Parameters Of Equation Research Articles

Tsallis holographic dark energy model in an anisotropic universe with linearly varying deceleration parameter and its correspondence with phantom

In this paper, we investigate a spatially homogeneous and anisotropic Bianchi type-I universe filled with cold dark matter and Tsallis holographic dark energy within the framework of General Relativity. We assume the cold dark matter and the Tsallis holographic dark energy to be non-interacting and obtain exact solutions of the Einstein field equations by considering a linearly varying deceleration parameter. The physical and kinematical properties of the resulting model are explored through different parameters of cosmological importance such as the Hubble parameter, the anisotropy parameter, the equation of state parameter, the jerk parameter, the snap parameter, the lerk parameter etc. The validation or violation of the four energy conditions are also checked. We find that the Weak and Dominant Energy Conditions are fulfilled but the Null and Strong Energy Conditions are violated at late time which supports the accelerated expansion of the universe. In addition, we establish the correspondence between phantom scalar field model and Tsallis holographic dark energy model.

Generative learning of the solution of parametric Partial Differential Equations using guided diffusion models and virtual observations

Generative learning of the solution of parametric Partial Differential Equations using guided diffusion models and virtual observations

Insightful Examination of Some Nonlinear Classifications Linked with Mathieu Oscillators

PurposeThe aim of the current study is to comparehend an analytical description of different classes of nonlinear Mathieu oscillators. The van der Pol-Duffing-Mathieu oscillator, and the generalized van der Pol-Duffing-Mathieu oscillator are examined. Additionally, the hybrid Rayleigh-van der Pol-Duffing-Mathieu oscillators, as well as the nonlinear Mathieu oscillator, are scrutinized.MethodThe non-perturbative approach (NPA) is utilized to convert the nonlinear ordinary differential equations (ODEs), of the illustrated dynamical systems, into linear ones. The approximate solutions are derived independently in the series expansion and without the use of conventional perturbation techniques. Therefore, the goal is to deviate from conventional perturbation techniques and get approximations of small amplitude parametric components without imposing any restrictions. The method is also expanded to determine the best solutions for the nonlinear immense amplitude of fluctuation.ResultsThe current method offers successive approximations of the solutions of parametric nonlinear fluctuations may be obtained by quickly estimating the frequency-amplitude relationship. The resulting parametric equations are validated, showing high degree of agreement with the original equation. Stability behavior is analyzed under various circumstances. The transition curves, bifurcation diagram, Poincaré map, and phase portrait are also examined using the Floquet theory.ConclusionThe stability regions are found to be diminishing with the rise of the natural frequency, and the excited frequency. Moreover, the achieved regions are found to be growing with the rise of the damping coefficient and the excitation amplitude. The stability settings have been examined by considering the effects of various factors in both the damped and un-damped phases for each situation. In the un-damped state, PolarPlots are examined of the transition curves of the two corresponding solutions, namely Cos- and Sin-oscillations. The conclusions of the acquired results suggested that the approach presented here is highly efficient, robust, founded on solid premises, and remarkably intuitive.

Predictive Equations for Modeling Shear Behavior in RC Columns with Non-Seismic Details for Enhanced Seismic Performance Assessment

Accurate modeling of the hysteretic behavior of reinforced concrete columns with non-seismic detailing is crucial for assessing the seismic performance of reinforced concrete structures. While shear spring models are commonly used to capture nonlinear shear deformation in shear-critical reinforced concrete columns, determining appropriate parameters for these models remains challenging and often relies on expert judgment. This study proposes predictive equations for shear spring parameters that can be integrated into fiber-section elements to enhance the accuracy of modeling reinforced concrete columns. Specifically, empirical equations are developed for the nonlinear modeling parameters of the Pinching4 model implemented in OpenSees. Experimental data of flexural–shear and shear-critical reinforced concrete columns are used to calibrate the modeling parameters for predicting strength, stiffness, pinching, and cyclic degradation behavior. The predictive equations, derived through regression analysis, are validated against experimental data and demonstrated relatively high accuracy in capturing the shear behavior. To evaluate the impact of the shear behavior of reinforced concrete columns on the seismic fragility of non-conforming reinforced concrete frames, two types of numerical models are established with and without the Pinching4 material using the proposed predictive equations. Analytical results indicate that neglecting the appropriate shear behavior of reinforced concrete columns could lead to an overestimation of the seismic performance of non-conforming reinforced concrete frames.

