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In this paper we consider the fractional parts of a general sequence, for example the sequence $\alpha \sqrt{n}$ or $\alpha n^2$. We give a general method, which allows one to show that long-range correlations (correlations where the support of the test function grows as we consider more points) are Poissonian. We show that these statements about convergence can be reduced to bounds on associated Weyl sums. In particular we apply this methodology to the aforementioned examples. In so doing, we recover a recent result of Technau-Walker (2020) for the triple correlation of $\alpha n^2$ and generalize the result to higher moments. For both of the aforementioned sequences this is one of the only results which indicates the pseudo-random nature of the higher level ($m \ge 3$) correlations., Comment: 132 pages

The paper introduces a new weak approximation scheme for Kolmogorov type hypoelliptic diffusions based on an operator splitting method and a Malliavin calculus approach. The approximation works even if the test function is not smooth, in other words, irregular functionals of hypoelliptic diffusions appearing in various fields are effectively approximated. A simple numerical algorithm is provided, which is easily implemented by a simulation method with minimum cost on random number generation. The effectiveness of the scheme is confirmed through numerical examples for irregular functionals of hypoelliptic diffusions.

Transcranial magnetic stimulation (TMS) is a noninvasive technology used to treat certain brain disorders. As an important part of the TMS system, the stimulation coil induces the electric field (EF) in the brain, which can change the excitability of nerve tissue. First, a three-layer multicoil (TLMC) is proposed on the basis of the principle of superposition and cancellation of magnetic field (MF). The simulation results in COMSOL show that this structure can significantly improve focality and stimulation strength, but it reduces the depth of stimulation. Second, a novel improved particle swarm optimizer (PSO) is proposed with three improved strategies. The novel improved PSO has higher convergence accuracy and stability in the test function. Finally, we utilized the novel improved PSO to optimize the current configuration and rotation angle of the TLMC three times. The numerical results show that the TLMC after the second optimization can increase stimulation strength by 19.10% on the basis of satisfying the stimulation depth of 15 mm. Compared with the unoptimized, the optimized TLMC can better balance the focality and stimulation depth.

Under investigation in this letter is an (3+1)-dimensional Hirota–Satsuma–Ito-like equation, which provide strong support for studying the dynamic behavior of nonlinear waves. Based on a special Cole–Hopf transformation and Hirota bilinear method, the bilinear form of the equation is obtained and this form has never been given. High-order breather solutions, lump solutions and mixed solutions are obtained by using complex conjugate parameters and long-wave limit method. Then, the influence of the coefficient $$g_{t}(t)$$ of the bilinear equation on the interaction of these solutions is analyzed by means of images. It can be found that $$g_{t}(t)$$ changes the interaction of the solutions by influencing the positions and trajectories of higher-order breather solutions, lump solutions and mixed solutions. We find that different values of g(t) make the interaction of solutions different. Finally, the mixed solution of the equation including a breather wave and a line rogue wave is obtained by using the test function, and its dynamic properties are illustrated by means of images.

In this paper, we study the Cauchy problems for quasilinear hyperbolic inequalities with nonlocal singular source term and prove the nonexistence of global weak solutions in the homogeneous and nonhomogeneous cases by the test function method.

In this paper, we study the Cauchy problem for the linear and semilinear Moore-Gibson-Thompson (MGT) equation in the dissipative case. Concerning the linear MGT model, by utilizing WKB analysis associated with Fourier analysis, we derive some L 2 estimates of solutions, which improve those in the previous research [51] . Furthermore, asymptotic profiles of the solution and an approximate relation in a framework of the weighted L 1 space are derived. Next, with the aid of the classical energy method and Hardy's inequality, we get singular limit results for an energy and the solution itself. Concerning the semilinear MGT model, basing on the obtained sharp L 2 estimates and constructing time-weighted Sobolev spaces, we investigate global (in time) existence of Sobolev solutions with different regularities. Finally, under a sign assumption on initial data, nonexistence of global (in time) weak solutions is proved by applying a test function method.

The paper introduces a new weak approximation algorithm for stochastic differential equations (SDEs) of McKean–Vlasov type. The arbitrary order discretization scheme is available and is given using Malliavin weights, certain polynomial weights of Brownian motion, which play a role as correction of the approximation. The new weak approximation scheme works even if the test function is not smooth. In other words, the expectation of irregular functionals of McKean–Vlasov SDEs such as probability distribution functions are approximated through the proposed scheme. The effectiveness of the higher order scheme is confirmed by numerical examples for McKean–Vlasov SDEs including the Kuramoto model.

