Linear Equations Research Articles

A novel Fourier domain scheme for three‐dimensional magnetotelluric modelling in anisotropic media

SUMMARYThis study presents a novel algorithm that combines the Lorenz gauge equations with the Fourier domain technique to simulate magnetotelluric responses in three‐dimensional conductivity structures with general anisotropy. The method initially converts the Helmholtz equations governing vector potentials into one‐dimensional differential equations in the wave number domain via the horizontal two‐dimensional Fourier transform. Subsequently, a one‐dimensional finite element method employing quadratic interpolation is applied to obtain three five‐diagonal linear equation systems. Upon solving these equations, the spatial domain fields are obtained via the inverse Fourier transform. This process guarantees the computational efficiency, memory efficiency and high parallelization of the algorithm. Moreover, an anisotropic medium iteration operator guarantees stable convergence of the method. The correctness, competence and applicability of the algorithm are verified using some synthetic models. The results demonstrate that the new method is efficient and performs well in anisotropic undulating terrain and complex structures. Compared to other Fourier domain methods and the latest edge‐based finite element algorithm, the proposed method exhibits superior computing performance. Finally, the impact of the Euler angles on the magnetotelluric responses is analysed.

Dynamical and thermodynamical stability of a charged thin-shell wormhole

AbstractA study of the dynamical and thermodynamical stability of a charged thin-shell wormhole built by gluing two Reissner–Nordström geometries is presented. The charge on the shell is linearly related to the matter content. For the dynamical stability, a concise inequality is obtained, valid for any barotropic equation of state that relates the pressure with the energy density at the throat. A thermodynamical description of the system is introduced, which leads to the temperature and the electric potentials. Adopting a linear equation of state for the pressure and a definite form for the entropy function, the set of equilibrium configurations that are both dynamically and thermodynamically stable is found.

Petrov-Galerkin method for small deflections in fourth-order beam equations in civil engineering

Abstract This study explores the Petrov–Galerkin method’s application in solving a linear fourth-order ordinary beam equation of the form u ″ ″ + q u = f u^{\prime\prime} ^{\prime\prime} +qu=f . The equation entails two distinct boundary conditions: pinned–pinned conditions on u u and u ′ u^{\prime} , and clamped–clamped conditions on u u and u ″ {u}^{^{\prime\prime} } . To satisfy these boundary conditions, we have built two sets of basis functions. The explicit forms of all spectral matrices were reported. The nonhomogeneous boundary conditions were easily handled using perfect transformations, ensuring the numerical solution’s accuracy. Detailed analysis of the method’s convergence was studied. Some numerical examples were presented, accompanied by comparisons with other existing methods in the literature.

Alternating Projection Method for Intersection of Convex Sets, Multi-Agent Consensus Algorithms and Averaging Inequalities

The history of the alternating projection method for finding a common point of several convex sets in Euclidean space goes back to the well-known Kaczmarz algorithm for solving systems of linear equations, which was devised in the 1930s and later found wide applications in image processing and computed tomography. An important role in the study of this method was played by I.I. Eremin’s, L.M. Bregman’s, and B.T. Polyak’s works, which appeared nearly simultaneously and contained general results concerning the convergence of alternating projections to a point in the intersection of sets, assuming that this intersection is nonempty. In this paper, we consider a modification of the convex set intersection problem that is related to the theory of multi-agent systems and is called the constrained consensus problem. Each convex set in this problem is associated with a certain agent and, generally speaking, is inaccessible to the other agents. A group of agents is interested in finding a common point of these sets, that is, a point satisfying all the constraints. Distributed analogues of the alternating projection method proposed for solving this problem lead to a rather complicated nonlinear system of equations, the convergence of which is usually proved using special Lyapunov functions. A brief survey of these methods is given, and their relation to the theorem ensuring consensus in a system of averaging inequalities recently proved by the first author is shown (this theorem develops convergence results for the usual method of iterative averaging as applied to the consensus problem).

New design of robust H∞ controllers for descriptor discrete time-varying delay equations with bounded disturbances

In this paper, a linear matrix inequality (LMI)-based design of robust [Formula: see text] controllers for linear descriptor discrete-time delay equations is proposed. The existing works of FTS and [Formula: see text] control are extended to the linear descriptor equations subjected to time-varying delays and bounded disturbances. First, sufficient LMI-based conditions are given to ensure the admissibility of such system, and then the conditions are applied to design [Formula: see text] controllers. A numerical example with simulations is given to illustrate the proposed results.

