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Consider the Dirichlet problem for Laplace’s/Poisson’s equation in a bounded simply-connected domain S . The source function q ln | P Q * ¯ | is a fundamental solution (FS), and it can be found in many physical problems. The singularity occurs when the boundary value data affected by q ln | P Q * ¯ | as the source node Q * is located near the boundary Γ ( = ∂ S ) . So far, there is no comprehensive study on this kind of singularity. In this paper, the solution singularity is explored and the reduced convergence rates are derived for the method of particular solutions (MPS) and the method of fundamental solutions (MFS). Classic domains, such as disks, ellipses and polygons, are discussed for analysis and computation. For this new kind of solution singularity, the convergence rates of the MFS and the MPS are very low. The errors caused by numerical integration are critical to the solution accuracy. A new analytic framework for the collocation Trefftz method (CTM) involving numerical integration is established in this paper; this is an advanced development of our previous study [19]. Since the numerical solutions are poor in accuracy, removal techniques are essential in applications. New removal techniques are proposed for a node Q * located near Γ . In this paper, an additional FS as, d 0 ln | P Q 0 ¯ | , is added to the original source nodes in the traditional MFS, and the point charge d 0 ( = q ) and the source node Q 0 are unknowns to be sought by nonlinear solvers (such as the secant method). When the source node Q * is located inside S but near Γ , both simple domains (such as disks, ellipses and squares) and complicated domains (such as amoeba-like domains) are studied. The validity of the new removal techniques is supported by numerical experiments. The removal techniques in this paper may also be applied to solve source identification problems. A comprehensive study has been completed in this paper for the solitary source function q ln | P Q * ¯ | as the source node Q * is located near Γ .

To solve the problem that the traditional hyperspectral image classification method cannot effectively distinguish the boundary of objects with a single scale feature, which leads to low classification accuracy, this paper introduces the idea of guided filtering into hyperspectral image classification, and then proposes a multi-scale guided feature extraction and classification (MGFEC) algorithm for hyperspectral images. Firstly, the principal component analysis theory is used to reduce the dimension of hyperspectral image data. Then, guided filtering algorithm is used to achieve multi-scale spatial structure extraction of hyperspectral image by setting different sizes of filtering windows, so as to retain more edge details. Finally, the extracted multi-scale features are input into the support vector machine classifier for classification. Several practical hyperspectral image datasets were used to verify the experiment, and compared with other spectral feature extraction algorithms. The experimental results show that the multi-scale features extracted by the MGFEC algorithm proposed in this paper are more accurate than those extracted by only using spectral information, which leads to the improvement of the final classification accuracy. This fully shows that the proposed method is not only effective, but also suitable for processing different hyperspectral image data.

Multi-label classification is a typical supervised machine learning problem and widely applied in text classification and image recognition. When there are redundant attributes in the data, the efficiency of classification will be reduced. However, the existing attribute reduction algorithms have high computational complexity. This paper aims to design an efficient attribute reduction algorithm. The k pairs of boundary samples were selected from the positive and negative classes respectively, and the distance between each pair was calculated as the evaluation of attributes. By maximizing the evaluation function, the definition of reduction and the design of the algorithm were established. The comparison experiment is carried out on eight generic multi-label data. The experimental results show that the attribute importance evaluation defined in this paper can better represent the classification performance of the attribute for multi-label classification. The boundary samples can better reflect the classification effect of attributes. The proposed model avoids the point-by-point statistics of all samples’ information and improves the computational efficiency.

Support vector domain description (SVDD) is a data description method inspired by support vector machine (SVM). This classifier describes a set of data points with a sphere that encloses the majority of them and has a minimal volume. The boundary of this sphere is used to classify new samples. SVDD has been successfully applied to many challenging classification problems and has shown a good generalization capability. However, this classifier still has some major weaknesses. This paper focuses on two of them: The first regards the large amount of memory and computational time required by SVDD in the training step. This problem manifests most strongly when dealing with large-size datasets and can hinder or prevent its use. This paper presents an approximate solution to this problem that permits to apply SVDD to large-scale datasets. This new version is based on divide-and-conquer strategy and it processes in two steps: It begins by dividing the whole large-size dataset into random subsets that each can be described efficiently with a small sphere using SVDD. Then, it applies our new algorithm that can find the smallest sphere that encloses the minimal spheres built in the previous step. The second weak point of standard SVDD concerns its static learning process. This classifier must be re-trained with the whole dataset each time when new training samples are available. This paper proposes a new dynamic approach that only trains the new samples with SVDD and incorporates the resulting minimal sphere with the previous one (s) to construct the smallest sphere that encloses all the samples. Like Support Vector Domain Description, the proposed approach can be extended to non-linear classification cases by using kernel functions. Experimental results on artificial and real datasets have successfully validated the performance of our approach.

