoders, there are only a few theoretical results supporting these architectures, usually stated in the form of universal approximation theorems. In particular, although the existing literature provides users with guidelines for designing convolutional autoencoders, the subsequent challenge of learning the latent features has been barely investigated. Furthermore, many practical questions remain unanswered, e.g., the number of snapshots needed for convergence or the neural network training strategy. In this work, using recent techniques from sparse high-dimensional function approximation, we fill some of these gaps by providing a new practical existence theorem for CNN-based autoencoders when the parameter-to-solution map is holomorphic. This regularity assumption arises in many relevant classes of parametric PDEs, such as the parametric diffusion equation, for which we discuss an explicit application of our general theory. [ABSTRACT FROM AUTHOR]

This paper treats a well-established public evaluation problem, which is the analysis of the funded research projects. We specifically deal with the collection of the research actions funded by the European Union over the 7th Framework Programme for Research and Technological Development and Horizon 2020. The reference period is 2007–2020. The study is developed through three methodological steps. First, we consider the networked scientific institutions by stating a link between two organizations when they are partners in the same funded project. In doing so, we build yearly complex networks. We compute four nodal centrality measures with relevant, informative content for each of them. Second, we implement a rank-size procedure on each network and each centrality measure by testing four meaningful classes of parametric curves to fit the ranked data. At the end of such a step, we derive the best fit curve and the calibrated parameters. Third, we perform a clustering procedure based on the best-fit curves of the ranked data for identifying regularities and deviations among years of research and scientific institutions. The joint employment of the three methodological approaches allows a clear view of the research activity in Europe in recent years. [ABSTRACT FROM AUTHOR]

The authors study the problem of identifying constant parameters involved in the right-hand sides of a linear, nonautonomous system of differential equations with first-order ordinary derivatives. The specificity of the problem lies in the fact that additional conditions for identifying parameters, firstly, are nonlocal, and secondly, they include derivatives of an unknown function. The authors examine the conditions for the existence and uniqueness of a solution to the problem and propose two different approaches to the numerical solution of the problem. They also present the results of computer experiments. [ABSTRACT FROM AUTHOR]

The paper analyzes the vibrational behavior of cylinders in the offset printing machine caused by a cylinder gap shock. Specifically, it assesses the stability of a system of two cylinders. The analysis of the proposed model is reduced to solving a set of Hill equations. The singularity of the obtained equations is the relationship between the natural frequencies of the system and modulation depth. Numerical simulations, along with the generalized Hill's determinant method, were employed to determine the critical parameters of parametric resonance, thereby establishing the conditions necessary for the stability of periodic vibrations. [ABSTRACT FROM AUTHOR]

The uniformity of the lapping trajectory is a crucial criterion to determine the processing quality. The relationship between the uniformity of the lapping trajectory and the surface roughness in precision lapping of cylindrical rollers was investigated. Based on the principle of geometry and kinematics, the parametric equations of lapping trajectory points are established, and the trajectory simulation is carried out. The standard deviation of trajectory density in different sections is employed to evaluate the trajectory uniformity. Subsequently, a prediction model for surface roughness Ra based on the cutting depth of undeformed abrasive particle was established, and its accuracy was validated through experiments. Finally, based on the insights gained from simulations and experiments, the correlation between the uniformity of the lapping trajectory and surface roughness was analyzed. It is found that the variation of trajectory density and distribution results in the difference of surface roughness uniformity under different motion parameters. The result shows that the standard deviation of trajectory uniformity can reflect the uniformity of roller surface roughness numerically. Under selected parameter combinations, the average surface roughness of the cylindrical roller significantly decreased from the initial 154.6 ± 15 to 25.3 ± 3.5 nm after 20 min of lapping. [ABSTRACT FROM AUTHOR]

We consider a nonlinear Dirichlet problem driven by the double phase operator. The reaction has the combined effects of parametric concave term and of an indefinite convex one ("concave-convex" problem with indefinite weight). Using the Nehari method we prove the existence of two bounded, positive ground state solutions. [ABSTRACT FROM AUTHOR]