Research and application of a new method for axial force test of prestressed high-strength concrete pipe piles.

In order to figure out actual shaft resistance, axial forces of pile shaft have to be calculated based on strain test, which is a challenge for heat and steam cured prestressed high-strength concrete pipe piles. Disadvantages of three common methods of strain test and the empirical linear equation between axial force and strain are analyzed. A new strain test method, namely the slot-forming wood method, is proposed and applied in laboratory and field test. Taking pipe pile PHC500AB-125 as an example, laboratory tests were performed to establish the nonlinear function between axial force and strain. Field tests were performed and strains were tested by the slot-forming wood method. Results calculated by the nonlinear function between axial force and strain were compared with that by traditional linear equation to reveal the difference. The tests indicated that the new method of strain test was applicable and useful since the survival rate of strain gauges was high and the accuracy has been improved. Overcoming the disadvantages of the traditional linear equation, the proposed function between axial force and strain is more accurate. The study provides theoretical basis for technology innovation in pile test and for revision of related technical standards.

Human identification via digital palatal scans: a machine learning validation pilot study.

This study aims to validate a machine learning algorithm previously developed in a training population on a different randomly chosen population (i.e., test set). The discrimination potential of the palatal intraoral scan-based geometric and superimposition methods was evaluated. A total of 23 participants (16 females and seven males) from different countries underwent palatal scans using the Emerald intraoral scanner. Geometric-based identification involved measuring the height, width, and depth of the palatal vault in each scan. These parameters were then input into Fisher's linear discriminant equations with coefficients determined previously on a training set. Sensitivity and specificity were calculated. For the superimposition method, scan repeatability was compared to between-subjects differences, calculating mean absolute differences (MAD) between aligned scans. Multiple linear regression analysis determined the effects of sex, longitude, and latitude of country of origin on concordance. The geometric-based method achieved 91.2% sensitivity and 97.1% specificity, consistent with the results from the training set, showing no significant difference. Latitude and longitude did not significantly affect geometric-based matches. In the superimposition method, the between-subjects MAD range (1.068-0.214mm) and the repeatability range (0.011-0.093mm) did not overlap. MAD was minimally affected by longitude and not influenced by latitude. The sex determination function recognized females over males with 69.0% sensitivity, similar to the training set. However, the specificity (62.5%) decreased. The assessment of geometric and superimposition discrimination has unequivocally demonstrated its robust reliability, remaining impervious to population. In contrast, the distinction between sexes carries only moderate reliability. The significant correlation observed among longitude, latitude, and palatal height suggests the feasibility of a comprehensive large-scale study to determine one's country of origin. Portable intraoral scanners can aid forensic investigations as adjunct identification methods by applying the proposed discriminant function to palatal geometry without population restrictions. The Clinicatrial.gov registration number is NCT05349942 (27/04/2022).

Global existence and below up of logarithmic non linear vscoelastic wave equation

In this paper, we consider the following nonlinear vscoelastic wave equations with internal constant delay term. Using the semigroup theory we proved the local existence , the global existence establish by the potential well method, and, we obtain the asymptotic behavior by applying the multiplier method.

Symmetric solutions to symmetric partial difference equations

This paper studies discrete systems defined by linear difference equations on the lattice Z n that are invariant under a finite group of symmetries, and shows that there always exist solutions to such systems that are also invariant under this group of symmetries.

PICL: Physics informed contrastive learning for partial differential equations

Neural operators have recently grown in popularity as Partial Differential Equation (PDE) surrogate models. Learning solution functionals, rather than functions, has proven to be a powerful approach to calculate fast, accurate solutions to complex PDEs. While much work has been performed evaluating neural operator performance on a wide variety of surrogate modeling tasks, these works normally evaluate performance on a single equation at a time. In this work, we develop a novel contrastive pretraining framework utilizing generalized contrastive loss that improves neural operator generalization across multiple governing equations simultaneously. Governing equation coefficients are used to measure ground-truth similarity between systems. A combination of physics-informed system evolution and latent-space model output is anchored to input data and used in our distance function. We find that physics-informed contrastive pretraining improves accuracy for the Fourier neural operator in fixed-future and autoregressive rollout tasks for the 1D and 2D heat, Burgers’, and linear advection equations.