In this paper, we present an empirical comparison of some Gradient Descent variants used to solve globaloptimization problems for large search domains. The aim is to identify which one of them is more suitable for solving an optimization problem regardless of the features of the used test function. Five variants of Gradient Descent were implemented in the R language and tested on a benchmark of five test functions. We proved the dependence between the choice of the variant and the obtained performances using the khi-2 test in a sample of 120 experiments. Those test functions vary on convexity, the number of local minima, and are classified according to some criteria. We had chosen a range of values for each algorithm parameter. Results are compared in terms of accuracy and convergence speed. Based on the obtained results,we defined the priority of usage for those variants and we contributed by a new hybrid optimizer. The new optimizer is testedin a benchmark of well-known test functions and two real applications are proposed. Except for the classical gradient descent algorithm, only stochastic versions of those variants are considered in this paper.

In this two-part paper, a stable and efficient nodally-integrated reproducing kernel particle method (RKPM) is introduced for solving the governing equations of generalized thermomechanical theories. Part I investigates quadrature in the weak form using coupled and uncoupled classical thermoelasticity as model problems. It is first shown that nodal integration of these equations results in spurious oscillations in the solution many orders of magnitude greater than pure elasticity. A naturally stabilized nodal integration is then proposed for the coupled equations. The variational consistency conditions for nth order exactness and convergence in the two-field problem are then derived, and a uniform correction on the test function approximations is proposed to achieve these conditions. Several benchmark problems are solved to demonstrate the effectiveness of the proposed method. In the sequel, these methods are developed for generalized thermoelasticity and generalized finite-strain thermoplasticity theories of the hyperbolic type that are amenable to efficient explicit time integration.

Structural lightweight optimization is one of the key themes to continuously improve vehicle performance. However, most of the studies so far have not considered reliability and manufacturing issues brought about by the uncertainty of tolerances in structural lightweight optimization. In this study, a reliability-based design for lightweight vehicle structures with uncertain manufacturing accuracy is proposed. This method can provide not only the optimal design variables but also the maximum allowable deviation range. The main design variables lacking sample information are regarded as interval variables and other design parameters as probabilistic values. A progressive reliability-based design with uncertain manufacturing accuracy is established for lightweight vehicle structures by employing the transformation method of the interval model and the decoupling strategy of the probabilistic model. Firstly, the general test function is used to analyze the accuracy and efficiency of the reliability decoupling method. Then, lightweight optimization models of a classic cantilever beam and two vehicle structures (parking robot frame, energy absorption box) are established respectively, and the effectiveness of the present optimization method is verified. The optimized design results show that the optimal variables and the corresponding maximum allowable deviation range under acceptable manufacturing requirements can be obtained. The requirements for manufacturing accuracy can be reduced by appropriately increasing the deviation range of the design variables, which may lead to a reduction in manufacturing cost. This study aims to provide feasible design options by optimizing lightweight structures considering uncertain manufacturing accuracy.

In this paper, we investigate the nonexistence of nontrivial global solutions for a fractional integro-differential problem in the space of absolutely continuous functions. We provide criteria under which no nontrivial global solutions exist. It is shown that a dissipation of order between zero and one or even a (frictional) dissipation of order one does not help providing global nontrivial solutions. The test function method is used with several derived estimations. Examples with numerical computations are given to illustrate the results.

To overcome the limitations that active disturbance rejection control (ADRC) system of a bearingless induction motor (BIM) has difficulty in tuning parameters depending on experience to select parameters, an ADRC strategy based on improved particle swarm optimization–genetic algorithm (IPSO-GA) is proposed. Based on the orientation of the air-gap magnetic field, the first-order and the second-order ADRC are, respectively, designed for the BIM rotation and suspension parts according to the different order of the BIM system. Then, the parameters of the basic particle swarm optimization algorithm are optimized by considering the characteristics of the basic particle swarm algorithm parameters, and the crossover and mutation operations are introduced to enhance global search capability. Meanwhile, through the performance test based on the test function, the performance of IPSO-GA is verified, and the parameters of ADRC are adjusted by IPSO-GA. In addition, this strategy is analyzed with simulation in MATLAB/Simulink and verified on an experimental prototype. Both simulation and experimental results show that the proposed strategy not only effectively improves the starting performance and antidisturbance ability of the BIM but also reduces the maximum radial offset of the rotor and improves the suspension precision of the motor.

