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A new approach for the calculation of eigenstates with the state-averaged (multi-layer) multi-configurational time-dependent Hartree (MCTDH) approach is presented. The approach is inspired by the recent work of Larsson [J. Chem. Phys. 151, 204102 (2019)]. It employs local optimization of the basis sets at each node of the multi-layer MCTDH tree and successive downward and upward sweeps to obtain a globally converged result. At the top node, the Hamiltonian represented in the basis of the single-particle functions (SPFs) of the first layer is diagonalized. Here p wavefunctions corresponding to the p lowest eigenvalues are computed by a block Lanczos approach. At all other nodes, a non-linear operator consisting of the respective mean-field Hamiltonian matrix and a projector onto the space spanned by the respective SPFs is considered. Here, the eigenstate corresponding to the lowest eigenvalue is computed using a short iterative Lanczos scheme. Two different examples are studied to illustrate the new approach: the calculation of the vibrational states of methyl and acetonitrile. The calculations for methyl employ the single-layer MCTDH approach, a general potential energy surface, and the correlation discrete variable representation. A five-layer MCTDH representation and a sum of product-type Hamiltonian are used in the acetonitrile calculations. Very fast convergence and order of magnitude reductions in the numerical effort compared to the previously used block relaxation scheme are found. Furthermore, a detailed comparison with the results of Avila and Carrington [J. Chem. Phys. 134, 054126 (2011)] for acetonitrile highlights the potential problems of convergence tests for high-dimensional systems. [ABSTRACT FROM AUTHOR]

The Neumann–Neumann method is a commonly employed domain decomposition method for linear elliptic equations. However, the method exhibits slow convergence when applied to semilinear equations and does not seem to converge at all for certain quasilinear equations. We therefore propose two modified Neumann–Neumann methods that have better convergence properties and require fewer computations. We provide numerical results that show the advantages of these methods when applied to both semilinear and quasilinear equations. We also prove linear convergence with mesh-independent error reduction under certain assumptions on the equation. The analysis is carried out on general Lipschitz domains and relies on the theory of nonlinear Steklov–Poincaré operators. [ABSTRACT FROM AUTHOR]

In this study, the results obtained from forest vegetation via the research project on plant sociology which was conducted in the Black Sea region of Türkiye is evaluated with the help of a fuzzy similarity measures approach. Via this project, the plant sociology in an area which has not been studied in the Black Sea region of Türkiye is performed to investigate the plant communities and ecological and sociological relationships with each other. The similarity relations among the plant communities and relevés (sampling areas) which they covered are investigated. The issue of fuzzy similarity of sets and elements in sets is studied. According to this point of view, the fuzzy similarity among the plant communities and among the relevés is introduced. This joint study is carried out in a fuzzy environment, considering the classical results found in the project in question to be obtained in more detail for application. It is also understood that such studies can only be best performed with an interdisciplinary working group. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the nonlinear Meyer-König and Zeller operators based on q-integers. Firstly, we describe the q-Meyer-König and Zeller operators of max-product type. Then, we give an error estimation for the q-Meyer-König and Zeller operators of max-product kind by using a modulus of continuity. [ABSTRACT FROM AUTHOR]

This paper focuses on the complex version of the Cahn-Hilliard-Oono equation with Neumann boundary conditions, which is used to capture long-range nonlocal interactions in the phase separation process. The first part of the paper establishes the well-posedness of the corresponding stationary problem associated with the equation. Subsequently, a numerical model is constructed using a finite element discretization in space and a backward Euler scheme in time. We demonstrate the existence of a unique solution to the stationary problem and obtain error estimates for the numerical solution. This, in turn, serves as proof of the convergence of the semi-discrete scheme to the continuous problem. Finally, we establish the convergence of the fully discrete problem to the semi-discrete formulation. [ABSTRACT FROM AUTHOR]

In this paper double, quadruple and octuple primes are discussed. Computational and theoretical properties as well as examples of quadruple and octuple primes are presented, where the octuple primes in most cases have primes between the defining quadruples although the existence of a pure quadruple prime has also been demonstrated computationally. The possibility of sixtentuple primes is also briefly discussed. [ABSTRACT FROM AUTHOR]

Fractional Kinetic equations (FKEs) including a wide variety of special functions are widely and successfully applied in describing and solving many important problems of physics and astrophysics. In this work, the solutions of the FKEs of the generalized Hurwitz-Lerch Zeta function using the Laplace transform are derived and examined. [ABSTRACT FROM AUTHOR]

The aim of this paper is to examine the stability of a class of nonlinear stochastic operators on a finite‐dimensional simplex. We identify a number of properties of the set of the class of stochastic operators, including the stability of each operator. We then use these operators to define non‐homogeneous Markov measures that rely on initial data and carry out an investigation into the absolute continuity and singularity of these measures. [ABSTRACT FROM AUTHOR]

