Linear Equations Research Articles

The Presence of Chaos in a Viscoelastic Harmonically Forced Von Mises Truss

Abstract This work investigates how viscoelasticity affects the dynamic behavior of a lumped-parameter model of a bistable von Mises truss. The system is controlled by a linear first-order equation and a second-order nonlinear Duffing equation with a quadratic nonlinearity that governs mechanical behavior. The second-order equation controls mechanical oscillations, while the linear first-order equation controls viscoelastic force evolution. Combined, the two equations form a third-order jerk equation that controls system dynamics. Viscoelasticity adds time scales and degrees-of-freedom to material behavior, distinguishing it from viscosity-only systems. Due to harmonic excitation, the system exhibits varied dynamic responses, from periodic to quasi-periodic to chaotic. We explore the dynamics of a harmonically forced von Mises truss with viscous damping to address this purpose. We demonstrate this system's rich dynamic behavior due to driving amplitude changes. This helps explain viscoelastic system behavior. A viscoelastic unit replaces the viscous damper, and we show that, although viscous damping merely changes how fast the trajectory decays to an attractor, viscoelasticity modifies both the energy landscape and the rate of decay. In a conventional linear solid model, three viscoelastic parameters control the system's behavior instead of one, as in pure viscous damping. This adds degrees-of-freedom that affect system dynamics. We present the parameter space for chaotic behavior and the shift from regular to irregular motion. Finally, Melnikov's criteria identify the regular-chaotic threshold. The system's viscous and elastic components affect the chaotic threshold amplitude

Energetics of the H-Bond Network in Exiguobacterium sibiricum Rhodopsin.

Exiguobacterium sibiricum rhodopsin (ESR) functions as a light-driven proton pump utilizing Lys96 for proton uptake and maintaining its activity over a wide pH range. Using a combination of methodologies including the linear Poisson-Boltzmann equation and a quantum mechanical/molecular mechanical approach with a polarizable continuum model, we explore the microscopic mechanisms underlying its pumping activity. Lys96, the primary proton uptake site, remains deprotonated owing to the loss of solvation in the ESR protein environment. Asp85, serving as a proton acceptor group for Lys96, does not form a low-barrier H-bond with His57. Instead, deprotonated Asp85 forms a salt-bridge with protonated His57, and the proton is predominantly located at the His57 moiety. Glu214, the only acidic residue at the end of the H-bond network exhibits a pKa value of ∼6, slightly elevated due to solvation loss. It seems likely that the H-bond network [Asp85···His57···H2O···Glu214] serves as a proton-conducting pathway toward the protein bulk surface.

Superconvergence of UWLDG Method for One-Dimensional Linear Sixth-Order Equations

Superconvergence of UWLDG Method for One-Dimensional Linear Sixth-Order Equations

Unique Solvability of the Zaremba Problem for Linear Second Order Elliptic Equations with Drift

Unique Solvability of the Zaremba Problem for Linear Second Order Elliptic Equations with Drift

The Dirac equation as a linear tensor equation for one component

The Dirac equation is one of the most fundamental equations of modern physics. It is a spinor equation, but some tensor equivalents of the equation were proposed previously. Those equivalents were either nonlinear or involved several components of the Dirac field. On the other hand, the author showed previously that the Dirac equation in electromagnetic field is equivalent to a fourth-order equation for one component of the Dirac spinor. The equivalency is used in this work to derive a linear tensor equivalent of the Dirac equation for just one component. This surprising result can be used in applications of the Dirac equation, for example, in general relativity or for lattice approximation of the Dirac field, and can improve our understanding of the Dirac equation.