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As a core component of vehicle navigation and autonomous driving, high-precision maps are characterized by high accuracy, sensitive information, and high cost. Ensuring their security and confidentiality is crucial. Digital watermarking, an important tool in the field of information security, can provide authentication and protection for data. However, due to the current lack of watermarking algorithms specifically designed for high-precision maps, these maps face a series of security issues during transmission and application. Aiming at the characteristics of high-precision maps, this paper proposes a zero-watermarking algorithm for high-precision maps. First, the watermarking index is determined based on the number of parametric cubic curves used to represent lane information in high-precision maps. Secondly, the watermarking information is determined based on the parity (odd or even) of the number of parametric cubic curves in two lane information nodes. Finally, combined with the watermarking index, a feature matrix is constructed according to the voting principle and the XOR operation is performed with the encrypted watermarking image to generate a zero-watermarking. Experimental results show that this algorithm can still extract relatively complete watermarking information after being subjected to high-level geometric attacks and addition and deletion attacks, demonstrating the good robustness. [ABSTRACT FROM AUTHOR]

This paper investigates the influence of multi-decadal climatic variability on the temporal evolution of root zone storage capacities (Sr,max) and its implications for streamflow predictions in the Meuse basin. Through a comprehensive analysis of 286 catchments across Europe and the US that are hydro-climatically comparable to the Meuse basin, we construct inter-decadal distributions of past deviations in evaporative ratios (IE) from expected values based on catchment aridity (IA). These distributions of ΔIE were then used to estimate inter-decadal changes in Sr,max and to quantify the associated consequences for streamflow predictions in the Meuse basin. Our findings reveal that, while catchments do not strictly adhere to their specific parametric Budyko curves over time, the deviations in IE are generally very minor, with an average ΔIE=0.01 and an interquartile range (IQR) of -0.01 to 0.03. Consequently, these minor deviations lead to limited inter-decadal changes in Sr,max , mostly ranging between -10 and +21 mm (-5 % to +10 %). When these changes (ΔSr,max) are accounted for in hydrological models, the impact on streamflow predictions in the Meuse basin is found to be marginal, with the most significant shifts in monthly evaporation and streamflow not exceeding 4 % and 12 %, respectively. Our study underscores the utility of parametric Budyko-style equations for first-order estimates of future Sr,max in hydrological models, even in the face of climate change and variability. This research contributes to a more nuanced understanding of hydrological responses to changing climatic conditions and offers valuable insights for future climate impact studies in hydrology. [ABSTRACT FROM AUTHOR]

This paper presents a general analytical solution for the problem of locating points in planar regions with an arbitrary geometry at the boundary. The proposed methodology overcomes the traditional solutions used for polygonal regions. The method originated from the explicit evaluation of the contour integral using the Residue and Cauchy theorems, which then evolved toward a technique very similar to the winding number and, finally, simplified into a variant of ray-crossing approach slightly more informed and more universal than the classic approach, which had been used for decades. The very close relation of both techniques also emerges during the derivation of the solution. The resulting algorithm becomes simpler and potentially faster than the current state of the art for point locations in arbitrary polygons because it uses fewer operations. For polygonal regions, it is also applicable without further processing for special cases of degeneracy, and it is possible to use in fully integer arithmetic; it can also be vectorized for parallel computation. The major novelty, however, is the extension of the technique to virtually any shape or segment delimiting a planar domain, be it linear, a circular arc, or a higher order curve. [ABSTRACT FROM AUTHOR]

Systematic observations were conducted to investigate the spatiotemporal structures of blowing snow. Along a line perpendicular to the dominant wind direction on the lee side of a flat field, 15 snow particle counters (SPCs) and ultrasonic anemometers (USAs) were placed 1.5 m apart. Data were recorded at high frequencies of 100 kHz for SPCs and 1 kHz for USAs. The horizontal mass flux distributions, representing the spatiotemporal variability of blowing snow, exhibited non-uniformity in both time and space and manifested periodic changes akin to snow waves. Additionally, the presence of "snow snakes", meandering near the snow surface, was observed. Quadrant analysis revealed predominant snow fluxes in quadrants Q1 (u′>0 , w′>0) and Q4 (u′>0 , w′<0). However, a more detailed parametric curve analysis indicated the existence of ejection events in Q2 (u′<0 , w′>0) before snow waves and in front of snow snakes, shifting to Q1 and Q4 afterward, implying the consideration of both top-down and bottom-up mechanisms for burst sweep events. [ABSTRACT FROM AUTHOR]