Segmentation process is very popular in Speech recognition, word count, speaker indexing and speaker diarization process. This paper describes the speaker segmentation system which detects the speaker change point in an audio recording of multi speakers with the help of feature extraction and proposed distance metric algorithms. In this new approach, pre-processing of audio stream includes noise reduction, speech compression by using discrete wavelet transform (Daubechies wavelet 'db40' at level 2) and framing. It is followed by two feature extraction algorithms pyknogram and nonlinear energy operator (NEO). Finally, the extracted features of each frame are used to detect speaker change point which is accomplished by applying dissimilarity measures to find the distance between two frames. To realize it, a sliding window is moved across the whole data stream to find the highest peak which corresponds to the speaker change point. The distance metrics incorporated are standard "Bayesian Information Criteria (BIC)", "Kullback Leibler Divergence (KLD)", "T-test" and proposed algorithm to detect the speaker boundaries. At the end, threshold value is applied and their results are evaluated with Recall, Precision and F-measure. Best result of 99.34% is shown by proposed distance metric with pyknogram as compare to BIC, KLD and T-test algorithms. [ABSTRACT FROM AUTHOR]

This paper aims to investigate new coincidence point theorems for self‐maps under an extended rational contraction. Our approach uses the p‐adic distance in partially ordered ultrametric spaces. Furthermore, to illustrate our major discoveries, we give a comprehensive numerical analysis. This study expands on prior research in partially ordered ultrametric spaces to provide a more comprehensive understanding of such spaces using rational contraction. [ABSTRACT FROM AUTHOR]

We study an infinite system of algebraic equations with monotone and concave nonlinearity. This system arises in the various discrete problems of the mathematical models of the natural sciences. In particular, these systems of such structure, with specific instances of nonlinearity and the corresponding infinite matrix, are encountered in the theory of radiative transfer, the kinetic theory of gases, and the mathematical theory of epidemic diseases. Under certain conditions on the entries of the corresponding infinite matrix and nonlinearity, we prove the existence and uniqueness theorems of a coordinatewise nonnegative nontrivial solution in the space of bounded sequences. Also, we obtain a uniform estimate of the corresponding successive approximations and prove that the so-constructed solution tends at infinity with speed to a positive fixed point of the function describing the nonlinearity of its system. The main tools are the Krasnoselskii method on constructing invariant cone segments for the corresponding nonlinear operator and the methods of the theory of discrete convolution operators, together with some geometric inequalities for concave and monotonic functions. Furthermore, we exhibit some particular examples of the corresponding infinite matrix and nonlinearity that satisfy all imposed hypotheses. [ABSTRACT FROM AUTHOR]

We introduce the notion of localized operators. We extend the boundedness of the localized operators from the weighted Lebesgue spaces to the weighted Herz spaces. The localized operators include the Hardy operator, the Riemann–Liouville fractional integrals, the general Hardy‐type operators, the geometric mean operator, and the one‐sided maximal function. Therefore, this paper extends the mapping properties of the the Hardy operator, the Riemann–Liouville fractional integrals, the general Hardy‐type operators, the geometric mean operator, and the one‐sided maximal function to the weighted Herz spaces. [ABSTRACT FROM AUTHOR]

In this paper, the structure of the incremental quasistatic contact problem with Coulomb friction in linear elasticity (Signorini–Coulomb problem) is unraveled and sharp existence results are proved for the most general two-dimensional problem with arbitrary geometry and elasticity modulus tensor. The problem is reduced to a variational inequality involving a nonlinear operator which handles both elasticity and friction. This operator is proved to fall into the class of the so-called Leray–Lions operators, so that a result of Brézis can be invoked to solve the variational inequality. It turns out that one property in the definition of Leray–Lions operators is difficult to check and requires proving a new fine property of the linear elastic Neumann-to-Dirichlet operator. This fine property is only established in the case of the two-dimensional problem, limiting currently our existence result to that case. In the case of isotropic elasticity, either homogeneous or heterogeneous, the existence of solutions to the Signorini–Coulomb problem is proved for arbitrarily large friction coefficient. In the case of anisotropic elasticity, an example of nonexistence of a solution for large friction coefficient is exhibited and the existence of solutions is proved under an optimal condition for the friction coefficient. [ABSTRACT FROM AUTHOR]

A new non-linear stationary process for time series of counts is introduced. The process is composed of the survival and innovation component. The survival component is based on the generalized zero-modified geometric thinning operator, where the innovation process figures in the survival component as well. A few probability distributions for the innovation process have been discussed, in order to adjust the model for observed series with the excess number of zeros. The conditional maximum likelihood and the conditional least squares methods are investigated for the estimation of the model parameters. The practical aspect of the model is presented on some real-life data sets, where we observe data with inflation as well as deflation of zeroes so we can notice how the model can be adjusted with the proper parameter selection. [ABSTRACT FROM AUTHOR]