Motivated by some recent potential theoretic results on subordinate killed L\'evy processes in open subsets of the Euclidean space, we study processes in an open set $D\subset {\mathbb R}^d$ defined via Dirichlet forms with jump kernels of the form $J^D(x,y)=j(|x-y|)\mathcal{B}(x,y)$ and critical killing functions. Here $j(|x-y|)$ is the L\'evy density of an isotropic stable process (or more generally, a pure jump isotropic unimodal L\'evy process) in $\mathbb{R}^d$. The main novelty is that the term $\mathcal{B}(x,y)$ tends to 0 when $x$ or $y$ approach the boundary of $D$. Under some general assumptions on $\mathcal{B}(x,y)$, we construct the corresponding process and prove that non-negative harmonic functions of the process satisfy the Harnack inequality and Carleson's estimate. We give several examples of boundary terms satisfying those assumptions. The examples depend on four parameters, $\beta_1, \beta_2, \beta_3$, $\beta_4$, roughly governing the decay of the boundary term near the boundary of $D$. In the second part of this paper, we specialise to the case of the half-space $D=\mathbb{R}_+^d=\{x=(\widetilde{x},x_d):\, x_d>0\}$, the $\alpha$-stable kernel $j(|x-y|)=|x-y|^{-d-\alpha}$ and the killing function$\kappa(x)=c x_d^{-\alpha}$, $\alpha\in (0,2)$, where $c$ is a positive constant. Our main result in this part is a boundary Harnack principle which says that, for any $p>(\alpha-1)_+$, there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, and the constant $c$ such that non-negative harmonic functions of the process must decay at the rate $x_d^p$ if they vanish near a portion of the boundary. We further show that there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, for which the boundary Harnack principle fails despite the fact that Carleson's estimate is valid., Comment: This is a corrected version of the published paper https://doi.org/10.1007/s11118-021-09947-8. Proofs of Lemmas 9.3 and 9.4 had minor gaps (estimates of integrals I_2 and IV in Lemma 9.3 and integral IV in Lemma 9.4 were incorrect). These are now fixed. We also corrected some typos

In this paper, we consider the Laplace operator in a flat band whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude.

Since some oil fields have entered the late stage of high water-cut development, the use of polymer flooding and other oil recovery technologies makes the fluid be of non-Newtonian characteristics, but the existing elastic outer boundary percolation model ignores this point. Firstly, this paper establishes a non-Newtonian power-law fluid percolation model with the elastic outer boundary condition. Secondly, the analytical solution of the percolation model is obtained in the Laplace space by the structural method. Thirdly, the double logarithmic characteristic curves are drawn. Finally, the sensitivity analysis of the parameters is carried out. The results show that the wellbore storage coefficient and skin factor play important roles in the early stage of the double logarithmic curves. For the radial flow stage and the later flow stage, the main influence factors are the power-law index, the elastic coefficient and outer boundary radius, respectively. It is concluded that the percolation model established in this paper provides a theoretical basis for reasonably solving the problem that the previous data deviation could not explain the reservoir parameters well. Meanwhile, the application of the structural method can provide a scientific calculation method for the well test analysis of non-Newtonian power-law fluids.

In this paper, we formulate mathematical foundations of applications of boundary integral equations with strongly singular integrals understood in the sense of finite Hadamard value to numerical solution of certain boundary-value problems. We describe numerical schemes for solving boundary strongly singular equations based on quadrature formulas and the collocation method. Also, we make references to known results on the mathematical justification of the numerical methods described in the paper.

