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This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, Zhen-Qing Chen and Masatoshi Fukushima cover the essential elements and applications of the theory of symmetric Markov processes, including recurrence/transience criteria, probabilistic potential theory, additive functional theory, and time change theory. The authors develop the theory in a general framework of symmetric quasi-regular Dirichlet forms in a unified manner with that of regular Dirichlet forms, emphasizing the role of extended Dirichlet spaces and the rich interplay between the probabilistic and analytic aspects of the theory. Chen and Fukushima then address the latest advances in the theory, presented here for the first time in any book. Topics include the characterization of time-changed Markov processes in terms of Douglas integrals and a systematic account of reflected Dirichlet spaces, and the important roles such advances play in the boundary theory of symmetric Markov processes. This volume is an ideal resource for researchers and practitioners, and can also serve as a textbook for advanced graduate students. It includes examples, appendixes, and exercises with solutions.

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.

This paper proposes a new method for section an interval in a given ratio intended for minimizing unimodal functions. The ratio section search is capable of quickly recognizing monotone functions and functions with a flat bottom, which contributes to increasing its performance, as measured by the number of minimized function evaluations. The method is implemented as passive and active algorithms. A comparison of the performance of the developed method with that of the classical methods of bisection search and the golden section search was performed on the basis of the data used to minimize twenty unimodal functions of various types. For all types of functions, the passive algorithm is 2.26 times faster than the bisection search and 1.72 times faster than the golden section method. Thus, the proposed method turned out to be the fastest of the known methods of cutting off segments intended for minimizing unimodal functions. The active algorithm is faster: for all types of functions, these indicators are 3.31 and 2.52, respectively. The fastest combined Brent’s method was also modernized. After the golden section procedure is replaced with a procedure for dividing a segment in a given ratio, a numerical experiment is conducted. The modernized method is 1.69 times faster than its prototype. Moreover, the performance of the active algorithm for dividing a segment at a given ratio exceeds that of the Brent’s method by 1.48 times for all types of functions. The modernized Brent’s method is approximately 4 times faster than the bisection search and 3 times faster than the golden section method.

Chlorinated solvents, common groundwater contaminants, can cause coexistence of the original contaminant and its degradation products during the transport process. Practically applicable analytical models for reactive transport are essential for simulating the plume migration of chlorinated solvent contaminants and their degradation products within a complex chemical mixture. Although several analytical models have been developed to solve advection–dispersion equations coupled with a series of decay reactions for simulating transport of the coexisting chlorinated solvent contaminants, the majority assume static, time-invariant inlet boundary conditions. Such time-invariant inlet boundary conditions may fail to adequately represent the temporal evolution of dissolved source discharge concentration concerning mass reduction, especially in the context of diverse DNAPL source remediation strategies. This study seeks to derive analytical models for three-dimensional reactive transport of multiple contaminants, specifically addressing the challenges posed by dynamical, time-varying inlet boundary conditions. The model development incorporates two distinct inlet functions: exponentially decaying and piecewise constant. Analytical solutions are obtained using three integral transform techniques. The accuracy of the newly developed analytical models is verified by comparing them with solutions derived from existing literature using multiple illustrative examples. By incorporating two distinct time-varying inlet boundary conditions, the models exhibit strong capabilities in capturing the complex transport dynamics and fate of contaminants within groundwater systems. These features make the models valuable tools for improving the understanding of subsurface contaminant behavior and for quantitatively evaluating and optimizing a range of remediation strategies.

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, we discuss the existence of solutions to a boundary value problem of differential equations of variable order, which is a piecewise constant function. Our results are based on the Schauder fixed point theorem. Then, under some assumptions on the nonlinear term, we obtain a generalized Lyapunov-type inequality to the two-point boundary value problem considered. To the best of our knowledge, there is no paper dealing with Lyapunov-type inequalities for boundary value problems in term of variable order. In addition, some examples of the obtained inequalities are given.

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.

In this paper, we provide an estimate for approximating the generalized-Euler-constant function γ(z)=∑k=1∞zk−1(1k−lnk+1k)\\(\\gamma (z)=\\sum_{k=1}^{\\infty }z ^{k-1} (\\frac{1}{k}-\\ln \\frac{k+1}{k} )\\) by its partial sum γN−1(z)\\(\\gamma _{N-1}(z)\\) when 0

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