Globally positive unbounded solutions, with zero derivative at infinity, are here considered for ordinary differential equations involving the generalized Euclidean mean curvature operator. When p ≥ 2 , the results highlight an analogy with an auxiliary equation with the p-Laplacian operator. The results are obtained using some comparison criteria for the principal solutions of a class of associated half-linear equations. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we construct a nonlinear regularized Hermitian and skew-Hermitian splitting (NRHSS) iteration method for a class of nonlinear saddle-point systems. The NRHSS iteration method can be seen as a generalization of the Peaceman-Rachford splitting method with monotone plus skew-symmetric splitting (PRSM/MSS). Besides, if the nonlinear operator reduces to the linear case, the NRHSS iteration method reduces to the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration method. Hence, the NRHSS iteration method can also be seen as a generalization of the RHSS iteration method. To overcome the heavy workload when direct solution methods are used, we also propose an inexact version of the NRHSS iteration method. Detailed theoretical analysis regarding the convergence properties is given. Numerical experiments on image deblurring problems are carried out to illustrate the feasibility and efficiency of the NRHSS iteration method. [ABSTRACT FROM AUTHOR]

In this study, the dynamics of a reaction–diffusion equation on a bounded interval at the first dynamic transition point is investigated. The main difference of this study from the existing ones in the dynamic transition literature is that in this work, no specific form of the nonlinear operator is assumed except that it is an analytic function of u$u$ and ux$u_x$. The linear operator is also assumed to be a general operator with sinusoidal eigenvectors. Moreover, we assume that there is a first transition as a real simple eigenvalue changes sign. With this general framework, we make a rigorous analysis of the dependence of the first transition dynamics on the coefficients of the Taylor expansion of the nonlinear operator. The main tool we use in this study is the center manifold reduction combined with the classification of the dynamic transition theory. This study is a generalization of a recent work where the nonlinear operator contains only low‐order (quadratic and cubic) nonlinearities. The current generalization requires certain technical difficulties such as the validity of the genericity conditions on the Taylor coefficients of the nonlinear operator and a bootstrapping argument using these genericity conditions. The results of this work can be generalized in various directions, which are discussed in the conclusions section. [ABSTRACT FROM AUTHOR]

We establish pointwise bounds expressed in terms of a nonlinear potential of a generalized Wolff type for 풜 -superharmonic functions with nonlinear operator 풜 : Ω × ℝ n → ℝ n having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls estimates from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfy conditions expressed in the natural scales. Finally, we give a variant of Hedberg–Wolff theorem on characterization of the dual of the Orlicz space. [ABSTRACT FROM AUTHOR]

We give a characterization of the singular points of the free boundary ∂{u>0} for viscosity solutions of the nonlinear equation F(D²u)=-χ {u>0}, where F is a fully nonlinear elliptic operator and χ is the characteristic function. This equation models the propagation of a nerve impulse along an axon. We analyze the structure of the free boundary ∂{u>0} near the singular points where u and ∇u vanish simultaneously. Our method uses the stratification approach developed in Dipierro and the author's 2018 paper. In particular, when n=2 we show that near a flat singular free boundary point, ∂{u>0} is a union of four C 1 arcs tangential to a pair of crossing lines. [ABSTRACT FROM AUTHOR]

In this paper, based on the Ulm-Chebyshev iterative procedure, we present a multi-step Ulm-Chebyshev-like method to solve systems of nonlinear equations F (x) = 0 , { y n = x n − B n F ( x n) , z n = y n − B n F ( y n) , x n + 1 = z n − B n F (z n) , B ¯ n = 2 B n − B n A n + 1 B n , B n + 1 = B ¯ n + B ¯ n (2 I − A n + 1 B ¯ n) (I − A n + 1 B ¯ n) , n = 0 , 1 , 2 , ... , where A n + 1 is an approximation of the derivative F ′ ( x n + 1). This method does not contain inverse operators in its expression, and does not require computing Jacobian matrices for solving Jacobian equations. We have proved that the multi-step Ulm-Chebyshev-like method converges locally to the solution with R -convergence rate 4 under appropriate conditions. Some applications are given, compared with other existing methods, where the most important features of the method are shown. [ABSTRACT FROM AUTHOR]

This article investigates the finite-approximate controllability properties for semi-linear integrodifferential systems involving higher-order fractional derivatives in Riemann-Liouville sense excluding Lipschitz assumptions of nonlinear operators. We discuss the existence of mild solutions by utilizing the Schaefer fixed point principle and compactness condition on the fractional resolvent. Then we show that one can steer the system in an arbitrary neighbourhood of any given target state simultaneously obeying the finitely many constraints. Lastly, an illustrative example is presented to validate the proposed methodology. [ABSTRACT FROM AUTHOR]