Deflection of Light by a Reissner-Nordström Black Hole and Painlevé VI equation

Abstract We consider the bending angle of the trajectory of a photon incident from and deflected to infinity around a Reissner-Nordström black hole. We treat the bending angle as a function of the squared reciprocal of the impact parameter and the squared electric charge of the background normalized by the mass of the black hole. It is shown that the bending angle satisfies a system of two inhomogeneous linear partial differential equations with polynomial coefficients. This system can be understood as an isomonodromic deformation of the inhomogeneous Picard-Fuchs equation satisfied by the bending angle in the Schwarzschild spacetime, where the deformation parameter is identified as the background electric charge. Furthermore, the integrability condition for these equations is found to be a specific type of the Painlevé VI equation that allows an algebraic solution. We solve the differential equations both at the weak and strong deflection limits. In the weak deflection limit, the bending angle is expressed as a power series expansion in terms of the squared reciprocal of the impact parameter and we obtain the explicit full-order expression for the coefficients. In the strong deflection limit, we obtain the asymptotic form of the bending angle that consists of the divergent logarithmic term and the finite O(1) term supplemented by linear recurrence relations which enable us to straightforwardly derive higher order coefficients. In deriving these results, the isomonodromic property of the differential equations plays an important role. Lastly, we briefly discuss the applicability of our method to other types of spacetimes such as a spinning black hole.

A variational formulation for three-dimensional linear thermoelasticity with ‘thermal inertia’

AbstractA variational model has been developed to investigate the coupled thermo-mechanical response of a three-dimensional continuum. The linear Partial Differential Equations (PDEs) of this problem are already well-known in the literature. However, in this paper, we avoid the use of the second principle of thermodynamics, basing the formulation only on a proper definition (i) of kinematic descriptors (the displacement and the entropic displacement), (ii) of the action functional (with kinetic, internal and external energy functions) and (iii) of the Rayleigh dissipation function. Thus, a Hamilton–Rayleigh variational principle is formulated, and the cited PDEs have been derived with a set of proper Boundary Conditions (BCs). Besides, the Lagrangian variational perspective has been expanded to analyze linear irreversible processes by generalizing Biot’s formulation, namely, including thermal inertia in the kinetic energy definition. Specifically, this implies Cattaneo’s law for heat conduction, and the well-known Lord–Shulman model for thermo-elastic anisotropic bodies is then deduced. The developed variational framework is ideal for the perspective of analyzing the thermo-mechanical problems with micromorphic and/or higher-order gradient continuum models, where the deduction of a coherent system of PDEs and BCs is, on the one hand, not straightforward and, on the other hand, natural within the presented variational deduction.

Optimizing the noise control in a two-layer conduit

In the modern world, noise pollution is a major concern due to its prevalence. This work focuses on optimizing noise control in a two-layer conduit. A conduit comprises an inlet and double-outlet zones (DOZ). The upper walls of the DOZ are lined with perforated absorbent material while the lower walls are layered with fibrous. Mathematically, the physical problem is formulated with a field wave equation together with rigid and impedance boundary conditions in the respective zones. Such governing boundary value problem (BVP) leads to the Sturm-Liouville (SL) category in which standard orthogonality relations (OR) are indispensable. The system of the linear equations of the BVP is acquired with semi-analytical Mode-Matching (MM) approach by implementing the continuity conditions of sound pressures and velocities at the matching junction with the aid of OR. These systems are truncated and solved numerically with computation learning to obtain the reflected and transmitted modal amplitudes in respective zones. Due to the lining's perforated upper walls, fibrous lower walls, and reversal in the DOZ, the analysis of the reflected and transmitted powers versus frequencies is significantly observed and is shown in graphical findings. The algebraic derivation is validated by satisfying the conservation law of power flux and matching continuity conditions for impedance, perforated, and fibrous lined boundaries.

New discrete-time zeroing neural network for solving time-dependent linear equation with boundary constraint

Recently, continuous- and discrete-time models of a zeroing neural network (ZNN) have been developed to provide online solutions for the time-dependent linear equation (TDLE) with boundary constraint. This paper presents a novel approach to address the bound-constrained TDLE (BCTDLE) problem by proposing a new discrete-time ZNN (DTZNN) model. The proposed DTZNN model is designed using the Taylor difference formula to discretize the previous continuous-time ZNNN (CTZNN) model. Theoretical analysis indicates the computational property of the proposed DTZNN model, and numerical results further demonstrate its validity. The applicability of the proposed DTZNN model is finally confirmed via its application to the motion planning of a PUMA560 robotic arm.