We consider a natural model of inhomogeneous random graphs that extends the classical Erdős–Rényi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous (Ann Probab 25:812–854, 1997). In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic (Electron J Probab 3:1–59, 1998) have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Lévy-type processes. We, instead, look at the metric structure of these components and prove their Gromov–Hausdorff–Prokhorov convergence to a class of (random) compact measured metric spaces that have been introduced in a companion paper (Broutin et al. in Limits of multiplicative inhomogeneous random graphs and Lévy trees: the continuum graphs.arXiv:1804.05871, 2020). Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general “critical” regime, and relies upon two key ingredients: an encoding of the graph by some Lévy process as well as an embedding of its connected components into Galton–Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of Lévy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Lévy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via model- or regime-specific proofs, for instance: the case of Erdős–Rényi random graphs obtained by Addario-Berry et al. (Probab Theory Relat Fields 152:367–406, 2012), theasymptotic homogeneouscase as studied by Bhamidi et al. (Probab Theory Relat Fields 169:565–641, 2017), or thepower-lawcase as considered by Bhamidi et al. (Probab Theory Relat Fields 170:387–474, 2018).

Back-analysis can be regarded as the reverse process of forwarding analysis. The forward analysis in geotechnical engineering is to utilize the geometric dimensions, constitutive relations, material parameters, and boundary load conditions determined by the rock mass medium to solve the physical quantity information (strain, stress, Displacement) process. In this paper, a novel back analysis program based on BP neural network is proposed, which can realize automatic correction and adjustment of parameters and adapt to most tunnel projects. Subsequently, the analysis program is applied to the front and back joint modeling of the Nagasaki tunnel project. The mechanical parameters and initial stresses of the back analysis are utilized to conduct finite element calculations, and the displacements of all monitoring points of the multi-point displacement measurement are calculated. The results are in good agreement with the measured values. The proposed back-analysis method in this paper can also be applied to the determination of other tunnel rock mass parameters.

Purpose The purpose of this paper is to analytically predict the on-load field distribution and electromagnetic performance (induced voltage, electromagnetic torque, winding inductances and unbalanced magnetic force) of dual-stator consequent-pole permanent magnet (DSCPPM) machines using subdomain model accounting for tooth-tip effect. The finite element (FE) results are presented to validate the accuracy of this subdomain model. Design/methodology/approach During the preliminary design and optimization of DSCPPM machines, FE method requires numerous computational resources and can be especially time-consuming. Thus, a subdomain model considering the tooth-tip effect is presented in this paper. The whole field domain is divided into four different types of sub-regions, where the analytical solutions of vector potential in each sub-region can be rapidly calculated. The proposed subdomain model can accurately predict the on-load flux density distributions and electromagnetic performance of DSCPPM machines, which is verified by FE method. Findings The radial and tangential components of flux densities in each sub-region of DSCPPM machine can be obtained according to the vector potential distribution, which is calculated based on the boundary and interface conditions using variable separation approach. The tooth-tip effect is investigated as well. Moreover, the phase-induced voltage, winding inductances, electromagnetic torque and X-axis/Y-axis components of unbalanced magnetic forces are calculated and compared by FE analysis, where excellent agreements are consistently exhibited. Originality/value The on-load field distributions and electromagnetic performance of DSCPPM machines are analytically investigated using subdomain method, which can be beneficial in the process of initial design and optimization for such DSCPPM machines.

This paper is concerned with a nonparametric estimator of the regression function based on the local linear estimation method in a twice censoring setting. The proposed method avoid the problem of boundary effect and reduces the bias term. Under suitable assumptions, the strong uniform almost sure consistency with rate is established and the finite sample properties of the local linear regression smoother is investigated by means of a simulation study.

In this paper we address the problem of controlling chaos by using a selective filter inserted in a negative feedback loop. This has been achieved in driven double-well Duffing oscillator in a parameter region exhibiting both a boundary crisis and generalized bistability. The optimization of the filter response allows us to improve the reduction of the control signal when the periodic solution competing with the chaotic attractor is approached. The advantage in using a selective frequency approach with respect to the Pyragas's method mainly resides in the tunability of the maximum of its amplitude response and zero phase condition. Numerical simulations performed in the temporal domain match with the experiment and make us glimpse the potential of an automatic adaptive strategy adopting filter parameter variations in order to reach an optimal feedback signal reduction.