This paper introduces a generalization sequence of positive and linear operators of integral type based on two parameters to improve the order of approximation. First, the simultaneous approximation is studied and a Voronovskaja-type asymptotic formula is introduced. Next, an error of the estimation in the simultaneous approximation is found. Finally, a numerical example to approximate a test function and its first derivative of this function is given for some values of the parameters.

In this paper, we deal with Caputo-type fractional differential inequality where there is a low-order fractional derivative with the term polynomial source. We investigate the nonexistence of nontrivial global solutions in a suitable space via the test function technique and some properties of fractional integrals. Finally, we demonstrate three examples to illustrate our results. The presented results are more general than those in the literature, which can be obtained as particular cases.

Thermal analysis of natural convection in a cavity is solved by numerical approach. The direct meshless local Petrov–Galerkin method combined with an implicit artificial-compressibility model is proposed here to simulate laminar natural convective heat transfer in triangular cavities. The spatial terms of the governing equation are discretized using the direct meshless local Petrov–Galerkin method with the Dirac function as the test function. The method uses an artificial-compressibility scheme to couple the pressure with the velocity components directly. The semi-algebraic system is solved using an implicit backward Euler pseudo-time method. Kovasznay flow problem is presented to examine the accuracy of proposed method. The validation of the obtained results by using experimental data is also provided, and four test problems — natural convection in an equilateral triangular cavity, a right-angled triangular cavity, an isosceles triangular cavity, and an irregular triangular cavity — are solved by using the proposed method. The numerical results obtained using the proposed method showed excellent agreement with those obtained using the conventional methods available in the literature.

The Pigeon-Inspired Optimization (PIO) algorithm is an intelligent algorithm inspired by the behavior of pigeons returned to the nest. The binary pigeon-inspired optimization (BPIO) algorithm is a binary version of the PIO algorithm, it can be used to optimize binary application problems. The transfer function plays a very important part in the BPIO algorithm. To improve the solution quality of the BPIO algorithm, this paper proposes four new transfer function, an improved speed update scheme, and a second-stage position update method. The original BPIO algorithm is easier to fall into the local optimal, so a new speed update equation is proposed. In the simulation experiment, the improved BPIO is compared with binary particle swarm optimization (BPSO) and binary grey wolf optimizer (BGWO). In addition, the benchmark test function, statistical analysis, Friedman’s test and Wilcoxon rank-sum test are used to prove that the improved algorithm is quite effective, and it also verifies how to set the speed of dynamic movement. Finally, feature selection was successfully implemented in the UCI data set, and higher classification results were obtained with fewer feature numbers.

Sensitivity analysis helps identify which model inputs convey the most uncertainty to the model output. One of the most authoritative measures in global sensitivity analysis is the Sobol' total-order index, which can be computed with several different estimators. Although previous comparisons exist, it is hard to know which estimator performs best since the results are contingent on the benchmark setting defined by the analyst (the sampling method, the distribution of the model inputs, the number of model runs, the test function or model and its dimensionality, the weight of higher order effects or the performance measure selected). Here we compare several total-order estimators in an eight-dimension hypercube where these benchmark parameters are treated as random parameters. This arrangement significantly relaxes the dependency of the results on the benchmark design. We observe that the most accurate estimators are Razavi and Gupta's, Jansen's or Janon/Monod's for factor prioritization, and Jansen's, Janon/Monod's or Azzini and Rosati's for approaching the "true" total-order indices. The rest lag considerably behind. Our work helps analysts navigate the myriad of total-order formulae by reducing the uncertainty in the selection of the most appropriate estimator., Previous versions of this manuscript were titled "The Battle of Total-Order Sensitivity Indices". The version with the title "A comprehensive comparison of total-order estimators for global sensitivity analysis" has been accepted for publication in the International Journal for Uncertainty Quantification. Please kindly cite the journal version

The Nelder-Mead (NM) method is a popular derivative-free optimization algorithm owing to its fast convergence and robustness. However, it is known that the method often fails to converge or costs a long time for a large-scale optimization. In the present study, the NM method has been improved using direct inversion in iterative subspace (DIIS). DIIS is a technique to accelerate an optimization method, extrapolating a better intermediate solution from linear-combination of the known ones. We compared runtimes of the new method (NM-DIIS) and the conventional NM method using unimodal test functions with various dimensions. The NM-DIIS method showed better results than the original NM on average when the dimension of the objective function is high. Long tails of the runtime distributions in the NM method have disappeared when DIIS was applied. DIIS has also been implemented in the quasi-gradient method, which is an improved version of the NM method developed by Pham et al. [IEEE Trans. Ind. Informatics, 7 (2011) 592]. The combined method also performed well especially in an upwardly convex test function. The present study proposes a practical optimization strategy and proves the versatility of DIIS.