Rainfall Intensity–Duration–Frequency Curves Dataset for Brazil

This is a database containing rainfall intensity–duration–frequency equations (IDF equations) for 6550 pluviographic and pluviometric stations in Brazil. The database was compiled from 370 different publications and contains the following information: station identification, geographic position, size and period of the rainfall series used, parameters of the IDF equations, and literature references. The database is available on Mendeley Data (DOI: 10.17632/378bdcmnc8.1) in the form of spreadsheets and vector files. Since the launch of the Pluvio 2.1 software in 2006, which included 549 IDF equations obtained in the country, this is the largest and most accessible database of IDF equations in Brazil. The data provided may be useful, among other purposes, for designing hydraulic structures, controlling water erosion, planning land use, and water resource planning and management.

A Basic Approach to Equations of States for Studying the Real Behavior of Noble Gases

The relation between the pressure and molar concentration (in mol/L) of real gases in a low- to medium-pressure range is precisely described by a logarithmic two-parameter equation. Increasing the concentration caused an increase in pressure and also the weakening of the effect due to intermolecular interactions, forming the basis for an equation with an adjusted parameter. Exceeding a critical concentration by a further increase caused a switch to another set of parameters in the same equation. At high pressure, a second switch to an exponential term was observed. This equation of state, defined segment by segment, was attributed to three different structures of the medium. The validity of the equations found was verified with experimental data reported in the literature for helium, neon, argon, krypton, and xenon and is discussed in more detail for argon. The temperature dependence of the parameters of the equations is reported and the formation of a liquid phase is discussed.

Charged Gravastar Solutions in the Finch‐Skea Framework with f(Q)$f(\mathbb {Q})$ Gravity

AbstractIn this study, the features of a charged gravastar are investigated within the framework of gravity ( represents nonmetricity) using the Finch–Skea metric. This metric is applied to both the interior and shell regions of the charged gravastar and the field equations are derived accordingly. For the exterior regions, various black holes are considered, i.e., Reissner–Nordström, Bardeen, and Hayward regular black holes. The interior and exterior layers are matched using the Israel junction conditions, which help to determine the surface energy density and surface pressure for these black holes. Some physical properties such as proper length, entropy, energy, and the equation of state parameter are examined. The stability of the developed gravastar model is discussed through the effective potential, the causality condition and adiabatic index. It is concluded that the compact gravastar structure could be a viable alternative to black holes within this framework.

Compositional Sparsity, Approximation Classes, and Parametric Transport Equations

Abstract Approximating functions of a large number of variables poses particular challenges often subsumed under the term “Curse of Dimensionality” (CoD). Unless the approximated function exhibits a very high level of smoothness the CoD can be avoided only by exploiting some typically hidden structural sparsity. In this paper we propose a general framework for new model classes of functions in high dimensions. They are based on suitable notions of compositional dimension-sparsity quantifying, on a continuous level, approximability by compositions with certain structural properties. In particular, this describes scenarios where deep neural networks can avoid the CoD. The relevance of these concepts is demonstrated for solution manifolds of parametric transport equations. For such PDEs parameter-to-solution maps do not enjoy the type of high order regularity that helps to avoid the CoD by more conventional methods in other model scenarios. Compositional sparsity is shown to serve as the key mechanism for proving that sparsity of problem data is inherited in a quantifiable way by the solution manifold. In particular, one obtains convergence rates for deep neural network realizations showing that the CoD is indeed avoided.