In this paper, the isogeometric dual reciprocity boundary element method (IG-DRBEM) is proposed to solve three-dimensional transient heat conduction problems. It is well known that the error of traditional BEM mainly comes from element dispersion, and the introduction of isogeometric ideas makes BEM become a veritable high-precision numerical method. At present, most of the problems solved by isogeometric BEM (IGBEM) are time-independent. The reason is similar to the traditional BEM, which cannot avoid solving domain integrals when solving time-dependent problems. In this paper, based on the potential fundamental solution the boundary-domain integral equation is obtained by the weighted residual method, where the classic dual reciprocity method is adopted to transform domain integrals into boundary integrals. Meanwhile, a two-level time integration scheme is used to solve the discretized differential equations. In addition, the adaptive integration scheme, the radial integral transform method and the power series expansion method are adopted to solve the boundary regular, nearly singular and singular integrals. Several classical numerical examples show that the presented method has good numerical stability and high precision by considering different factors such as the approximation function, the time step, the number of interior points and so on.

This paper concerns the effect of the (separated/connected) protection zone for the evolution of an endangered species on the reaction–diffusion equation with the strong Allee effect and the free boundary. First, we describe the long-time dynamical behavior of the system of two types protection zones with the same length. Furthermore, the asymptotic profiles of solutions and the asymptotic spreading speed are estimated when spreading happens. Our results, together with those in previous papers Du et al. (J Differ Equ 266:7327–7356, 2019), Du and Lou (J Eur Math Soc 17:2673–2724, 2015) on two other closely related models, show that the protection zone and the free boundary play an important role in the evolution of the endangered species.

With the concept of sustainable development, enterprises are facing severe challenges in ecological protection and economic development. Approaches to improving effectiveness of the coordinated development strategy must continue to evolve to address uncertainty and hazards that may be encountered in the future. We propose a coordinated development strategy model based on the combination of soft fuzzy and rough set theory and construct its prediction model. For the multistrategy dataset in the paper, parameter for each kind shall be selected through converting the multistrategy data into two prediction datasets. An algorithm transformed by SFRC shall be subject to weighted average for each parameter. Furthermore, we use training methods and soft fuzzy rough sets’ learning algorithm to calculate, and the evaluating indicator rough set is constructed with a three-tier model structure. After the final rough set training is completed, test results show that the rough set model which has a higher rating accuracy builds a better completed business performance evaluation. By comparing the prediction effect, both SVM algorithm and multistrategy prediction model for the soft fuzzy rough set in the paper can realize effective prediction for the enterprise’s coordinated development strategy. Moreover, the prediction result obtained at the time of adopting boundary to get the expected value is superior to that of giving one fixed threshold. It shows that the prediction performance of the algorithm in the paper is more excellent and represents the advantage of the algorithm prediction performance at the time of adopting boundary to get the expected value. The model provides support methods to assist enterprise management in making more efficient and scientific decisions for enterprise’s coordinated development.