We prove nonexistence theorems of nonnegative nontrivial solutions for a system of ordinary differential inequalities in bounded domains with singular points on the boundary. The proofs are based on the test function using a method developed by Mitidieri and Pohozaev. We also give the examples demonstrating that the conditions obtained are sharp in the case of the problem under consideration.

Multi-dimensional advection diffusion equations involving fractional diffusion flux are studied with finite element formulation. It has been demonstrated that the Galerkin finite element formulation may suffer spatial instability and nodal oscillations due to the fractional diffusion flux in addition to the advection term. To resolve this issue, a fractional streamline upwind Petrov-Galerkin finite element formulation is developed. In this formulation, an artificial viscosity is added to the test function to eliminate the spatial oscillations. By taking into account both the fractional diffusion flux and the advection term, a decomposition algorithm is proposed to formulate the artificial viscosity to avoid the cross-wind diffusion effect for multi-dimensional problems. Numerical examples with regular/irregular domains discretized by uniform/nonuniform meshes for both steady state and time-dependent fractional advection diffusion equations are presented to thoroughly demonstrate the effectiveness of the proposed methodology.

The computing method of the average optimal position is one of the most important factors that affect the optimization performance of the QPSO algorithm. Therefore, a particle position weight computing method based on particle fitness value grading is proposed, which is called HWQPSO (hierarchical weight QPSO). In this method, the higher the fitness value of a particle, the higher the level of the particle, and the greater the weight. Particles at different levels have different weights, while particles at the same level have the same weight. Through this method, the excellent particles have higher average optimal position weight, and at the same time, the absolute weight of a few particles is avoided, so that the algorithm can quickly and stably converge to the optimal solution, and improve the optimization ability and efficiency of the algorithm. In order to verify the effectiveness of the method, five standard test functions are selected to test the performance of HWQPSO, QPSO, DWC-QPSO and LTQPSO algorithm, and the algorithms are applied to the task scheduling of the cloud computing platform. Through the test experiment and application comparison, the results show that the HWQPSO algorithm can converge to the optimal solution of the test function faster than the other three algorithms. It can also find the task scheduling scheme with the shortest time consumption and the most balanced computing resource load in the cloud platform. In the experiment, compared with QPSO, DWC-QPSO and LTQPSO algorithm, HWQPSO execution time of the maximum task scheduling was reduced by 35%, 23% and 21% respectively.

Aiming at the three-dimensional path planning of unmanned aerial vehicle (UAV) in the complex environment of material delivery in earthquake-stricken areas, this paper proposes an improved adaptive grey wolf optimization algorithm (AGWO) based on the grey wolf optimization algorithm (GWO). There are two main contributions of the proposed method. Firstly, we propose an adaptive convergence factor adjustment strategy and an adaptive weight factor to update the individual’s position. The effectiveness of the improved algorithm is verified by the convergence analysis and the test function simulation experiment. Secondly, the improved algorithm is applied to UAV path planning, the environmental map model is established by integrating digital elevation map and equivalent mountain threat model, and the performance evaluation function is established by fitting the calculated track length. The simulation results show that the improved AGWO is superior to the traditional intelligent algorithm in convergence precision, speed and stability performance, and it is effective for 3D trajectory optimization in complex environment.

It is well known that most classical test functions to solve nonlinear partial differential equations can be constructed via single hidden layer neural network model by using Bilinear Neural Network Method (BNNM). In this paper, the neural network model of test function for the (3+1)-dimensional Jimbo–Miwa equation is extended to the “4-2-3” model. By giving some specific activation functions, new test function is constructed to obtain analytical solutions of the (3+1)-dimensional Jimbo–Miwa equation. Rogue wave solutions and the bright and dark solitons are obtained by giving some specific parameters. Via curve plots, three-dimensional plots, contour plots and density plots, dynamical characteristics of these waves are exhibited.