The design of fixed offshore structures for wind turbines with a common joint configuration has been widely adopted for the past few decades. A completely overlapped tubular joint (COJ) is usually proposed as a replacement of a common joint in the design as it possesses a higher ultimate joint capacity. However, very few detailed guidelines were available as recommended design practices, so were relative research studies on COJs. This study aims at deriving a new design equation of COJ capacity based on a failure mechanic model of through-wall plastification. A symmetry of three-hinge mechanic model was applied in the derivation. An empirical study of effective width of the joint was achieved through parametric study on a validated finite element model. The influence of geometric parameters upon the joint capacity was investigated. The through-brace inclined angle, the lapped-through-brace diameter ratio and thickness ratio were found to be proportional to the joint capacity. Meanwhile, the gap of short segment and the slenderness ratio of the through brace were inversely proportional to the joint capacity. An assessment on the proposed design equation revealed that it obeys the finite element capacity on the conservative side and appears to be more accurate than the existing TY-joint equation in other design codes and parametric equations available in the literature. [ABSTRACT FROM AUTHOR]

B‐spline curve interpolation is a fundamental algorithm in computer‐aided geometric design. Determining suitable parameters based on data points distribution has always been an important issue for high‐quality interpolation curves generation. Various parameterization methods have been proposed. However, there is no universally satisfactory method that is applicable to data points with diverse distributions. In this work, a hybrid parametrization method is proposed to overcome the problem. For a given set of data points, a classifier via supervised learning identifies an optimal local parameterization method based on the local geometric distribution of four adjacent data points, and the optimal local parameters are computed using the selected optimal local parameterization method for the four adjacent data points. Then a merging method is employed to calculate global parameters which align closely with the local parameters. Experiments demonstrate that the proposed hybrid parameterization method well adapts the different distributions of data points statistically. The proposed method has a flexible and scalable framework, which can includes current and potential new parameterization methods as its components. [ABSTRACT FROM AUTHOR]

Due to the widespread applications of 2‐dimensional spherical designs, there has been an increasing requirement of modeling on the 핊2 manifold in recent years. Due to the non‐Euclidean nature of the sphere, it has some challenges to find a method to represent 2D regions on 핊2 manifold. In this paper, a skeleton‐based representation method of regions on 핊2, disk B‐spline(DBSC) on 핊2 is proposed. Firstly, we give the definition and basic algorithms of DBSC on 핊2. Then we provide the calculation method of DBSC on 핊2, which includes calculating the boundary points, internal points and their corresponding derivatives. Based on that, we give some modeling methods of DBSC on 핊2, including approximation, deformation. In the end, some stunning application examples of DBSC on 핊2 are shown. This work lays a theoretical foundation for further applications of DBSC on 핊2. [ABSTRACT FROM AUTHOR]

To address the inaccurate contour error calculation problem in traditional cross-coupled control (CCC) methods for biaxial motion systems, this paper presents a novel CCC method based on a recently developed tangential velocity tracking (TVT) strategy. It has the advantage that existing high-precision algorithms for searching the foot point can be directly integrated to obtain the excellent estimation accuracy of contour error. The cumbersome parameter tuning for position controllers of each axis is unnecessary. Moreover, a velocity interpolator for parametric curves is developed to extend the TVT strategy to the variable feedrate case. The stability of the TVT-based CCC system is proved using a quadratic Lyapunov function. Comparative experiments are carried out, and the results indicate that the estimation deviation of contour error in the TVT-based CCC method can be constrained within 1 μm. The maximum contour error is significantly reduced by 65.32% and 50.10% compared with the traditional CCC methods based on tangential and circular approximations, respectively. [ABSTRACT FROM AUTHOR]