Fractal interpolation has been conventionally treated as a method to construct a univariate continuous function interpolating a given finite data set with the distinguishing property that the graph of the interpolating function is the attractor of a suitable iterated function system. On the one hand, attempts have been made to extend the univariate fractal interpolation from a finite data set to a countably infinite set. On the other hand, fractal interpolation in higher dimensions, particularly the theory of fractal interpolation surfaces (FISs), has received increasing attention for more than a quarter century. This article targets a two-fold extension of the notion of fractal interpolation by providing a general framework to construct FISs for a prescribed set consisting of countably infinite data on a rectangular grid. By using this as a crucial tool, we obtain a parameterized family of bivariate fractal functions simultaneously interpolating and approximating a prescribed bivariate continuous function. Some elementary properties of the associated nonlinear (not necessarily linear) fractal operators are established, thereby allowing the interaction of the notion of fractal interpolation with the theory of nonlinear operators.

To investigate the impact of non-linear interactions on dynamic coarse graining, we study a simplified model system featuring a tracer particle in a complex environment. Using a projection operator formalism and computer simulations, we systematically derive generalized Langevin equations (GLEs) describing the dynamics of this particle. We compare different kinds of linear and non-linear coarse-graining procedures to understand how non-linearities enter reconstructed GLEs and how they influence the coarse-grained dynamics. For non-linear external potentials, we show analytically and numerically that the non-Gaussian parameter and the incoherent intermediate scattering function will not be correctly reproduced by the GLE if a linear projection is applied. This, however, can be overcome by using non-linear projection operators. We also study anharmonic coupling between the tracer and the environment and demonstrate that the reconstructed memory kernel develops an additional trap-dependent contribution. Our study highlights some open challenges and possible solutions in dynamic coarse graining. [ABSTRACT FROM AUTHOR]

We prove the convergence of the newly defined iteration method to a common fixed point of a finite nonexpansive mappings family in CAT(0) space. A numerical example is given to check the convergence of the newly generalized iteration process. Many known results are extended and improved in this article. [ABSTRACT FROM AUTHOR]

The convergence issues and getting trapped in local optimal points are two of the major concerns in the field of optimization. For this purpose, improving the standard algorithms to reach better performance in facing complex optimization problems is considered as one of the main challenges in the field of optimization. In this paper, the performance improvement of metaheuristic algorithms is considered while the applicability of the improved and standard algorithms is evaluated through the weight optimization problem of truss structures. For this purpose, the recently proposed Coot optimization algorithm is utilized as the main algorithm which is inspired by different movement types of Coot birds in the water in order to reach food supplies. Regarding the fact that the standard Coot algorithm utilizes random movement in the main search loop, a new improving methodology is utilized in this paper by replacing these random movements with Levy flight as a stochastic procedure with step length defined by levy distribution. The performance of the standard and improved Coot optimization algorithms is investigated in dealing with the problem of optimizing the shape and size of truss structures. Based on the best and statistical results, it is concluded that the improved Coot algorithm is capable of providing better results that the standard Coot algorithm while the capability of the improving methods in increasing the overall performance of the standard algorithm is demonstrated. [ABSTRACT FROM AUTHOR]

This paper presents an investigation of the existence and the unique feature of solutions for nonlinear Langevin equations involving two fractional orders with anti-periodic boundary conditions (APBCs). As a result of employing some fixed point theorems like Schauder and contraction mapping principles, the existence and uniqueness of solutions are examined. On top of this, the stability within the scope of Ulam-Hyers of solutions to this problem is also considered. The distinctive features of the present study are its similar variant and the existence of derivatives of Caputo and Riemann in the problem structure. Finally, to illustrate the result of the study, an example is presente. [ABSTRACT FROM AUTHOR]

In this paper, we established the Boyd-Wong type and Meir-Keeler type contractions in a new generalized b-metric space. Two types of fixed point theorems are proven, which extend the same results in the metric and b-metric spaces. Some examples and one application are also discussed to show the applicability of the results. [ABSTRACT FROM AUTHOR]

This paper introduces a hybrid spectral-conjugate gradient (SCG) method to solve nonlinear monotone operator equations efficiently. The proposed method incorporates a hybrid parameter that encompasses the Polak--Ribière--Polyak (PRP), Liu-Storey (LS), Fletcher-Reeves (FR), and conjugate descent (CD) methods as particular instances. Additionally, we derive the spectral parameter to ensure that the search direction adheres to the sufficient descent condition. The search direction is also designed to be bounded, and under specific conditions, we demonstrate that the sequence produced by our hybrid SCG algorithm converges toward a solution. Furthermore, to underscore the effectiveness of our proposed method, we conducted extensive numerical experiments comparing its performance against that of existing algorithms. These experiments were based on a selection of benchmark nonlinear monotone operator equations, highlighting our proposed algorithm's superior efficiency and potential in practical applications. [ABSTRACT FROM AUTHOR]