The structure of information technology optimization of complex dynamic systems is developed. The structure includes a block of systems models, a module of integration methods, a block for calculating system quality criteria, a block of optimization methods, a module of data structures and a block for presenting information. A functional model of information technology is presented. The block of systems models contains models of dynamic systems – control objects, regulators, information control systems. This makes it possible to simulate information control systems of NPP power units, a quadrocopter control system, generators of electrical impulses and other dynamic systems. In the module of integration methods, methods for integrating systems of differential equations are collected: the matrix exponential method for integrating linear systems, system methods for integrating nonlinear systems, and other methods. In the block for calculating system quality criteria, programs have been created for calculating various criteria for information control systems: stability of control processes, identification of parameters, direct indexes of the quality of systems, integral quadratic estimates. The block of optimization methods has been developed. This block contains modules of methods of one-dimensional search, multivariate unconditional optimization, global search, genetic algorithms, sum of squares minimization, conditional optimization, vector optimization. To test methods, test function modules are created. These modules include test functions, their databases and testing programs. A module of common data structures has been developed. Global data structures are proposed – the structure of constant data of optimization problems and methods, the structure of the state variables of the optimization process and the structure of functions to coordinate the interaction of optimization methods. The block for presenting information for solving problems allows the optimization process to be presented in the form of tables and graphs.

The performance of differential evolution (DE) algorithm highly depends on the selection of mutation strategy. However, there are six commonly used mutation strategies in DE. Therefore, it is a challenging task to choose an appropriate mutation strategy for a specific optimization problem. For a better tackle this problem, in this paper, a novel DE algorithm based on local fitness landscape called LFLDE is proposed, in which the local fitness landscape information of the problem is investigated to guide the selection of the mutation strategy for each given problem at each generation. In addition, a novel control parameter adaptive mechanism is used to improve the proposed algorithm. In the experiments, a total of 29 test functions originated from CEC2017 single-objective test function suite which are utilized to evaluate the performance of the proposed algorithm. The Wilcoxon rank-sum test and Friedman rank test results reveal that the performance of the proposed algorithm is better than the other five representative DE algorithms.

We study the nonexistence of global solutions for new classes of nonlinear fractional differential inequalities. Namely, sufficient conditions are provided so that the considered problems admit no global solutions. The proofs of our results are based on the test function method and some integral estimates.

Taking the real-world Internet platform “Tao Factory” of Alibaba as an example, the order matching mechanism between manufacturers and retailers is studied. This platform first collects multi-variety small batch orders from retailers and modular production resources from manufacturers, and then matches production orders according to the needs of both parties. Considering the characteristics of the matching process, we take the qualifications of all manufacturers matched to the orders and total transportation costs as optimization goals and establish an order matching model. In order to expand the search space algorithm, we introduce the niche strategy, which can remove crowded particles and make the particle swarm evenly distributed. In addition, the local search method is combined to improve the particle swarm algorithm, and the double-object ZDT test function set is used to verify the superiority of the improved multi-objective particle swarm algorithm (IMOPSO). Finally, based on the order matching model and IMOPSO, simulation experiments are conducted to determine the factors that affect the use of manufacturers’ idle capacity and provide some constructive suggestions for small and medium-sized manufacturers (SMEs).

Energy conservations are studied for inhomogeneous incompressible and compressible Euler equations with general pressure law in a torus or a bounded domain. We provide sufficient conditions for a weak solution to conserve the energy. By exploiting a suitable test function, the spatial regularity for the density is only required to be of order $2/3$ in the incompressible case, and of order $1/3$ in the compressible case. When the density is constant, we recover the existing results for classical incompressible Euler equation., Comment: The new version deals with general pressure law for compressible equations. By using a new test function, we allow the density to be zero instead of excluding completely vacuum. Required regularities for the density in the compressible case are also reduced

We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached to all connected reductive groups over $p$-adic local fields with $p\geq 5$. In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme., Minor corrections; added {\S}{\S}7-8 on the test function conjecture for modified local models