ОЦІНЮВАННЯ ВПЛИВУ ЗАСОБІВ ПІДКЛЮЧЕННЯ РЕЗЕРВУ НА ПОКАЗНИКИ НАДІЙНОСТІ СИСТЕМИ ПЕРЕДАВАННЯ ДАНИХ

Structural redundancy is included in modern data transmission systems to ensure the specified reliability indicators. Structural redundancy gives the system fault tolerance and significantly increases its reliability. To manage the redundant resource, it is necessary to have additional tools that will connect the redundant equipment to the main configuration of the data transmission system. The redundant equipment is a separate technical means and, from the point of view of reliability, it is connected in series with other modules of the data transmission system. Thus, the inclusion of redundant connection facilities in the data transmission system reduces its reliability. In addition, even with perfect reliability, redundant connections may fail or malfunction and, under certain conditions, cause the transmission system to fail. To determine the impact of reserve connection facilities on the reliability of the transmission system, the article applies an original technology for building models of discrete-continuous stochastic systems. ASNA software was used as a software tool for modeling the reliable behavior of the data transmission system. This software provides automated construction of the model in the form of a graph of states and transitions based on a structural automaton model. Based on the graph of states and transitions, an analytical model is automatically generated in the form of a system of linear differential Kolmogorov-Chapman equations. Using the developed model, the article quantitatively analyzes the impact of redundant connection means (switches) on the reliability indicators of the data transmission system. In the study, the proposed models, unlike the existing ones, take into account both reliability indicators (probability of failure) and functionality indicators (probability of switching, speed) of the reserve connection means. On the basis of the developed model, the limit values of reliability and functionality of the redundant connection means (switches) were determined, at which they will minimize the reliability of the fault-tolerant data transmission system. Based on the analysis of the obtained results, practical recommendations for the selection of reliability indicators of the backup connection means, their speed and probability of switching are formed.

The learning curve for minimally invasive Achilles repair using the “lumbar puncture needle and oval forceps” technique

IntroductionAn acute Achilles tendon rupture represents a common tendon injury, and its operative methods have been developed over the years. This study aimed to quantify the learning curve for the minimally invasive acute Achilles tendon rupture repair.MethodsFrom May 2020 to June 2022, sixty-seven patient cases who received minimally invasive tendon repair were reviewed. Baseline data and operative details were collected. The cumulative summation (CUSUM) control chart was used for the learning curve analyses. Achilles tendon rupture score (ATRS), American Orthopedic Foot and Ankle Society (AOFAS) ankle/hindfoot score, and visual analog scale (VAS) at 3/6/9/12 months were calculated to assess the clinical outcomes.ResultsThirty-six cases underwent at least a year of follow up and were enrolled in this study. The gender ratio and average age were 80.5% and 32.5 years. The linear equation fitted well (R2 = 0.95), and CUSUM for operative time peaked in the 12th case, which was divided into the learning phase (n = 12) and master phase (n = 24). No significant difference was detected between the two groups in clinical variables, except for the operative time (71.1 ± 13.2 min vs 45.8 ± 7.2 min, p = 0.004). Moreover, we detected one case with a suture reaction and treated it properly.ConclusionMinimally invasive Achilles repair provides an opportunity for early rehabilitation. Notably, the learning curve showed that the “lumbar puncture needle and oval forceps” technique was accessible to surgeons.

Analog quantum simulation of partial differential equations

Abstract Quantum simulators were originally proposed for simulating one partial differential equation in particular -- Schrodinger's equation. Can quantum simulators also efficiently simulate other partial differential equations? While most computational methods for partial differential equations -- both classical and quantum -- are digital (they must be discretised first), partial differential equations have continuous degrees of freedom. This suggests that an analog representation can be more natural. While digital quantum degrees of freedom are usually described by qubits, the analog or continuous quantum degrees of freedom can be captured by qumodes. Based on a method called Schrodingerisation, we show how to directly map D-dimensional linear partial differential equations onto a (D+1)-qumode quantum system where analog or continuous-variable Hamiltonian simulation on D+1 qumodes can be used. This very simple methodology does not require one to discretise partial differential equations first, and it is not only applicable to linear partial differential equations but also to some nonlinear partial differential equations and systems of nonlinear ordinary differential equations. We show some examples using this method, including the Liouville equation, heat equation, Fokker-Planck equation, Black-Scholes equations, wave equation and Maxwell's equations. We also devise new protocols for linear partial differential equations with random coefficients, important in uncertainty quantification, where it is clear how the analog or continuous-variable framework is most natural. This also raises the possibility that some partial differential equations may be simulated directly on analog quantum systems by using Hamiltonians natural for those quantum systems.