In this manuscript, we study the existence of mild solutions to initial value problems for hybrid fractional semi-linear evolution equations. On the other hand, we prove four different types of Ulam-Hyers stability results for mild solutions. The existence of mild solutions is proved by the Dhage fixed point theorem. Finally, an example is given to illustrate our results. [ABSTRACT FROM AUTHOR]

The interest in incorporating environmentally friendly and renewable sources of energy, like photovoltaic (PV) technology, into electricity grids has grown significantly. These sources offer benefits, such as reduced power losses and improved voltage stability. To optimize these advantages, it is essential to determine optimal placement and management of these energy resources. This paper proposes an Improved RUNge–Kutta optimizer (IRUN) for allocating PV-based distributed generations (DGs) and Battery Energy Storage (BES) in distribution networks. IRUN utilizes three strategies to avoid local optima and enhance exploration and exploitation phases: a non-linear operator for smoother transitions, a Chaotic Local Search for thorough exploration, and diverse solution updates for refinement. The efficacy of IRUN is evaluated using 10 benchmark functions from the CEC'20 test suite, followed by statistical analysis. Next, IRUN is used to optimize the allocation of PVDG and BES to minimize energy losses in two standard IEEE distribution networks. The optimization problem is divided into two stages. In the first stage, the optimal size and the location of PV systems are calculated to meet peak load demand. In the second stage, considering time-varying load demand and intermittent PV generation, effective energy management of BES is employed. The effectiveness of IRUN is compared against the original RUN and other well-known optimization algorithms through simulation results. The comprehensive analysis demonstrates that IRUN outperforms the compared algorithms, making it a leading solution for optimizing PV distributed generation and BES allocation in distribution networks and the results show that the energy loss reduction reaches 63.54% and 68.19% when using PVand BES in IEEE 33-bus and IEEE 69 bus respectively. [ABSTRACT FROM AUTHOR]

It is known that translation-invariant Gibbs measures of a model with an uncountable set of spin values can be described by positive fixed points of a nonlinear integral operator of Hammerstein type. Significant results have been obtained on positive fixed points of a Hammerstein-type operator with a degenerate kernel, but the existence of Gibbs measures corresponding to the fixed points have not been proved for constructed kernels. We construct new degenerate kernels of the Hammerstein operator in the context of the theory of Gibbs measures, and show that each positive fixed point of the operator gives a translation-invariant Gibbs measure. [ABSTRACT FROM AUTHOR]

Research on particle filters has been progressing with the aim of applying them to high-dimensional systems, but alleviation of problems with ensemble Kalman filters (EnKFs) in nonlinear or non-Gaussian data assimilation is also an important issue. It is known that the deterministic EnKF is less robust than the stochastic EnKF in strongly nonlinear regimes. We prove that if the observation operator is linear the analysis ensemble perturbations of the local ensemble transform Kalman filter (LETKF) are uniform contractions of the forecast ensemble perturbations in observation space in each direction of the eigenvectors of a forecast error covariance matrix. This property approximately holds for a weakly nonlinear observation operator. These results imply that if the forecast ensemble is strongly non-Gaussian the analysis ensemble of the LETKF is also strongly non-Gaussian, and that strong non-Gaussianity therefore tends to persist in high-frequency assimilation cycles, leading to the degradation of analysis accuracy in nonlinear data assimilation. A hybrid EnKF that combines the LETKF and the stochastic EnKF is proposed to mitigate non-Gaussianity in nonlinear data assimilation with small additional computational cost. The performance of the hybrid EnKF is investigated through data assimilation experiments using a 40- variable Lorenz-96 model. Results indicate that the hybrid EnKF significantly improves analysis accuracy in high-frequency data assimilation with a nonlinear observation operator. The positive impact of the hybrid EnKF increases with the increase of the ensemble size. [ABSTRACT FROM AUTHOR]

In this paper, we have introduced a functional approach for approximating nonparametric functions and coefficients in the presence of multivariate and functional predictors. By utilizing the Fisher scoring algorithm and the cross-validation technique, we derived the necessary components that allow us to explain scalar responses, including the functional index, the nonlinear regression operator, the single-index component, and the systematic component. This approach effectively addresses the curse of dimensionality and can be applied to the analysis of multivariate and functional random variables in a separable Hilbert space. We employed an iterative Fisher scoring procedure with normalized B-splines to estimate the parameters, and both the theoretical and practical evaluations demonstrated its favorable performance. The results indicate that the nonparametric functions, the coefficients, and the regression operators can be estimated accurately, and our method exhibits strong predictive capabilities when applied to real or simulated data. [ABSTRACT FROM AUTHOR]

The goal of this paper is to explore the asymptotic behaviour of anisotropic problems governed by operators of the pseudo p-Laplacian type when the size of the domain goes to infinity in different directions. [ABSTRACT FROM AUTHOR]