Uncertainty analysis of MHD oscillatory flow of ternary nanofluids through a diverging channel: a comparative study of nanofluid composites

Purpose This study aims to investigate the heat and mass transfer characteristics of a temperature-sensitive ternary nanofluid in a porous medium with magnetic field and the Soret–Dufour effect through a tapered asymmetric channel. The ternary nanofluid consists of Boron Nitride Nanotubes (BNNT), silver (Ag) and copper (Cu) nanoparticles, with a focus on understanding the thermal behaviour and performance across mono, hybrid and tri-hybrid nanofluids. This paper also examines the thermal behaviour of MHD oscillatory nanofluid flow and carries out an uncertainty analysis of the model using the Taguchi method. Design/methodology/approach The governing equations for this system are transformed into coupled linear partial differential equations using non-similarity transformations and solved numerically with the Crank–Nicolson scheme. The impact of temperature sensitivity at three distinct temperatures (5°C, 20°C and 60°C) is incorporated to analyse variations in viscosity and Prandtl number. The study also examines the combined effects of Soret–Dufour numbers and thermal radiation on heat and mass transfer within the nanofluid. Findings The results demonstrate that the inclusion of BNNT, Ag and Cu nanoparticles significantly enhances heat and mass transfer rate, with copper nanoparticles showing superior performance in terms of skin friction and heat transfer rates. The Soret and Dufour effects play critical roles in modulating heat and mass diffusion within tri-hybrid nanofluids. The study reveals that temperature sensitivity alters heat and mass transfer characteristics depending on the temperature range, with pronounced variations at elevated temperatures. The influence of thermal radiation and the Peclet number is found to significantly impact temperature distribution and overall heat transfer performance within the asymmetric channel. Originality/value To the best of the authors’ knowledge, this study is the first to analyse the heat and mass diffusion in a ternary nanofluid composed of BNNT, Ag and Cu nanoparticles, considering porous media, oscillatory flow and thermal radiation within a tapered asymmetric channel. The research extends to a novel examination of temperature sensitivity in mono, hybrid and tri-hybrid nanofluids at varying temperature gradients. Furthermore, a comparative analysis of skin friction and heat transfer rates between copper, alumina and ferro composites is presented for optimising the nanofluid performance.

Clinical characteristics of hemodynamics and laboratory parameters of recipients during orthotopic heart transplantation

Objective. To study the clinical and laboratory characteristics of recipients during orthotopic heart transplantation according to intraoperative monitoring. Materials and methods. The clinical and laboratory data of 38 patients who underwent orthotopic heart transplantation for vital indications in the period from 2022 to 2024 were studied. The average age of patients was (38.5 ± 13.5) years. There were 35 men (92.1%) and 3 women (7.9%). Peripheral oxygen saturation, arterial and central venous pressure were measured in each recipient delivered to the operating room at 5 stages of the operation during orthotopic heart transplantation. Results. The main cause of chronic heart failure with subsequent heart transplantation in all 38 (100%) patients was dilated cardiomyopathy. Statistically significant (p < 0.05) acidosis and hypercapnia in recipients during heart transplantation were most pronounced at the first stage of surgery, before the connection of a heart–lung machine. Acidosis during heart transplantation had a mixed metabolic and respiratory genesis with hypercapnia and lactatacidosis. A statistically significant (p < 0.05) increased lactate content at the first and second stages and statistically significant (p < 0.05) hyperglycemia at all stages of surgery were found. Central venous pressure was statistically significantly (p<0.05) higher than the reference values at the stages of sternum cutting and before the connection of the heart–lung machine. A direct statistically significant (p<0.05) correlation between the level of arterial blood pH and the level of central venous pressure (r=0.334) was established, with the corresponding linear regression equation: pH=4.7043+central venous pressure × 0.02391. Conclusions. Clinical and laboratory characteristics of recipients undergoing orthotopic heart transplantation demonstrated a direct correlation between mixed metabolic and respiratory acidosis and central venous pressure. Peripheral hemodynamic parameters in orthotopic heart transplantation were characterized by instability at the stage of heart–lung machine disconnection.

The solution of a class of linear differential equations

The solution of a class of linear differential equations

An Analytical Study of Tsunamis Generated by Submarine Landslides

In this work, the problem of tsunamis generated by underwater landslides is addressed. Two new solutions are derived in the framework of the linear shallow water equations and linear potential wave theory, respectively. Those solutions are analytical (1 + 1D) and another is semi-analytical (2 + 1D). The 1 + 1D model considers a solid body sliding over a sloping beach at a constant speed, and the 2 + 1D model considers a solid landslide that moves at a constant velocity on a flat bottom. The solution 1 + 1D is checked numerically using a different finite scheme. The 2 + 1D model examines the kinematic and geometric features of the landslide at a constant ocean depth and its influence on the generation of tsunamis. Landslide geometry significantly influences run-up height. Our results reveal a power law relationship between normalized run-up and landslide velocity within a realistic range and a negative power law for the landslide length–thickness. Additionally, a critical aspect ratio between the length and width of the sliding body is identified, which enhances the tsunamigenic process. Finally, the results show that the landslide shape does not have a decisive influence on the pattern of tsunami wave generation and propagation.