Дослідження є продовженням наших зусиль щодо застосування спеціально побудованих наборів послідовних експоненціальних функцій як пробних (базисних) функцій в методі зважених нев’язок (МЗН) на прикладі класичних задач конструкційної механіки. Стаття не направлена на отримання нових результатів, а присвячена обґрунтуванню ефективності запропонованого методу. Розглядаються статична деформація та вільні коливання ізотропної тонкостінної квадратної пластини. Особливістю роботи є вибір вагових (перевірочних) функцій втрьох варіантах: як пробних функцій (метод Гальоркіна, МГ); як функцій, що є результатом застосування диференціального оператора до пробних функцій (метод найменших квадратів, МНК); як функцій, які є добутком других похідних від пробних функцій по x і y (метод моментів, ММ). Розв’язок будується як добуток двох незалежних множин функцій відносно координат x та y. Кожна множина є комбінацією п'яти послідовних експоненціальних функцій, де перший коефіцієнт дорівнює 1, а чотири інші коефіцієнти визначаються з граничних умов на протилежних сторонах пластини. Довільним параметром в методі є коефіцієнт масштабування в показниках, розумний діапазон якого ретельно досліджено і показано його вплив на результати. Статична деформація досліджена на прикладі простої шарнірно-опертої пластини, коли зовнішнє навантаження або симетричне і зосереджене поблизу центру пластини, або зміщене від центра добудь-якої кутової точки. Продемонстровано, що результати сходяться до точного рішення швидше, ніж у класичному методі Нав'є. ММ та МНК дають кращу точність при визначенні згинаючих моментів, ніж МГ. Запропонований метод застосований до аналізу вільних коливань пластини, точність результатів визначення власних частот є відмінною навіть при невеликій кількості членів ряду. Проаналізовано порівняно складний випадок – вільні коливання защемленої на всіх сторонах пластини, досягнуто дуже хороших результатів щодо ефективності та точності. The paper is continuation of our efforts on application of the properly constructed sets of exponential functions as the trial (basic) functions in weighted residuals method, WRM, on example of classical tasks of structural mechanics. The purpose of thispaper is justification of new method’s efficiency as opposed to getting new results. So, static deformation and free vibration of iso-tropic thin – walled plate are considered here. Another peculiarity of paper is choice of weight (test) functions, where three options are investigated: it is the same as trial one (Galerkin method); it is taken as results of application of differential operator to trial function (least square method); it equals to the second derivative of trial function with respect to both x and y coordinate (moment method). Solution is considered as product of two independent sets of functions with respect to x or y coordinates. Each set is the combination of five consequent exponential functions, where coefficient at first function is equal to one, and four other coefficients are to satisfy two boundary conditions at each opposite boundary. The only arbitrary value in this method is the scaling factor at exponents, the reasonable range of which was carefully investigated and was shown to have a negligible impact on results. Static deformation was investigated on example of simple supported plate when outer loading is either symmetrical and concentrated near the center or is shifted to any corner point. It was demonstrated that results converge to correct solution much quickly than in classical Navier method, while moment method seems to be a best choice. Then method was applied to free vibration analysis, andagain the accuracy of results on frequencies and mode shape were excellent even at small number of terms. At last the vibration of relatively complicated case of clamped-clamped plate was analyzed and very encouraged results as to efficiency and accuracy were achieved. Данное исследование является продолжением наших усилий по применению специально построенных наборов последовательных экспоненциальных функций как пробных (базисных) функций в методе взвешенных невязок (МВН) на примере классических задач структурной механики. Статья ненаправлена на получение новых результатов, а посвящена обоснованию эффективности предлагаемого метода. Рассматриваются статическая деформация и свободные колебания изотропной тонкостенной квадратной пластины. Особенностью работы является выбор весовых (поверочных) функций в трех вариантах: как пробных функций (метод Галеркина, МГ); как функций, являющихся результатом применения диффе-ренциального оператора к пробным функций (метод наименьших квадратов, МНК); как функций, являющихся произведениями вторых производных от пробных функций по x и y (методмоментов, ММ). Решение строится как произведение двух независимых множеств функций относительно координат x и y. Каждое множество представляет собой комбинацию пяти последовательных экспоненциальных функций, в которой первый коэффициент равен 1, а четыре другие коэффициента определяются из граничных условий на противоположных сторонах пластины. Произвольным параметром в методе является коэффициент масштабирования в показателях, разумный диапазон которого тщательно исследовался, и показано его влияние на результаты. Статическая деформация исследована на примере простой шарнирно-опертой пластины, когда внешняя нагрузка или симметрична и сосредоточена в близи центра пластины, или смещена от центра к произвольной угловой точке. Продемонстрировано, что результаты сходятся к точному решению быстрее, чем в классическом методе Навье. ММ и МНК дают лучшую точность при определении изгибающих моментов, чем МГ. Предложенный метод применен к анализу свободных колебаний пластины, точность результатов определения собственных частот является отличной даже при небольшом количестве членов ряда. Проанализированы сравнительно сложный случай - свободные колебания защемленной по всем сторонам пластины, достигнуты очень хорошие результаты по эффективности и точности.

Purpose This paper aims to investigate the influence of droplet infiltration and sliding on the deposition size and make a uniform deposition by controlling the interaction between droplets, using the three-dimensional lattice Boltzmann method (LBM) based on the actual working condition. Design/methodology/approach D3Q19 Shan-Chen LB approach is developed and optimized based on the metal droplet deposition. The Carnahan-Starling equation of state and transition layers are introduced to maintain the greater stability and low pseudo velocities. In addition, an additional collision term is adopted to implement immersed moving boundary scheme to deal with no-slip boundaries on the front of the phase change. Findings The numerical results show that the new¬ incoming droplet wet and slide off the solidified surface and the rejection between droplets are the reasons for the deviation of the actual deposition length. The total length of the longitudinal section negatively correlates with the deposition distance. To improve the dimensional accuracy, the deposition distance and repulsion rate need to be guaranteed. The optimal deposition distance is found to have a negative linear correlation with wettability. Originality/value The numerical model developed in this paper will help predict the continuous metal droplet deposition and provide guidance for the selection of deposition distance.