As the complexity of optimization problems continues to rise, the demand for high-performance algorithms becomes increasingly urgent. This paper addresses the challenges faced by the Aquila Optimizer (AO), a novel swarm-based intelligent optimizer simulating the predatory behaviors of Aquila in North America. While AO has shown good performance in prior studies, it grapples with issues such as poor convergence accuracy and a tendency to fall into local optima when tackling complex optimization tasks. To overcome these challenges, this paper proposes a multi-strategy boosted AO algorithm (PGAO) aimed at providing enhanced reliability for global optimization. The proposed algorithm incorporates several key strategies. Initially, a chaotic map is employed to initialize the positions of all search agents, enriching population diversity and laying a solid foundation for global exploration. Subsequently, the pinhole imaging learning strategy is introduced to identify superior candidate solutions in the opposite direction of the search domain during each iteration, accelerating convergence and increasing the probability of obtaining the global optimal solution. To achieve a more effective balance between the exploration and development phases in AO, a nonlinear switching factor is designed to replace the original fixed switching mechanism. Finally, the golden sine operator is utilized to enhance the algorithm's local exploitation trends. Through these four improvement strategies, the optimization performance of AO is significantly enhanced. The proposed PGAO algorithm's effectiveness is validated across 23 classical, 29 IEEE CEC2017, and 10 IEEE CEC2019 benchmark functions. Additionally, six real-world engineering design problems are employed to assess the practicability of PGAO. Results demonstrate that PGAO exhibits better competitiveness and application prospects compared to the basic method and various advanced algorithms. In conclusion, this study contributes to addressing the challenges of complex optimization problems, significantly improving the performance of global optimization algorithms, and holds both theoretical and practical significance. [ABSTRACT FROM AUTHOR]

Piezoelectric ceramic, an indispensable micro-nano actuator, has advantages of small size, high sensitivity and strong controllability, which is widely applied in various fields like micro-nano processing, medicine, chemistry and materials. However, its nonlinear and asymmetric hysteresis features directly affect its positioning accuracy. The classical and generalized Prandtl-Ishlinskii models with linear symmetry operators are commonly used to deal with these issues, but they still face difficulties in asymmetric systems. Therefore, a piecewise modified Prandtl-Ishlinskii (PMPI) model is proposed to compensate nonlinear and asymmetric hysteresis in this paper. The new nonlinear operator and piecewise nonlinear envelope function are introduced into the PMPI model. Independent right and left parts are adopted to describe hysteresis increasing and decreasing stages, while the threshold and weight vectors correspond to them. Weight vectors are extended and a new diagonal matrix is constructed for avoiding excessive parameters and computation. Simulations and experiments are shown that the PMPI model has good compensation performance and effectiveness, which has better fitting, higher accuracy, fewer parameters and computation, and can be well applied to the precise nano-positioning systems with less nonlinear and asymmetric errors. [ABSTRACT FROM AUTHOR]

ABSTRACT This article proposes a novel method to accelerate the boundary feedback control design of cascaded parabolic difference equations (PDEs) through DeepONet. The backstepping method has been widely used in boundary control problems of PDE systems, but solving the backstepping kernel function can be time‐consuming. To address this, a neural operator (NO) learning scheme is leveraged for accelerating the control design of cascaded parabolic PDEs. DeepONet, a class of deep neural networks designed for approximating nonlinear operators, has shown potential for approximating PDE backstepping designs in recent studies. Specifically, we focus on approximating gain kernel PDEs for two cascaded parabolic PDEs. We utilize neural operators to map only two kernel functions, while the other two are computed using the analytical solution, thus simplifying the training process. We establish the continuity and boundedness of the kernels, and demonstrate the existence of arbitrarily close DeepONet approximations to the kernel PDEs. Furthermore, we demonstrate that the DeepONet approximation gain kernels ensure stability when replacing the exact backstepping gain kernels. Notably, DeepONet operator exhibits computation speeds two orders of magnitude faster than PDE solvers for such gain functions, and their theoretically proven stabilizing capability is validated through simulations. [ABSTRACT FROM AUTHOR]

We consider a nonlinear Dirichlet problem driven by the double phase differential operator and a reaction that has the competing effects of a parametric concave term and of a convective perturbation. Using truncation and comparison techniques and the theory of nonlinear operators of monotone type, we show that for all small values of the parameter, the problem has a bounded positive solution. [ABSTRACT FROM AUTHOR]