In order to explore the global optimizing ability of the surrogate-based stacking sequence optimization of composite shells with multiple cutouts, a multi-fidelity competitive sampling (MFCS) method is developed in this paper, which is composed of a low-fidelity sampling level and a high-fidelity sampling level. In the low-fidelity sampling level, competitive sampling points are determined by means of the prior screening criterion and the further screening criterion in sequence. Based on the POD-based buckling method, a novel reduced-order model is established as the low-fidelity model for fast linear buckling analysis of composite shells with multiple cutouts. In the high-fidelity sampling level, the explicit dynamic method is employed to calculate the post-buckling response of competitive sampling points. Based on these sampling results, a surrogate model is built. Next, a surrogate-based stacking sequence optimization framework is established for minimum material cost of composite shells with multiple cutouts. A simple optimization example of Mishra's Bird test function is performed and the effectiveness of the MFCS method is demonstrated. Finally, the proposed method is used for the surrogate-based stacking sequence optimization of composite shells with multiple cutouts. The optimal result of total material cost by the MFCS method decreases by 23.0% than the optimal result by the traditional sampling method, which indicates that the MFCS method contributes to improving the global optimizing ability of the surrogate-based stacking sequence optimization of composite shells with multiple cutouts.

We consider three types of semilinear equations (elliptic, parabolic and hyperbolic) posed in the N-dimensional exterior domain R N \ D , where N ≥ 2 and D is the closed unit ball in R N . A nontrivial Robin boundary condition is imposed on the boundary of D. Using a test function approach with judicious choices of the test functions, we show that the considered problems share a common critical behavior. We discuss separately the cases N = 2 and N ≥ 3 . Moreover, in the case N ≥ 3 , the dependence of the critical exponent on initial data is discussed. To the best our knowledge, the study of the critical behavior in an exterior domain with a nontrivial Robin boundary condition has never been studied in the literature.

Stochastic gradient descent (SGD) optimization algorithms are key ingredients in a series of machine learning applications. In this article we perform a rigorous strong error analysis for SGD optimization algorithms. In particular, we prove for every arbitrarily small $\varepsilon \in (0,\infty)$ and every arbitrarily large $p\in (0,\infty)$ that the considered SGD optimization algorithm converges in the strong $L^p$-sense with order $\frac{1}{2}-\varepsilon$ to the global minimum of the objective function of the considered stochastic approximation problem under standard convexity-type assumptions on the objective function and relaxed assumptions on the moments of the stochastic errors appearing in the employed SGD optimization algorithm. The key ideas in our convergence proof are, first, to employ techniques from the theory of Lyapunov-type functions for dynamical systems to develop a general convergence machinery for SGD optimization algorithms based on such functions, then, to apply this general machinery to concrete Lyapunov-type functions with polynomial structures, and, thereafter, to perform an induction argument along the powers appearing in the Lyapunov-type functions in order to achieve for every arbitrarily large $ p \in (0,\infty) $ strong $ L^p $-convergence rates. This article also contains an extensive review of results on SGD optimization algorithms in the scientific literature.

In this paper, we focus on the lump and interaction solutions for a generalized (3 + 1)-dimensional propagation model of nonlinear waves in fluid dynamics. With symbolic computation, some new exact...

The paper analyzes the Morley element method for the Cahn–Hilliard equation. The objective is to prove the numerical interfaces of the Morley element method approximate the Hele-Shaw flow. It is achieved by establishing the optimal error estimates which depend on 1/ε polynomially, and the error estimates should be established from lower norms to higher norms progressively. If the higher norm error bound is derived by choosing test function directly, we cannot obtain the optimal error order, and we cannot establish the error bound which depends on 1/ε polynomially either. Different from the discontinuous Galerkin (DG) space [Feng et al. SIAM J. Numer. Anal. 54 (2016) 825–847], the Morley element space does not contain the finite element space as a subspace such that the projection theory does not work. The enriching theory is used in this paper to overcome this difficulty, and some nonstandard techniques are combined in the process such as the a priori estimates of the exact solution u, integration by parts in space, summation by parts in time, and special properties of the Morley elements. If one of these techniques is lacked, either we can only obtain the sub-optimal piecewise L∞(H2) error order, or we can merely obtain the error bounds which are exponentially dependent on 1/ε. Numerical results are presented to validate the optimal L∞(H2) error order and the asymptotic behavior of the solutions of the Cahn–Hilliard equation.

We classify Bogolyubov endomorphisms of the CCR algebra over a general test function space. This classification requires a non-commutative extension of the usual hyperbolic functions. We consider semi-groups of such endomorphisms and characterize their generators under the assumption of norm continuity.