This paper proposes a new boundary uncertainty-based estimation method that has significantly higher accuracy, scalability, and applicability than our previously proposed boundary uncertainty estimation method. In our previous work, we introduced a new classifier evaluation metric that we termed “boundary uncertainty.” The name “boundary uncertainty” comes from evaluating the classifier based solely on measuring the equality between class posterior probabilities along the classifier boundary; satisfaction of such equality can be described as “uncertainty” along the classifier boundary. We also introduced a method to estimate this new evaluation metric. By focusing solely on the classifier boundary to evaluate its uncertainty, boundary uncertainty defines an easier estimation target that can be accurately estimated based directly on a finite training set without using a validation set. Regardless of the dataset, boundary uncertainty is defined between 0 and 1, where 1 indicates whether probability estimation for the Bayes error is achieved. We call our previous boundary uncertainty estimation method “Proposal 1” in order to contrast it with the new method introduced in this paper, which we call “Proposal 2.” Using Proposal 1, we performed successful classifier evaluation on real-world data and supported it with theoretical analysis. However, Proposal 1 suffered from accuracy, scalability, and applicability limitations owing to the difficulty of finding the location of a classifier boundary in a multidimensional sample space. The novelty of Proposal 2 is that it locally reformalizes boundary uncertainty in a single dimension that focuses on the classifier boundary. This convenient reduction with a focus toward the classifier boundary provides the new method’s significant improvements. In classifier evaluation experiments on Support Vector Machines (SVM) and MultiLayer Perceptron (MLP), we demonstrate that Proposal 2 offers a competitive classifier evaluation accuracy compared to a benchmark Cross Validation (CV) method as well as much higher scalability than both CV and Proposal 1.

This paper endeavours to provide an analysis for nanofluid boundary layer that develops beneath a generalized vortex flow subjected to viscous dissipation effects. Tangential flow high above the disk is assumed to vary with radial distance [Formula: see text] as [Formula: see text]. Hybrid nanofluid comprises alumina-[Formula: see text] and copper-[Formula: see text] nanoparticles with water as base liquid. A generalized version of von Kármán relations proposed in a recent paper is opted to present self-similar solutions. In presence of viscous dissipation term, self-similar solutions are possible only when temperature difference is proportional to [Formula: see text]. Otherwise, the solutions are only locally similar. Present model reduces to two special situations namely rigid body rotation [Formula: see text] and potential vortex [Formula: see text]. It is perceived that inclusion of nanoparticles markedly affects the boundary layer development under the prescribed vortex flow, and the associated heat transfer process. Also, viscous dissipation term has important implications on the resulting heat transfer process.

The development of fast and reliable optimization algorithms is required in order to obtain real-time optimal trajectory on-board spacecraft. In addition, the wide spread of small satellites, due to their low costs, is leading to a greater number of satellite formations in space. This paper presents an Improved version of the Magnetic Charged System Search (IMCSS) metaheuristic algorithm to compute time-suboptimal manoeuvres for satellite formation flying. The proposed algorithm exploits some strategies aimed at improving the convergence to the optimum, such as the chaotic local search and the boundary handling technique, and it is able to self-tune its internal parameters and coefficients. Moreover, the inverse dynamics technique and the differential flatness approach, through the B-splines curves, are used to approximate the trajectory. The optimization procedure is applied to the circular J2 relative model developed by Schweighart and Sedwick and to the elliptical relative motion model developed by Yamanaka and Ankersen. The results of this paper show that the convergence is better achieved by using the proposed tools, thus proving the efficiency and reliability of the algorithm in solving some space engineering problems.