This work presents the mathematical/theoretical framework of the "nth-Order Feature Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems" (abbreviated as "nth-FASAM-L"), which enables the most efficient computation of exactly obtained mathematical expressions of arbitrarily-high-order (nth-order) sensitivities of a generic system response with respect to all of the parameters (including boundary and initial conditions) underlying the respective forward/adjoint systems. Responses of linear models can depend simultaneously on both the forward and the adjoint state functions. This is in contradistinction to responses for nonlinear systems, which can depend only on the forward state functions since nonlinear operators do not admit bona-fide adjoint operators. Among the best-known model responses that depend simultaneously on both the forward and adjoint state functions are Lagrangians used for system optimization, the Schwinger and Roussopoulos functionals for analyzing reaction rates and ratios thereof, and the Rayleigh quotient for analyzing eigenvalues and/or separation constants. The sensitivity analysis of such responses makes it necessary to treat linear models/systems in their own right, rather than treating them just as particular cases of nonlinear systems. The unparalleled efficiency and accuracy of the nth-FASAM-L methodology stems from the maximal reduction of the number of adjoint computations (which are "large-scale" computations) for computing high-order sensitivities, since the number of large-scale computations when applying the nth-FASAM-N methodology is proportional to the number of model features as opposed to the number of model parameters (which are considerably more than the number of features). The mathematical framework underlying the nth-FASAM-L is developed in linearly increasing higher-dimensional Hilbert spaces, as opposed to the exponentially increasing "parameter-dimensional" spaces in which response sensitivities are computed by other methods (statistical, finite differences, etc.), thus providing the basis for overcoming the curse of dimensionality in sensitivity analysis and all other fields (uncertainty quantification, predictive modeling, etc.) which need such sensitivities. [ABSTRACT FROM AUTHOR]

We show that there exists only one solution (the trivial identically zero solution) for some nonlinear elliptic Dirichlet problems, involving the p-Laplacian operator and nonlinear terms with supercritical growth, in bounded contractible ramified domains, that is domains of ℝn with n ≥ 2, close to a prescribed subset of ℝn, which is contractible in itself and consists of a finite number of smooth curves. In dimension n = 2 we expect that this result might be extended to cover all the bounded contractible domains of ℝ2. On the contrary, this extension is not possible in dimension n ≥ 3 because of some counterexamples concerning existence and multiplicity of nontrivial solutions in some contractible domains that may be even arbitrarily close to starshaped domains (where the well-known Pohozaev nonexistence result holds). However, also for n ≥ 3 our result allows us to prove nonexistence of nontrivial solutions in bounded contractible domains that may be very different from the starshaped ones and even arbitrarily close to some noncontractible domains where there exist many positive and nodal solutions. [ABSTRACT FROM AUTHOR]

In this paper, we examine the existence and analyze properties of inertial manifolds of a nonlinear semigroup of operators in a Hilbert space. This questions were studied in a general setting that allows generalizing results of the well-known works of K. Foias, J. Sell, and R. Temam. Our reasoning is based on the scheme of proofs of similar assertions proposed earlier by S. Sternberg and F. Hartman for ordinary autonomous differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a method for efficiently obtaining an approximate solution for constrained nonlinear monotone operator equations. The search direction of the proposed method closely aligns with the Broyden-Fletcher-Goldfarb-Shanno (BFGS) direction, known for its low storage requirement. Notably, the search direction is shown to be sufficiently descent and bounded without using the line search condition. Furthermore, under some standard assumptions, the proposed method converges globally. As an application, the proposed method is applied to solve image restoration problems. The efficiency and robustness of the method in comparison to other methods are tested by numerical experiments using some test problems. [ABSTRACT FROM AUTHOR]

The purpose of the present work is investigating a degenerate parabolic equation with anisotropic p-Laplacian operator and strongly nonlinear source under boundary conditions of Dirichlet type; the existence of a periodic non-negative solution is shown. The proof is based on the Leray–Schauder topological degree, which is tricky to work with in this kind of equations. [ABSTRACT FROM AUTHOR]

In 2021, Z. Fu, Y. Chen and B. Han introduced an inexact Newton regularization (REGINN-IT) using an idea involving the non-stationary iterated Tikhonov regularization scheme for solving nonlinear ill-posed operator equations. In this paper, we suggest a simplified version of the REGINN-IT scheme by using the Bregman distance, duality mapping and a suitable parameter choice strategy to produce an approximate solution. The method is comprised of inner and outer iteration steps. The outer iterates are stopped by a Morozov-type stopping rule, while the inner iterate is executed by making use of the non-stationary iterated Tikhonov scheme. We have studied convergence of the proposed method under some standard assumptions and utilizing tools from convex analysis. The novelty of the method is that it requires computation of the Fréchet derivative only at an initial guess of an exact solution and hence can be identified as more efficient compared to the method given by Z. Fu, Y. Chen and B. Han. Further, in the last section of the paper, we discuss test examples to inspect the proficiency of the method. [ABSTRACT FROM AUTHOR]