State estimators such as Kalman filters compute an estimate of the instantaneous state of a dynamical system from sparse sensor measurements. For spatio-temporal systems, whose dynamics are governed by partial differential equations (PDEs), state estimators are typically designed based on a reduced-order model (ROM) that projects the original high-dimensional PDE onto a computationally tractable low-dimensional space. However, ROMs are prone to large errors, which negatively affects the performance of the estimator. Here, we introduce the reinforcement learning reduced-order estimator (RL-ROE), a ROM-based estimator in which the correction term that takes in the measurements is given by a nonlinear policy trained through reinforcement learning. The nonlinearity of the policy enables the RL-ROE to compensate efficiently for errors of the ROM, while still taking advantage of the imperfect knowledge of the dynamics. Using examples involving the Burgers and Navier-Stokes equations with parametric uncertainties, we show that in the limit of very few sensors, the trained RL-ROE outperforms a Kalman filter designed using the same ROM and yields accurate instantaneous estimates of high-dimensional states corresponding to unknown initial conditions and physical parameter values. The RL-ROE opens the door to lightweight real-time sensing of systems governed by parametric PDEs. [ABSTRACT FROM AUTHOR]

Previous research has revealed that vertical cantilever beam structures, after considering the effects of gravity, can exhibit a 1:3 internal resonance between two transverse modes. However, whether a similar phenomenon would still occur in the system when a proof body is attached to the free end of cantilever beams, and how to perform parameter design to deliberately induce internal resonance are still open issues. The present study conducts research on the natural frequencies of a vertical cantilever beam with a tip mass. Within the framework of the Euler–Bernoulli beam theory, Hamilton's principle is utilized to derive the integral-partial differential equation of free vibration of the system. The corresponding frequency equation is then solved using the Chebyshev pseudospectral method, with an algebraic equation of the natural frequency obtained. Moreover, an explicit analytical expression for the fundamental frequency is obtained using parameter fitting, which is then used to reveal the relationship among parameters under the critical state of buckling and evaluate the extent of gravity's impact on the fundamental frequency. The study finally investigates the ratio of the first two natural frequencies. A parameter design method is proposed, which derives a parametric curve in terms of the fundamental frequency, ensuring a 1:3 ratio between the first two natural frequencies. The free vibration time history at the midpoint of the beam is acquired by directly solving the system's partial differential equation through the finite element method. Fourier transformation of this time history verifies that the fundamental frequency aligns with a predetermined value, and the ratio between the first two natural frequencies is 1:3, thus confirming the efficacy of the proposed method. It is found that the addition of a tip mass provides more flexibility in adjusting parameters to achieve internal resonance, which facilitates the miniaturization of actual structures. [ABSTRACT FROM AUTHOR]

For systems with higher control accuracy requirements, the requirements for convergence time are also higher, such as requiring the system to maintain stability in a finite time. This paper studies the finite-time control for input-saturated system based on the event-triggered strategy and the parametric Lyapunov equation. A finite-time state feedback controller and a finite-time output feedback controller are designed by introducing a time-varying parameter which can be obtained by solving a differential equation. The proposed methods make the closed-loop system stable in finite time and save the system resources. The designed controller overcomes the problem of slow convergence speed caused by low-gain feedback control. The numerical simulation results demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper describes a project assigned in a multivariable calculus course. The project showcases many fundamental concepts studied in a typical course, including the distance formula, equations of lines and planes, intersection of planes, Lagrange multipliers, integrals in both Cartesian and polar coordinates, parametric equations, and arc length. [ABSTRACT FROM AUTHOR]

For the ship stability estimation and ship design, it is helpful to know the probability of the roll amplitude exceeding a certain threshold. Therefore, it is necessary to obtain the probability density function of the roll amplitude. In this study, first, we derive the moment values of roll amplitude by combining the moment equations and the linearity of expectation. With this, we propose a method to estimate the probability density function of the roll amplitude by using the obtained moment values. The results of our proposed method are compared to those obtained from Monte Carlo simulations. When the higher-order cumulant neglect closure method is used, the moment values resulting from the moment equations are close to the results of corresponding Monte Carlo simulations. In addition, our proposed method for deriving the probability density function of the roll amplitude is validated by comparison with Monte Carlo simulation results. In conclusion, we can state that the proposed methods for deriving the moments and the probability density function of the roll amplitude are available for practical use cases. [ABSTRACT FROM AUTHOR]