The extraction of effective signals of microseismic event caused by water inrush in tunnels is an important prerequisite for accurate location of event. In this paper, based on the characteristics of the main frequency components of the tunnel water burst signal, the variational modal method (VMD) with good frequency decomposition ability is adopted to effectively extract the high-frequency components of the water inrush signal. The effective modes are identified by the method of detrend analysis, and the modal decomposition number K is determined. At the same time, the boundary between noise and effective signal is determined. The method proposed in this paper solves the problem that noise in some modal components is mixed with effective signals, and further improves the ability of extracting weak high-frequency signals. Through the analysis of case data, the validity of the research results in this paper is verified, and an effective signal extraction method of tunnel water inrush microseismic event based on VMD-DFA principle is formed.

In the paper, we study a high order numerical boundary scheme for solving the complex moving boundary problem on a fixed Cartesian mesh, and numerically investigate the moving rigid body with the complex boundary under the impingement of an inviscid shock wave. Based on the high order inverse Lax-Wendroff (ILW) procedure developed in the previous work (TAN, S. and SHU, C. W. A high order moving boundary treatment for compressible inviscid flows. Journal of Computational Physics, 230(15), 6023–6036 (2011)), in which the authors only considered the translation of the rigid body, we consider both translation and rotation of the body in this paper. In particular, we reformulate the material derivative on the moving boundary with no-penetration condition, and the newly obtained formula plays a key role in the proposed algorithm. Several numerical examples, including cylinder, elliptic cylinder, and NACA0012 airfoil, are given to indicate the effectiveness and robustness of the present method.

PurposeThe purpose of the study is to solve numerically the inverse problem of determining the time-dependent convection coefficient and the free boundary, along with the temperature in the two-dimensional convection-diffusion equation with initial and boundary conditions supplemented by non-local integral observations. From the literature, there is already known that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data.Design/methodologyFor the numerical discretization, this paper applies the alternating direction explicit finite-difference method along with the Tikhonov regularization to find a stable and accurate numerical solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB routine lsqnonlin. Both exact and numerically simulated noisy input data are inverted.FindingsThe numerical results demonstrate that accurate and stable solutions are obtained.Originality/valueThe inverse problem presented in this paper was already showed to be locally uniquely solvable, but no numerical solution has been realized so far; hence, the main originality of this work is to attempt this task.

The connectivity of a graph is the minimum number of nodes, whose removal will cause the graph disconnected. It is one of the basic and important properties of the graph, and can be used as a metric of the graph’s robustness. Since the connectivity of a graph is upper-bounded by its minimal node-degree, which can be small, the notion of conditional connectivity has been proposed in the past to more realistically reflect a graph’s robustness. The extra connectivity is such a conditional connectivity. In this paper, we introduce a new approach to finding the extra connectivity of the hypercube, a well-known regular graph model for networks of parallel computers. We will show that the results on the isoperimetric inequalities for hypercubes can be used to obtain their extra connectivity. That is, using isoperimetric inequalities, for an n -dimensional hypercube Q n with n ≥ 5 , the h -extra connectivity is equal to its minimum ( h + 1 ) -vertex boundary number for 1 ≤ h ≤ n − 4 and n + 1 ≤ h ≤ 2 n − 4 . For n − 3 ≤ h ≤ n , the h -extra connectivity of hypercube equals its minimum ( n − 2 ) -vertex boundary number. The paper’s result for the first time establishes a relationship between the isoperimetric inequalities and the extra connectivity of graphs.

The paper presents a study of a supersonic boundary layer while heat supply to the layer’s narrow internal region. Calculations were performed for a flow with Mach number M = 2 and the dimensionless temperature (normalized to the temperature at a layer boundary edge Tw = 3.9 (630 K). It was demonstrated that heat supply causes a twofold increase in the 2D disturbance increments, while the frequency range expands by factor of three with a heat-insulated plate (zero heat input) taken as a reference. For a heated plate (with Tw = 3.9), the heat input into this layer delays the disturbance growth; that is, this facilitates the boundary layer stabilization and the disturbance increments reduce by almost four times. Along with 2D waves, the oblique waves were investigated in this paper. It was found that for low-frequency disturbance, we observe an intense growth of 3D (oblique) waves. Meanwhile, 2D disturbances enhance for a high-frequency range. These facts comply with the numerous data on boundary layer stability. However, these factors for boundary layer stability become less significant in the case of heat input to the supersonic boundary layer (compared with the zero heat input). The study also observes the longitudinal structures with a downstream decay. The heat supply makes these structures decrement lower in the supersonic boundary layer (meaning a slower decay). The results for the study of boundary layer disturbance are compared to similar data for the flow with M ≪ 1.