This paper presents a new fast Intelligent Programmed Genetic Algorithm (IPGA) based evolutionary optimization algorithm which requires lesser number of objective function evaluation for reaching optima. The proposed algorithm, apart from using probabilistic genetic operator, i.e. crossover and mutation, also uses a deterministic diversity creating operator for generating new solution in the current population. This is done by first projecting objective surface from higher dimension to lower dimension for visualization purpose and then deterministically generates new solution using some predefined rules in the region with higher objective function value. As the newly generated solution is in lower-dimensional space, these solutions are again projected back to higher dimensional space and then the objective function is evaluated at that point. The proposed IPGA is tested on three different categories of standard test functions viz. Unimodal function (2 Test Function), Unrotated Multimodal function (6 Test Function) and Rotated Multimodal function (5 Test Function). Simulation results were compared with that obtained using Binary Coded GA, Real Coded GA, recently proposed GA with Differential Evolution crossover operator (GA–DEx) and another success-history-based adaptive GA with aging mechanism (GA–aDExSPS) in terms of mean and standard deviation of the objective function, average number of objective function evaluation required to reach optima and algorithmic complexity. Simulation results clearly demonstrate better performance of the proposed IPGA when compared with other variants of GAs.

The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series representation in the space of level 1 automorphic forms of a split classical group $G$ over $\mathbf {Z}$ , and provide numerical applications in absolute rank $\leq 8$ . Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of $\mathrm{GL}_{n}$ over $\mathbf {Q}$ ( $n$ arbitrary) whose motivic weight is $\leq 24$ . In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existence of certain level one algebraic cusp forms on $\mathrm{GL}_{n}$ , and that we push further on in this paper. We use these vanishing results to obtain an arguably “effortless” computation of the elliptic part of the geometric side of the trace formula of $G$ , for an appropriate test function. Thoses results have consequences for the computation of the dimension of the spaces of (possibly vector-valued) Siegel modular cuspforms for $\mathrm{Sp}_{2g}(\mathbf {Z})$ : we recover all the previously known cases without relying on any, and go further, by a unified and “effortless” method.

This paper is devoted to proving some new nonexistence theorems for a class of quasilinear parabolic differential inequalities with a singular potential term and nonlocal source term in the case of homogeneous and non-homogeneous by the test function method.

As part of the development of an automated virtual design classification approach for nonlinear structural dynamics, alternative excitation functions are evaluated with respect to their overall performance and efficiency in feature-based response analysis. Robust design of nonlinear structures requires analysis of extensive parameter variations. Both the character of the stimulus and feature metrics used are central to the performance of a response classification approach. The main purpose of this study is to compare stimulus candidates with respect to their efficiency in response classification. A deterministic multilevel, multifrequency stepped-sine periodic test function is used as a baseline. Order-wise differences between generalized and linearized system frequency response functions are evaluated by a selected feature metric to allow categorization into primary, sub and super harmonic responses, as well as odd and even order response distortions. An alternative excitation function type is the pseudorandom phase multi-sine. Its robust variant estimates the best linear approximation of the generalized frequency response function and related nonlinear and noise variances, which can be used for response classification. The fast variant of this method further detects and classifies occurring even and odd order nonlinear responses using a hypothesis test. This article describes the application of these three methods to a virtually running two-piece rotor shaft model. Time response signals from simulated test parameter variations are used to calculate selected nonlinear feature metric values. The total simulation and measurement time, as well as the predictive performance in a few typical nonlinear response cases are evaluated.

It is common for papers on surrogate fitting to select test functions for testing algorithms. This raises the issue of how well the algorithms generalize to other functions. This editorial proposes as a possible complement to use a Gaussian process (GP) for generating test functions. The GP is defined by spatial decay rates of correlation between function values at different locations. The selection of these decay rates is tantamount to selection of the wave lengths of the functions that can be approximated by the surrogate. A possible complement to a given test function is a GP with similar decay rates to those of the test function, which would generate random functions with similar waviness. Two approaches are considered. In the simpler one, the GP is used to generate random test functions at a predetermined set of points. The more elaborate scheme targets situations where some samples are already available, and the GP is restricted to generate random functions that interpolate the samples. The process is illustrated for a one-dimensional case, where we seek to measure the correlation between the 95% confidence interval of the GP fit and the actual error. It is hoped that the editorial will stimulate discussion of the problem of algorithms customized to particular test functions and other options besides the one proposed here.