Lean Mass Longitudinally Confounds Sedentary Time and Physical Activity With Blood Pressure Progression in 2513 Children.

Randomized controlled trials have reported no effect of moderate-to-vigorous physical activity (MVPA) on reducing blood pressure (BP) in youth, probably due to short trial durations. This study examined the longitudinal effect of sedentary time (ST), light PA (LPA) and MVPA on BP in 11-year-old children followed up for 13 years to determine the confounding and mediating role of body composition. Data included 2513 children from the Avon Longitudinal Study of Parents and Children (ALSPAC), UK birth cohort who had data on at least one time-point measure of accelerometer-based movement behaviour across the follow-up and complete BP measures at ages 11, 15 and 24 years. Body composition was assessed with dual-energy x-ray absorptiometry at all time points. Multivariate adjusted generalized linear mixed-effect model and structural equation causal mediation model analyses were conducted. Among 2513 participants (61% female, mean [SD] age 11.72 [0.21] years), ST steadily increased from ~6 h/day in childhood (age 11 years) to ~9 h/day in young adulthood (age 24 years), whereas LPA and MVPA decreased, but BP had an inverted U-shaped increase. In the longitudinal analysis, after full adjustment, a 1-min cumulative ST from ages 11 to 24 years was positively associated with increased systolic BP (0.009 mmHg [95% CI 0.007-0.011]; p < 0.001) and diastolic BP. A 1-min cumulative LPA was inversely associated with systolic BP (-0.007 mmHg [-0.009 to -0.004]; p < 0.001), but not diastolic BP. In isotemporal substitution analyses, longitudinal replacement of 10 min of ST with equal time in LPA during childhood, adolescence and young adulthood cumulatively decreased systolic BP by -2.63 mmHg [95% CI -3.17 to -2.08] (p < 0.0001) and diastolic BP by -1.93 mmHg [95% CI -2.36 to -1.50] (p < 0.0001). Replacing 10 min of ST with 10 min of MVPA had no statistically significant effect due to an absolute confounding effect of lean mass. The association of ST with systolic BP was fully mediated by increased lean mass (93% mediation). Increased total fat mass partially mediated (19%-27%) the inverse associations of cumulative MVPA with cumulative systolic and diastolic BP. Theoretically replacing 10 min/day spent sedentary with 10 min/day of LPA during growth from childhood through young adulthood may lower systolic BP by -3 mmHg and diastolic BP by -2 mmHg. Lean mass seems more significant than fat mass in the relations of ST and PA with BP and should be accounted for in future intervention studies in the paediatric and young adult population.

A new tool for estimation of the in vivo distribution of intake radionuclides.

This study developed a more universal tool for estimating the in vivo distribution of intake radionuclides. The biokinetic models and transfer rates for intake radionuclides were cited from the publications of the International Commission on Radiological Protection (ICRP), and the models were transformed into a series of linear differential equations and numerically solved using the Scipy algorithm in Python. The user interfaces for model selection, data input and in vivo distribution calculations were designed using PyQt5. The calculation results of the new tool are completely consistent with those of the data viewer released by the ICRP and are also very similar to other reported values. The new tool allows users to select and change the biokinetic models and transfer rates and display the in vivo distribution of radionuclides in an image format. It is much more user friendly.

Explicit Runge–Kutta Numerical Manifold Method for Solving the Burgers’ Equation via the Hopf–Cole Transformation

This paper presents an efficient numerical manifold method for solving the Burgers’ equation. To improve accuracy and streamline the solution process, we apply a nonlinear function transformation technique that reformulates the original problem into a linear diffusion equation. We utilize a dual cover mesh along with an explicit multi-step time integration method for spatial and temporal discretization, respectively. Constant cover functions are employed across mathematical covers, interconnected by a linear weight function for each manifold element. The full discretization formulation is derived using the Galerkin weak form. To efficiently compute the inverse of the symmetric positive definite mass matrix, we employ the Crout algorithm. The performance and convergence of our method are rigorously evaluated through several benchmark numerical tests. Extensive comparisons with exact solutions and alternative methods demonstrate that our approach delivers an accurate, stable, and efficient computational scheme for the Burgers’ equation.

