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Consider a sequence of real data points X₁, . . . , Xn with underlying means θ*₁, . . . ,θ*n . This paper starts from studying the setting that θ*i is both piecewise constant and monotone as a function of the index i. For this, we establish the exact minimax rate of estimating such monotone functions, and thus give a nontrivial answer to an open problem in the shape-constrained analysis literature. The minimax rate under the loss of the sum of squared errors involves an interesting iterated logarithmic dependence on the dimension, a phenomenon that is revealed through characterizing the interplay between the isotonic shape constraint and model selection complexity. We then develop a penalized least-squares procedure for estimating the vector θ*= (θ*₁, . . . ,θ*n )T. This estimator is shown to achieve the derived minimax rate adaptively. For the proposed estimator, we further allow the model to be misspecified and derive oracle inequalities with the optimal rates, and show there exists a computationally efficient algorithm to compute the exact solution.

In this paper, the existence, uniqueness and stability of solutions to a boundary value problem of nonlinear FDEs of variable order are established. To do this, we first investigate some aspects of variable order operators of Hadamard type. Then, with the help of the generalized intervals and piecewise constant functions, we convert the variable order Hadamard FBVP to an equivalent standard Hadamard BVP of the fractional constant order. Further, two fixed point theorems due to Schauder and Banach are used and, finally, the Ulam–Hyers–Rassias stability of the given variable order Hadamard FBVP is examined. These results are supported with the aid of a comprehensive example.

In this paper, the existence of the solution and its stability to the fractional boundary value problem (FBVP) were investigated for an implicit nonlinear fractional differential equation (VOFDE) of variable order. All existence criteria of the solutions in our establishments were derived via Krasnoselskii’s fixed point theorem and in the sequel, and its Ulam–Hyers–Rassias (U-H-R) stability is checked. An illustrative example is presented at the end of this paper to validate our findings.

The semigroup properties of the Riemann–Liouville fractional integral have played a key role in dealing with the existence of solutions to differential equations of fractional order. Based on some results of some experts’, we know that the Riemann–Liouville variable order fractional integral does not have semigroup property, thus the transform between the variable order fractional integral and derivative is not clear. These judgments bring us extreme difficulties in considering the existence of solutions of variable order fractional differential equations. In this work, we will introduce the concept of approximate solution to an initial value problem for differential equations of variable order involving the derivative argument on half-axis. Then, by our discussion and analysis, we investigate the unique existence of approximate solution to this initial value problem for differential equation of variable order involving the derivative argument on half-axis. Finally, we give examples to illustrate our results.

We provide here an optimal method of approximating a signal by piecewise constant functions. To this end, we minimize over the signal subdomains a fidelity term between the signal and its corresponding piecewise approximations; subdomains being determined by the number of approximations samples used for. An optimal recursive relationship is then obtained and proven, which helps us to derive the proposed approximation algorithm. The complexity of the algorithm is O(MN2), where N is the number of samples of the processed signal and M is the number of piecewise constant approximation functions. There are different techniques to approximate a signal using piecewise constant functions, wavelet decomposition is one of them by means of a Haar wavelet. Our approach is then compared to linear and nonlinear wavelet-based approximations, and both qualitative and quantitative results are provided on various tested signals, showing the efficiency of the proposed approach.

In this article, we study the existence and uniqueness of solutions for a two- point boundary value problem of Caputo fractional di erential equation of variable order. The results are obtained by means of Banach's and Krasnoselskii's fixed point theorems. In addition, the obtained results are illustrated with the aid of a numerical example.

Using variable-order fractional derivatives in differential equations is essential. It enables more precise modeling of complex phenomena with varying memory and long-range dependencies, improving our ability to describe real-world processes reliably. This study investigates the properties of solutions for a two-point boundary value problem associated with φ-Caputo fractional derivatives of variable order. The primary objectives are to establish the existence and uniqueness of solutions, as well as explore their stability through the Ulam-Hyers concept. To achieve these goals, Banach’s and Krasnoselskii’s fixed point theorems are employed as powerful mathematical tools. Additionally, we provide numerical examples to illustrate results and enhance comprehension of theoretical findings. This comprehensive analysis significantly advances our understanding of variable-order fractional differential equations, providing a strong foundation for future research. Future directions include exploring more complex boundary value problems, studying the effects of varying fractional differentiation orders, extending the analysis to systems of equations, and applying these findings to real-world scenarios, all of which promise to deepen our understanding of Caputo fractional differential equations with variable order, driving progress in both theoretical and applied mathematics.

In this paper, we establish the existence of solutions to a nonlinear boundary value problem (BVP) of variable order at resonance. The main theorem in this study is proved with the help of generalized intervals and piecewise constant functions, in which we convert the mentioned Caputo BVP of fractional variable order to an equivalent standard Caputo BVP at resonance of constant order. In fact, to use the Mawhin’s continuation technique, we have to transform the variable order BVP into a constant order BVP. We prove the existence of solutions based on the existing notions in the coincidence degree theory and Mawhin’s continuation theorem (MCTH). Finally, an example is provided according to the given variable order BVP to show the correctness of results.

Partitioning a mesh surface into several semantic components is a fundamental task in geometry processing. This paper presents a new stable and effective segmentation method, which contains two stages. The first stage is a spectral clustering procedure, while the second stage is a variational refining procedure. For spectral clustering, we construct a new Laplacian matrix which reflects more semantic information than classical Laplacian matrices. By this new Laplacian, we introduce a simple and fast spectral clustering method, which gives quite satisfying segmentation results for most surfaces and provides a good initialization for the second stage. In the second stage, we propose a variational refining procedure by a new discretization of the classical non-convex Mumford–Shah model. The variational problem is solved by efficient iterative algorithms based on alternating minimization and alternating direction method of multipliers (ADMM). The first stage provides a good initialization for the second stage, while the second stage refines the result of the first stage well. Experiments demonstrated that our method is very stable and effective compared to existing approaches. It outperforms competitive segmentation methods when evaluated on the Princeton Segmentation Benchmark.

IL-6 signaling plays an important role in inflammatory processes in the body. While a number of models for IL-6 signaling are available, the parameters associated with these models vary from case to case as they are non-trivial to determine. In this study, optimal experimental design is utilized to reduce the parameter uncertainty of an IL-6 signaling model consisting of ordinary differential equations, thereby increasing the accuracy of the estimated parameter values and, potentially, the model itself. The D-optimality criterion, operating on the Fisher information matrix and, separately, on a sensitivity matrix computed from the Morris method, was used as the objective function for the optimal experimental design problem. Optimal input functions for model parameter estimation were identified by solving the optimal experimental design problem, and the resulting input functions were shown to significantly decrease parameter uncertainty in simulated experiments. Interestingly, the determined optimal input functions took on the shape of PRBS signals even though there were no restrictions on their nature. Future work should corroborate these findings by applying the determined optimal experimental design on a real experiment.