Projective Cohen-Macaulay monomial curves and their affine charts

Abstract In this paper, we explore when the Betti numbers of the coordinate rings of a projective monomial curve and one of its affine charts are identical. Given an infinite field k and a sequence of relatively prime integers $$a_0 = 0< a_1< \cdots < a_n = d$$ a 0 = 0 < a 1 < ⋯ < a n = d , we consider the projective monomial curve $$\mathcal {C}\subset \mathbb {P}_k^{\,n}$$ C ⊂ P k n of degree d parametrically defined by $$x_i = u^{a_i}v^{d-a_i}$$ x i = u a i v d - a i for all $$i \in \{0,\ldots ,n\}$$ i ∈ { 0 , … , n } and its coordinate ring $$k[\mathcal {C}]$$ k [ C ] . The curve $$\mathcal {C}_1 \subset \mathbb {A}_k^n$$ C 1 ⊂ A k n with parametric equations $$x_i = t^{a_i}$$ x i = t a i for $$i \in \{1,\ldots ,n\}$$ i ∈ { 1 , … , n } is an affine chart of $$\mathcal {C}$$ C and we denote by $$k[\mathcal {C}_1]$$ k [ C 1 ] its coordinate ring. The main contribution of this paper is the introduction of a novel (Gröbner-free) combinatorial criterion that provides a sufficient condition for the equality of the Betti numbers of $$k[\mathcal {C}]$$ k [ C ] and $$k[\mathcal {C}_1]$$ k [ C 1 ] . Leveraging this criterion, we identify infinite families of projective curves satisfying this property. Also, we use our results to study the so-called shifted family of monomial curves, i.e., the family of curves associated to the sequences $$j+a_1< \cdots < j+a_n$$ j + a 1 < ⋯ < j + a n for different values of $$j \in \mathbb {N}$$ j ∈ N . In this context, Vu proved that for large enough values of j, one has an equality between the Betti numbers of the corresponding affine and projective curves. Using our results, we improve Vu’s upper bound for the least value of j such that this occurs.

Constitutive Equation and Processing Parameter Sensitivities for Hot Deformation of Pure Mo

Constitutive Equation and Processing Parameter Sensitivities for Hot Deformation of Pure Mo

Hyperelastic constitutive modeling of healthy and enzymatically mediated degraded articular cartilage.

This research demonstrates a systematic curve fitting approach for acquiring parametric values of hyperelastic constitutive models for both healthy and enzymatically mediated degenerated cartilage to facilitate finite element modeling of cartilage. Several widely used phenomenological hyperelastic constitutive models were tested to adequately capture the changes in cartilage mechanics that vary with the differential/unequal abundance of matrix metalloproteinases (MMPs). Trauma and physiological conditions result in an increased production of collagenases (MMP-1) and gelatinases (MMP-9), which impacts the load-bearing ability of cartilage by significantly deteriorating its extracellular matrix (ECM). The material parameters in the constitutive equation of each hyperelastic model are significant for developing a comprehensive computational interpretation of MMP mediated degenerated cartilage. Stress-strain responses obtained from indentation test were fitted with selected Ogden, polynomial, reduced polynomial, and van der Waals hyperelastic constitutive models by optimizing their adjustable parameters (material constants). The goodness of fit of the 2nd order reduced polynomial and van der Waals model exhibited the closest data fitting with the experimental stress-strain distributions of healthy and degraded articular cartilage. The coefficient of the shear modulus for the 2nd order reduced polynomial decreased gradually by 21.9% to 80.1% with more enzymatic degradation of collagen fibril due to the relative abundance of MMP-1 (collagenases), and 28.5% to 69.2% for the van der Waals model. Our findings showed that the major materials coefficients of the models were reduced in the degenerated cartilages, and the reduction varied differentially with the relative abundance of MMPs-1 and 9, correlating the severity of degeneration. This work advances the understanding of cartilage mechanics and offers insights into the impact of biochemical (enzymatic) effects on cartilage degradation.