One loop effective actions in Kerr-(A)dS black holes

We compute new exact analytic expressions for one-loop scalar effective actions in Kerr (A)dS black hole (BH) backgrounds in four and five dimensions. These are computed by the connection coefficients of the Heun equation via a generalization of the Gelfand-Yaglom formalism to second-order linear ordinary differential equations with regular singularities. The expressions we find are in terms of Nekrasov-Shatashvili special functions, making explicit the analytic properties of the one-loop effective actions with respect to the gravitational parameters and the precise contributions of the quasinormal modes. The latter arise via an associated integrable system. In particular, we prove asymptotic formulas for large angular momenta in terms of hypergeometric functions and give a precise mathematical meaning to Rindler-like region contributions. Moreover, we identify the leading terms in the large distance expansion as the point particle approximation of the BH and their finite size corrections as encoding the BH tidal response. We also discuss the exact properties of the thermal version of the BH effective actions by providing a proof of the Denef-Hartnoll-Sachdev formula and explicitly computing it for new relevant cases. Although we focus on the real scalar field in dS-Kerr and (A)dS-Schwarzschild in four and five dimensions, similar formulas can be given for higher spin matter and radiation fields in more general gravitational backgrounds. Published by the American Physical Society 2024

On Deadlock Analysis and Characterization of Labeled Petri Nets with Undistinguishable and Unobservable Transitions

This work addresses the analysis and characterization of deadlocks in discrete-event systems modeled by labeled Petri nets (LPNs) with undistinguishable and unobservable transitions. To provide a solution for the notorious problem, it is essential to present an effective characterization in such a way that deadlock control and synthesis are technically and methodologically possible. To this end, we introduce the notion of dangerous implicit vectors (DIVs), which implicitly threaten the system deadlock-freedom. The set of dead markings is divided into two subsets: dead basis markings (DBMs) and dangerous implicit markings (DIMs). An algorithm is designed to compute the sets of DIVs and DIMs at a given basis state of a system. Moreover, by virtue of linear algebraic equations, we formulate sufficient conditions for identifying the existence of blocking markings in an LPN. Finally, an algorithm is developed to construct an observed graph that is a compendious presentation of the reachability graph of a net system, with respect to the existence of dead reaches. At the end of this paper, experiment results that illustrate the correctness and effectiveness of the reported solution are presented.

Abstract Su203: Development of Bioimpedance for the Measurement of Ventilation During Cardiopulmonary Resuscitation

Introduction: Previous studies have shown that ventilation during cardiopulmonary resuscitation (CPR) produces better patient outcomes. However, research regarding ventilation during the early stages of basic life support CPR is limited. Prior studies showed that thoracic bioimpedance, recorded by the LifePak 12 defibrillator device, can be used to count ventilations with a tidal volume at least 250 mL during CPR. However, the relationship between thoracic bioimpedance and tidal volume has yet to be studied for the Phillips MRx device, excluding a large data set from ventilation studies. Hypothesis: Bioimpedance, measured by the Phillips MRx device has a linear relationship with tidal volume. Aims: To determine the relationship between bioimpedance measured by the Phillips MRx device and tidal volume in males and females. Methods: We conducted the study in a respiratory laboratory at a University Hospital. We asked 27 volunteers to breathe fixed volumes (250 mL, 300 mL, 400 mL, 600 mL, and 800 mL) delivered by a mechanical ventilator while recording bioimpedance simultaneously with the Philips MRx monitor-defibrillator through electrode pads placed on the chest. Six measurements of bioimpedance waveform amplitudes (Ohms) were taken at each tidal volume. Results: Of 27 volunteers, 11 were female and 16 were male. In the female sample set, a linear equation of y=0.003x + 0.837 with an R^2 value of 0.9995 was calculated. In the male sample set, a linear equation of Y=0.0016x +0.1039 with a R^2 value of 0.995 was calculated. Females had a higher median bioimpedance amplitude than men at each tidal volume, and a steeper linear regression slope. At the minimum threshold tidal volume used for counting ventilations (250 mL), the median bioimpedance amplitude was 0.5 (0.4, 0.6) ohm for males and the median bioimpedance amplitude was 0.8 (0.6, 1) ohm for females. Conclusion: As measured and recorded with the Philips MRx monitor-defibrillator, there is a linear relationship between tidal volume and thoracic bioimpedance. The linear models created in this study can be used in future studies to count ventilations during CPR using the Philips MRx device.