The physically based deformable curve models are widely used to simulate thin one-dimensional objects in computer graphics, interactive simulation, and surgery simulation. These models consider objects to be rods described by an adapted frame curve that contains the rod's centerline as well as the orthonormal material frame of each point on the centerline. However, they pose challenges including fine discretization, redundancy in modeling slender rods, and maintaining accuracy and stability. In this paper, we propose a novel physically based deformable B-spline curve model that regards curves as rods consisting of parallel fibers and derive elastic potential energy only from curves' representations. Therefore, our model does not take rotation-based adapted frames into consideration and reduces degree of freedom. Our model divides the curves into infinitesimal elements in parameter space and derives the analytical relationship between elastic potential energy function and curves' representations through the change of total length of infinitesimal elements' fibers. Our model can support material attributes in the real world and maintain the reality and stability of the solution. We employ isogeometric analysis to solve the dynamic equations derived from our deformable model as isogeometric analysis is suitable to solve the dynamic equations of parametric models. We compare the scenarios in the real world, our model's simulation results, and other model's results to demonstrate the reality of our models. The results are in line with expectation. We design several examples to demonstrate our models' applications. [ABSTRACT FROM AUTHOR]

We address the prescribed-time stability of a class of nonlinear system with uncertainty/disturbance. With the help of the parametric Lyapunov equation (PLE), we designed a state feedback control to regulate the full-state of a controlled system within prescribed time, independent of initial conditions. The result illustrated that the controlled state converges to zero as t approaches the settling time and remains zero thereafter. It was further proved that the controller is bounded by a constant that depends on the system state. A numerical example is presented to verify the validity of the theoretical results [ABSTRACT FROM AUTHOR]

Climatic variability can considerably affect catchment-scale root zone storage capacity (Sumax), which is a critical factor regulating latent heat fluxes and thus the moisture exchange between land and atmosphere as well as the hydrological response and biogeochemical processes in terrestrial hydrological systems. However, direct quantification of changes in Sumax over long time periods and the mechanistic drivers thereof at the catchment scale are missing so far. As a consequence, it remains unclear how climatic variability, such as precipitation regime or canopy water demand, affects Sumax and how fluctuations in Sumax may influence the partitioning of water fluxes and therefore also affect the hydrological response at the catchment scale. Based on long-term daily hydrological records (1953–2022) in the upper Neckar River basin in Germany, we found that variability in hydro-climatic conditions, with an aridity index IA (i.e. EP/P) ranging between ∼ 0.9 and 1.1 over multiple consecutive 20-year periods, was accompanied by deviations ΔIE between - 0.02 and 0.01 from the expected IE inferred from the long-term parametric Budyko curve. Similarly, fluctuations in Sumax , ranging between ∼ 95 and 115 mm or ∼ 20 %, were observed over the same time period. While uncorrelated with long-term mean precipitation and potential evaporation, it was shown that the magnitude of Sumax is controlled by the ratio of winter precipitation to summer precipitation (p < 0.05). In other words, Sumax in the study region does not depend on the overall wetness condition as for example expressed by IA , but rather on how water supply by precipitation is distributed over the year. However, fluctuations in Sumax were found to be uncorrelated with observed changes in ΔIE. Consequently, replacing a long-term average, time-invariant estimate of Sumax with a time-variable, dynamically changing formulation of that parameter in a hydrological model did not result in an improved representation of the long-term partitioning of water fluxes, as expressed by IE (and fluctuations ΔIE thereof), or in an improved representation of the shorter-term response dynamics. Overall, this study provides quantitative mechanistic evidence that Sumax changes significantly over multiple decades, reflecting vegetation adaptation to climatic variability. However, this temporal evolution of Sumax cannot explain long-term fluctuations in the partitioning of water (and thus latent heat) fluxes as expressed by deviations ΔIE from the parametric Budyko curve over multiple time periods with different climatic conditions. Similarly, it does not have any significant effects on shorter-term hydrological response characteristics of the upper Neckar catchment. This further suggests that accounting for the temporal evolution of Sumax with a time-variable formulation of that parameter in a hydrological model does not improve its ability to reproduce the hydrological response and may therefore be of minor importance for predicting the effects of a changing climate on the hydrological response in the study region over the next decades to come. [ABSTRACT FROM AUTHOR]