In this paper, we mainly concern with the sharp weighted Trudinger-Moser inequalities with the Ln norm on the whole space (See Theorem 1.1 and 1.3). Most proofs in the literature of existence of extremals for the Trudinger-Moser inequalities on the whole space rely on finding a radially maximizing sequence through the symmetry and rearrangement technique. Obviously, this method is not efficient to deal with the existence of maximizers for the double weighted Trudinger-Moser inequality Eq. 1.4 because of the presence of the weight t and s. In order to overcome this difficulty, we first apply the method of change of variables developed by Dong and Lu (Calc. Var. Part. Diff. Eq. 55, 26–88, 2016) to eliminate the weight s. Then we can employ the method combining the rearrangement and blow-up analysis to obtain the existence of the extremals to the double weighted Trudinger-Moser inequality Eq. 1.4. By constructing a proper test function sequence, we also derive the sharpness of the exponent a of the Trudinger-Moser inequalities Eqs. 1.3 and 1.4 (see Theorem 1.2 and 1.4). This complements earlier results in Nguyen (2017); Li and Yang (J. Diff. Eq. 264, 4901–4943, 2018); Lu and Zhu (J. Diff. Eq. 267, 3046–3082, 2019).

In this paper, we obtain sufficient conditions for the nonexistence of global solutions for the system of \begin{document}$ q $\end{document} -difference inequalities. Our approach is based on the weak formulation of the problem, a particular choice of the test function, and some \begin{document}$ q $\end{document} -integral inequalities.

In this paper we give introduction to the concepts of Ubuntu and how we used Mechanism design concepts to construct Ubuntu as an optimisation algorithm. Ubuntu philosophy is old and consists of many oral proverbs that have been documented in recent years. This work thus introduces an incentive mechanism based on Ubuntu, thus called Ubuntu Incentive, which is modelled according to Mechanism Design principles. This incentive scheme is introduced as a fitness function which the algorithm tries to improve. To achieve this, the algorithm draws inspiration from Bantu proverbs that guide how individuals ought to behave within the Ubuntu community. Trust is an important element within these communities and it is shown how trust influences the obtaining of the Pareto efficiency. The algorithm is introduced with different mathematical configurations which are tested against each other. Ulimisana optimisation algorithm (UOA) manages to solve the benchmark test functions used in this work. This is found to be in accordance to the Ubuntu philosophy as used in the Ulimisana/Letsema practice amongst the Bantu people of Southern Africa. The UOA performed better in some benchmark test function when compared to other algorithms and coming second on most performance to PSO for most test benchmark function.

The main goal of this paper is to study the continuity of composition of linear canonical wavelet transform (LCWTs) on generalized test function spaces Lp,A, Gp,A and BA(R3). The boundedness result for composition of linear canonical wavelet transforms on Hps,A is given.

We study some new Liouville theorems for nonlinear differential inequalities with gradient terms and singular variable coefficients that have singularities on unbounded sets. The proofs are based on the test function method developed by Mitidieri and Pohozaev.

This paper concerns the question to what extent it can be efficiently determined whether an arbitrary program correctly solves a given problem. This question is investigated with programs of a very simple form, namely instruction sequences, and a very simple problem, namely the non-zeroness test on natural numbers. The instruction sequences concerned are of a kind by which, for each $n > 0$, each function from $\{0,1\}^n$ to $\{0,1\}$ can be computed. The established results include the time complexities of the problem of determining whether an arbitrary instruction sequence correctly implements the restriction to $\{0,1\}^n$ of the function from $\{0,1\}^*$ to $\{0,1\}$ that models the non-zeroness test function, for $n > 0$, under several restrictions on the arbitrary instruction sequence., 32 pages, minor revision with several obscurities made clear

In order to improve the stability and sensitivity of particle swarm optimization (PSO) algorithm and to solve the problem of premature convergence, in this re-search, a computer PSO algorithm based on thermodynamic motion mechanism is proposed based on the principle of thermodynamic motion mechanism. Firstly, the thermodynamic motion phenomenon, the diffusion law in kinematics and the standard PSO algorithm are introduced. Then, according to the basic idea of thermodynamic motion mechanism, the standardized PSO algorithm is optimized and its optimization process is introduced. Finally, the experimental results are analysed by setting the test function. The results show that among the five test functions, the computer PSO algorithm based on thermodynamic motion mechanism has a higher probability of jumping out of the local optimal solution. Its robustness and stability are much better than standard PSO algorithms. The evolution ability of the computer PSO algorithm based on thermodynamic motion mechanism is better than that of the standard PSO algorithm. The standard PSO algorithm is superior because it is based on thermodynamic motion mechanism. The research in this paper can provide good guidance for improving the performance of PSO algorithm.