A numerical technique based on Legendre wavelet for linear and nonlinear hyperbolic telegraph equation

A numerical technique based on Legendre wavelet for linear and nonlinear hyperbolic telegraph equation

Thermal Environment Analysis and Optimization for Large Space Buildings with Radiant Cooling Floors: A Case Study of Xianyang International Airport

Radiant cooling floors combined with ventilation systems have been widely applied in large space buildings. However, there has been a lack of research on system control strategies for their adaptation to weather changes. This study aimed to find control strategies for radiant cooling floors combined with displacement ventilation systems used in large space buildings in order to achieve energy conservation and environmental improvement. Supply air temperature and cooling surface temperature were determined to be the control variables. It was found that cooling capacity of the combined system and the comfort index, PMV (predicted mean vote), were linear in relation to the supply air temperature and cooling surface temperature. The linear equations regarding cooling capacity and PMV were established separately using environment data, and then the optimal region was determined. A case study on Terminal 3 of Xi’an Xianyang International Airport was conducted. The thermal environment was investigated through on-site measurements, questionnaires, and numerical simulations with CFD (computational fluid dynamics). It was found that supply air temperature and cooling surface temperature had a significant impact on PMV, and less impact on the cooling capacity. Therefore, it was determined that the supply air temperature should be altered first when the indoor temperature exceeds the upper limit, and then the cooling surface temperature should be changed if the indoor environment continues to overheat with the supply air temperature set to 18 °C. Thus, the supply air temperature was kept at 18 °C, and the floor surface temperature was set to be 22 °C on a high-temperature day. The average PMV was 0.87, and the cooling capacity of the combined system was 200 W/(m2·K), according to the CFD simulation. In addition, the surface heat transfer coefficient of the cooling floor was found to be 10.26 W/(m2·K). This research provides important references for the design and operational management of radiant cooling floors in large space buildings.

The New Integral Transform “Bayawa Transform”

In this paper a new integral transform namely Bayawa transform is proposed to solve linear ordinary differential equations with constant coefficients

An efficient meshless numerical method with the error estimate for two-dimensional Schrödinger equation

In this study, we present a capable meshfree numerical approach to numerical solving of the two-dimensional Schrödinger equation (SE). Firstly, the radial point interpolation is obtained to build the shape functions for discretization in the mentioned equation. These meshless shape functions have Kronecker delta attributes. Secondly, the local weak form is constructed over the local sub-domain for the purpose of diminishing the solution of the time-dependent SE in a two-dimensional case and transforming it into a system of algebraic linear equations. Also, semi and fully discrete schemes with their error estimates for this meshless method are obtained. We indicate the proposed method square of error has the order of O(h2m+τ2) in the L2 norm. Moreover, some numerical tests are provided, and their exact and finite difference solutions are compared to verify the efficiency of our algorithm.

Enzymatic and Algebraic Methodology to Determine the Contents of Kunitz and Bowman-Birk Inhibitors and Their Contributions to Total Trypsin or Chymotrypsin Inhibition in Soybeans.

Soybeans are the number one source of plant proteins for food and feed, but the natural presence of protein protease inhibitors (PIs), namely, the Kunitz trypsin inhibitor (KTI) and the Bowman-Birk inhibitor (BBI), exerts antinutritional effects. This communication describes a new methodology for simultaneously quantitating all parameters of PIs in soybeans. It consists of seven steps and featured enzymatically measuring trypsin and chymotrypsin inhibitory activities, respectively, and subsequently determining the contents of reactive KTI and BBI and the contributions of each toward total PI mass and total trypsin or chymotrypsin inhibition by solving a proposed system of linear equations with two variables (C = dB + eK and T = xB + yK). This enzymatic and algebraic (EA) methodology was based on differential inhibitions of KTI and BBI toward trypsin and chymotrypsin and validated by applications to a series of mixtures of purified KTI and BBI, two KTI-null and two conventional soybeans, and by sodium dodecyl sulfate polyacrylamide gel electrophoresis. The EA methodology allowed calculations of PI composition and the contributions of individual inhibitors toward total inhibition with ease. It was first found that although BBI constituted only about 30% of the total PI mass in conventional raw soybeans, it contributed about 80% toward total chymotrypsin inhibitor activity and about 45% toward trypsin inhibitor activity. Therefore, BBI caused more total protease inhibitions than those of KTI. Furthermore, the so-called KTI-null soybean mutants still contained measurable KTI content and thus should be named KTI-low soybeans.