Accelerating cosmological models in Hoyle–Narlikar gravity with observational constraints

In this paper, we explore the cosmic acceleration and the physical aspects of the dark energy nature in Hoyle–Narlikar’s creation-field theory, taking into account observational constraints. We find analytical solutions to the modified field equations for a specific choice of creation field [Formula: see text] with certain background fluid sources as a barotropic fluid and a radiation fluid in a flat Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime universe. From there, we use observational constraints with data from the cosmic chronometer (CC) Hubble data and the apparent magnitude from the Pantheon SNe sample to get the model parameters’ constrained values with [Formula: see text] and [Formula: see text] confidence levels. We look at the physical properties of the models we’ve made using these estimated values of model parameters. To do this, we look at the deceleration parameter [Formula: see text], the effective equation of state parameter [Formula: see text], the jerk parameter [Formula: see text], the snap parameter [Formula: see text], and the energy conditions. We also examine the cosmic behavior of square sound speed [Formula: see text] for the test of causality and viability of the models. We measure the present age of the universe and transition redshift [Formula: see text] for the signature-flipping point. We obtain two transit-phase accelerating universe models with [Formula: see text] Km/s/Mpc, [Formula: see text], [Formula: see text], and [Formula: see text], respectively, with barotropic and radiation fluid sources.

Zeta Potential Change and curing effects on Shield Muck solidified by Metal Ion Complex Catalysis

Using potentiometric testing, we investigated the zeta potential of shield muck curing materials’ particle surfaces, varying the concentration of metal ion complex. We analyzed the microscopic characteristics of shield muck curing products by using the electron microscopy, revealing the impact of metal ion complex on curing. Results showed that the metal ion complex significantly reduces the surface zeta potential of shield muck and conventional curing materials, with cement showing the most substantial effect, followed by shield muck, calcium carbonate, and calcium sulfate. The concentration of metal ion complex negatively correlates with the zeta potential within a range of 0 to 100 mmol/L. Comparing shield muck cured for 7 and 28 days, with and without metal ion complexes, indicated that metal ion complexes accelerate crystalline product formation, increasing the compressive strength of shield cinder cured products by over 35%. Fitted curves and empirical parameter equations of zeta potential and metal ion complex concentration align with experimental results. This study provides a reference for the efficient treatment and sustainable utilization of shield slag soil, holding significant practical engineering value.

Fully Distributed Cooperative Output Regulation of Heterogeneous Periodic Linear Multi‐Agent Systems With Input Saturation

ABSTRACTThis article delves into the cooperative output regulation of heterogeneous continuous‐time periodic linear multi‐agent systems with input saturation under undirected graphs. Leveraging characteristics inherent in the parametric differential Lyapunov equation and cooperative output regulation theory, we formulate an observer‐based protocol. Specifically, we design distributed state observers for both the leader and followers, aiming to estimate the leader state and the follower state itself, respectively. Utilizing these estimated states, we propose a class of fully distributed observer‐based protocols to address the considered problem. In addition, we make an extension to the reduced‐order observer‐based form. The most distinguishing contributions of our proposed protocols, in contrast to existing literature, lie in the fact that the considered dynamics of each agent are time‐varying and have input saturation constraints. Notably, our protocols eliminate the need for global information of the network, operating in a fully distributed method. Finally, we apply the proposed control protocol to the multi‐spacecraft elliptical orbit rendezvous problem.

Sustainability of the Rural Environment Based on a Tripartite Game Among Government, Enterprises, and Farmers Under the Prisoner’s Dilemma

In recent years, with rapid economic development, the importance of environmental governance has gradually been overlooked. The increasing pollution caused by rural sewage discharge has posed a significant threat to the local ecological environment. In this context, this paper treats the local governments, village enterprises, and farmers as the game subjects, integrating them with prisoner’s dilemma theory to construct a tripartite game model. By analyzing the equilibrium points and stability of the model, as well as the effects of the parameters in the dynamic equations, this paper reveals the evolution of strategic choices made by different participants. The results indicate that the optimal strategy for local governments is strict regulation, and village enterprises tend to adopt purification measures, while farmers weigh the costs and benefits and exhibit a negative attitude toward participating in governance. Finally, from a practical perspective, this paper offers suggestions for improving rural ecological environmental protection by engaging local governments, village enterprises, and farmers, aiming to promote the sustainable development of the ecological environment.

GPS Phase Integer Ambiguity Resolution Based on Eliminating Coordinate Parameters and Ant Colony Algorithm.