Many physical systems exhibit translational invariance, meaning that the underlying physical laws are independent of the position in space. Data driven approximations of the infinite dimensional but linear Koopman operator of non-linear dynamical systems need to be physically informed in order to respect such physical symmetries. In the current work, we introduce a translation invariant extended dynamic mode decomposition (tieDMD) for coupled non-linear systems on periodic domains. This is achieved by exploiting a block-diagonal structure of the Koopman operator in Fourier space. Variants of tieDMD are applied to data obtained on one-dimensional periodic domains from the non-linear phase-diffusion equation, the Burgers equation, the Korteweg–de Vries equation, and a coupled FitzHugh–Nagumo system of partial differential equations. The reconstruction capability of tieDMD is compared to existing linear and non-linear variants of the dynamic mode decomposition applied to the same data. For the regarded data, tieDMD consistently shows superior capabilities in data reconstruction. [ABSTRACT FROM AUTHOR]

Reconstructing a nonlinear dynamical system from empirical time series is a fundamental task in data-driven analysis. One of the main challenges is the existence of hidden variables; we only have records for some variables, and those for hidden variables are unavailable. In this work, the techniques for Carleman linearization, phase-space embedding, and dynamic mode decomposition are integrated to rebuild an optimal dynamical system from time series for one specific variable. Using the Takens theorem, the embedding dimension is determined, which is adopted as the dynamical system's dimension. The Carleman linearization is then used to transform this finite nonlinear system into an infinite linear system, which is further truncated into a finite linear system using the dynamic mode decomposition technique. We illustrate the performance of this integrated technique using data generated by the well-known Lorenz model, the Duffing oscillator, and empirical records of electrocardiogram, electroencephalogram, and measles outbreaks. The results show that this solution accurately estimates the operators of the nonlinear dynamical systems. This work provides a new data-driven method to estimate the Carleman operator of nonlinear dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper, a novel parabolic system involving nonlinear and nonhomogeneous partial differential operator with variable growth structure is introduced for investigating the image denoising and restoration. More precisely, our model is based on regularizing the classical models involving variable exponent operators by considering a nonlinear operator with double phase flux. We begin by investigating theoretically the solvability to the parabolic system under consideration. Under the setting of Musielak-Orlicz spaces, we build a suitable functional framework to study the proposed system. Therefore, we develop the Faedo-Galerkin approach to demonstrate the existence and uniqueness of a weak solution to our model. To illustrate our theoretical results in the context of image noise removal, we present various numerical implementations on some grayscale images. To enrich these simulations, we test the robustness efficiency of the proposed model in the so-called Magnetic Resonance Images (MRI). The obtained numerical results claim that our model is more efficient and robust against noise, in comparison (visually and quantitatively) to some existing state-of-the-art methods. • A novel nonlinear parabolic system is introduced. • The theoretical results are applied to the image denoising and restoration. • Various numerical implementations on some grayscale images are carried out. [ABSTRACT FROM AUTHOR]

Tautenhahn (2002) studied the Lavrentiev regularization method (LRM) to approximate a stable solution for the ill-posed nonlinear equation κ (u) = v , where κ : D (κ) ⊆ X ⟶ X is a nonlinear monotone operator and X is a Hilbert space. The operator in the example used in Tautenhahn's paper was not a monotone operator. So, the following question arises. Can we use LRM for ill-posed nonlinear equations when the involved operator is not monotone? This paper provides a sufficient condition to employ the Lavrentiev regularization technique to such equations whenever the operator involved is non-monotone. Under certain assumptions, the error analysis and adaptive parameter choice strategy for the method are discussed. Moreover, the developed theory is applied to two well-known ill-posed problems—inverse gravimetry and growth law problems. [ABSTRACT FROM AUTHOR]

In this paper we state some sharp maximum principle, i.e. we characterize the geometry of the sets of minima for supersolutions of equations involving the $k$ -th fractional truncated Laplacian or the $k$ -th fractional eigenvalue which are fully nonlinear integral operators whose nonlocality is somehow $k$ -dimensional. [ABSTRACT FROM AUTHOR]

Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most notable is the Moore–Penrose inverse, widely used in physics, statistics, and various fields of engineering. This work investigates generalized inversion of nonlinear operators. We first address broadly the desired properties of generalized inverses, guided by the Moore–Penrose axioms. We define the notion for general sets and then a refinement, termed pseudo-inverse, for normed spaces. We present conditions for existence and uniqueness of a pseudo-inverse and establish theoretical results investigating its properties, such as continuity, its value for operator compositions and projection operators, and others. Analytic expressions are given for the pseudo-inverse of some well-known, non-invertible, nonlinear operators, such as hard- or soft-thresholding and ReLU. We analyze a neural layer and discuss relations to wavelet thresholding. Next, the Drazin inverse, and a relaxation, are investigated for operators with equal domain and range. We present scenarios where inversion is expressible as a linear combination of forward applications of the operator. Such scenarios arise for classes of nonlinear operators with vanishing polynomials, similar to the minimal or characteristic polynomials for matrices. Inversion using forward applications may facilitate the development of new efficient algorithms for approximating generalized inversion of complex nonlinear operators. [ABSTRACT FROM AUTHOR]