Change in Dental Arch Parameters—Perimeter, Width and Length after Treatment with a Printed RME Appliance

One of the important parameters in orthodontics is the perimeter of the dental arch. Precise assessment is necessary in cases of maxillary constriction treated with a rapid maxillary expander (RME). The orthodontic software allows customization of the processes from diagnosis to manufacturing of the treatment device. The aim of the present study is to evaluate a relationship between the parameters of the dental arch—perimeter, width, and length—and to follow the changes during treatment. The study is based on the digital measurements of 3D models of 33 patients treated with a digitally planned and printed RME. In the results an increase of 3.99 mm in perimeter was achieved. The rest of the parameters were changed as follows: The width of the dental arch was increased in the premolar area by an average of 3.3 mm; in the area of the first molars, the increase was 4.41 mm; the length of the dental arch in the anterior segment was reduced by an average of 0.54 mm; and the whole length by 0.52 mm. Correlation between the studied variables was described by linear equations. In conclusion, rapid maxillary expansion is a reliable method for gaining predictable space in the dental arch.

Reduced-order modeling on a near-term quantum computer

Quantum computing is an advancing area of research in which computer hardware and algorithms are developed to take advantage of quantum mechanical phenomena. In recent studies, quantum algorithms have shown promise in solving linear systems of equations as well as systems of linear ordinary differential equations (ODEs) and partial differential equations (PDEs). Reduced-order modeling (ROM) algorithms for studying fluid dynamics have shown success in identifying linear operators that can describe flowfields, where dynamic mode decomposition (DMD) is a particularly useful method in which a linear operator is identified from data. In this work, DMD is reformulated as an optimization problem to propagate the state of the linearized dynamical system on a quantum computer. This reformulation was chosen as a means of facilitating implementation on a near-term quantum computer. Quadratic unconstrained binary optimization (QUBO), a technique for optimizing quadratic polynomials in binary variables, allows for quantum annealing algorithms to be applied. A quantum circuit model (quantum approximation optimization algorithm, QAOA) is utilized to obtain predictions of the state trajectories. Results are shown for the quantum-ROM predictions for flow over a 2D cylinder at Re = 220 and flow over a NACA0009 airfoil at Re = 500 and α=15∘. The quantum-ROM predictions are found to depend on the number of bits utilized for a fixed point representation and the truncation level of the DMD model. Comparisons with DMD predictions from a classical computer algorithm are made, as well as an analysis of the computational complexity and prospects for future, more fault-tolerant quantum computers.

A novel hybrid finite element – Finite difference approach for 3D Magnetotelluric modeling

A new method is developed to solve the Magnetotelluric (MT) forward problem in three dimensions (3D). The finite-differences (FD) and the finite-elements (FE) methods are considered to be popular numerical techniques to solve the partial differential equations in electromagnetics. The FD approach is known to be easily applicable to a given problem while the FE method is much more flexible when a topography is present, however, it is also computationally more expensive to solve the linear matrix equations for the FE due to the number of non-zero entries it populates when compared to the FD’s. For that reason, a new algorithm is offered to mix the FE and FD in a single mesh using hexahedra and rectangular prisms in 3D. By doing that, it is possible to take advantage of the beneficial aspects of these two methods. It was found out that the newly developed hybrid approach (HYB) is almost as fast as the FD method while being very memory efficient which is an important aspect due to the limited memory space of a GPU which is employed to solve the systems iteratively. It was found that the HYB method is as accurate as the FE method while it is a more reasonable choice for larger models due to the solution speed of the FE method as it does not scale well with the increased model size. On average, a speed-up of 1.6 is found between the FE and HYB methods with the three models tested and it increases with the increasing model size. Using a GPU during the iterative solution made it possible to solve the linear systems in seconds and with the help of the HYB method, the memory consumption was found to be less than half of what the FE method needed. This ultimately means that much larger models can be simulated on the limited memory that a GPU has. Finally, the technique shown to implement the HYB approach makes the deployment of it to existing FE codes so much easier.