Correctly fixing the integer ambiguity of GNSS is the key to realizing the application of GNSS high-precision positioning. When solving the float solution of ambiguity based on the double-difference model epoch by epoch, the common method for resolving the integer ambiguity needs to solve the coordinate parameter information, due to the influence of limited GNSS phase data observations. This type of method will lead to an increase in the ill-posedness of the double-difference solution equation, so that the fixed success rate of the integer ambiguity is not high. Therefore, a new integer ambiguity resolution method based on eliminating coordinate parameters and ant colony algorithm is proposed in this paper. The method eliminates the coordinate parameters in the observation equation using QR decomposition transformation, and only estimates the ambiguity parameters using the Kalman filter. On the basis that the Kalman filter will obtain the float solution of ambiguity, the decorrelation processing is carried out based on continuous Cholesky decomposition, and the optimal solution of integer ambiguity is searched using the ant colony algorithm. Two sets of static and dynamic GPS experimental data are used to verify the method and compared with conventional least squares and LAMBDA methods. The results show that the new method has good decorrelation effect, which can correctly and effectively realize the integer ambiguity resolution.

Environmental cosmic acceleration from a phase transition in the dark sector

A new degravitation mechanism within the framework of scalar tensor gravity is postulated and included by prescription. The mechanism eliminates all constant contributions from the potential to the Friedmann equation, leaving only the kinematic and the dynamic terms of the potential to drive cosmic acceleration. We explore a scenario involving a density-triggered phase transition in the late-time universe, and argue that the resulting effective energy density and equation of state parameter can explain late-time cosmology when extrapolated to a region of the parameter space.

Adjoint‐State Reflection Traveltime Tomography for Velocity and Interface Inversion With Its Application in Central California Near Parkfield

AbstractTraveltime tomography considering reflection arrivals is a promising approach for investigating interface topography and near‐interface velocity heterogeneity. In this study, we formulate this inverse problem as an eikonal equation‐constrained optimization problem, in which the traveltime field of the reflection wave is accurately described by a two‐stage eikonal equation. The novelty lies in deriving the Fréchet derivative with respect to interface topography. By employing the coordinate transformation technique to convert an irregular physical domain with an undulating interface to a regular computational domain, we successfully encode the interface topography into the anisotropic parameters in the eikonal equation. This approach enables us to derive explicit forms of the Fréchet derivatives related to interface topography and velocity based on the adjoint‐state method, which is not only computationally efficient but also avoids potential inaccuracy in ray tracing. Several numerical experiments are conducted to verify our new method. Finally, we apply this method to central California near Parkfield by inverting traveltimes of both first‐P and Moho‐reflected waves (named PmP). The low‐velocity anomalies imaged in the lower crust are consistent with the along‐strike variations of low‐frequency earthquakes (LFEs) beneath the San Andreas Fault (SAF), suggesting the presence of fluids that may influence the occurrence of LFEs in this region.

Reduction of singularities in Fuchsian equations, and Heun equations for perturbations of Kerr–Newman–de Sitter and anti-de sitter spacetimes

For Fuchsian differential equations with N singularities at finite points we display two simply expressed conditions on coefficients which guarantee equivalence, under a certain inversion transformation, to a differential equation with N−1 singularities at finite points. For the radial and angular differential equations that govern perturbations of Kerr–Newman–de Sitter and anti-de Sitter spacetimes we find that these two conditions are satisfied identically for arbitrary values of all the parameters in the differential equations, including pole locations, with regular singularities and non-zero cosmological constant. From this we generalize the results of Suzuki, Tarasugi, and Umetsu, Prog. Theor. Phys. 100, 491 (1998) and find 128 Heun equations which are equivalent to the equations which govern the perturbations of the KN(A)dS spacetimes.

Understanding the language of molecules: predicting pure component parameters for the PC-SAFT equation of state from SMILES

The SPT-PC-SAFT model predicts PC-SAFT parameters from the SMILES code of a molecule. High accuracy is achieved by training the parameters of the transformer model directly on experimental vapor pressure and density data.