Summary Solving the dynamic response of three-dimensional anisotropic media has always been rather challenging. Although many researchers have found solutions to homogeneous anisotropic half-spaces, there is little research into anisotropic layered media. In this paper, for the three-dimensional anisotropic layered media, the modified scaled boundary finite element method (SBFEM) is used to solve the dynamic impedance matrix of the anisotropic layered media after introducing a high-order transmitting boundary based on the continued-fraction method. The algorithm in this paper can solve not only isotropic and cross-anisotropic layered media, but also general anisotropic layered media. First, the algorithm is degraded to obtain the dynamic impedance matrix of cross-anisotropic layered media; the result is compared with that in the previous studies to verify the accuracy of our algorithm. Then, the solutions of our algorithm are given for general anisotropic media.

This paper is a continuation of the research distributed in [1], the theory of Boolean functions is considered from the point of view of universal algebras. This paper describes a class of functionally incomplete algebras, studies the main types of algebras and their location on the tiers of the signature cube. In the data of researches of universal Boolean algebras the 11-dimensional signal cube according to which 2048 algebras enter is created. Codification of these algebras has been offered. The notion of adjacent, boundary, and internal classes of functionally superficial and functionally incomplete algebras is introduced. Boolean algebras of the class are divided into four subclasses: a class of internal functionally incomplete algebras, a class of boundary functionally incomplete algebras, a class of boundary functionally superficial algebras, a class of internal functionally complete algebras. In this paper, an algorithm is proposed for finding boundary functionally complete algebras based on the expansion of signals of functionally incomplete algebras by Boolean operations. Subclasses of boundary algebras have been constructed for each of the eleven operations. The isomorphism of graphs of some classes of boundary algebras has been indicated. -graph of boundary functionally complete algebras was obtained on the basis of combining graphs.

This paper studies the water inflow problem when a tunnel is excavated in the area affected by a water-rich fault from an analytical perspective, and a seepage model considering the influence of the fault boundary is established. In the model, the inclined fault is equivalent to the vertical fault by regarding the volume of water inflow and the distribution of water head as the standard of measurement; then, the image superposition principle is used to transform the seepage problem with a finite boundary into a seepage model in semi-infinite space. In the process of solving the model, the stratum adopts a conformal mapping coordinate system, and the lining and grouting circle adopt the polar coordinate system, both of which are jointly solved. Then, the distribution formula of the water head in the stratum and the semi-analytical solution to water inflow are obtained, and the rationality of the semi-analytical solution is proven by degenerate checking, numerical simulation calculation and engineering case. Finally, the paper analyzes the sensitivity of the water inflow to various influencing factors and finds that the effect of the distance between the tunnel and the water table is less than that of the construction quality of the waterproof measures and more than that of the distance between the tunnel and the fault.

The availability of engineering models to estimate the risk from space debris is essential for space missions. According to current research, cumulative flux calculation is mostly carried out based on the equal-width interval discretization. The method discretizes the volume around the Earth into cells defined in earth centered inertial coordinates. The resulting debris flux onto a target object is shown to depend on the chosen size of the cells. To avoid a discretization error, this must be accounted for. In order to present reliable flux predictions for space mission, the algorithm improvement is an ongoing topic for the related research field. The aim of this study was to examine the discretization error during the cumulative flux determination process. Both the effect of interval step length and the orbital boundary are under investigation. Several typical orbits are selected as examples here and the 2018/01/03 TLE (Two Line Element) data published by the US Space Surveillance Network is used as the debris background in this paper. Furthermore, the Interval Distance-Based method for Discretization (IDD) is adopted in this paper. A position-centered flux determination method is introduced based on the IDD method. According to the example analysis, the IDD used in the flux calculation process provides results which are less affected by the interval step-size setup; and the orbital boundary has no effect on the calculation process. In other words, the discretization error is significantly reduced. The position-centered method provided a possible suggestion for the improvement of space debris environment models.

This paper proposes a high-performance bidirectional architecture for the quasi-comparison-free sorting algorithm. Our architecture improves the performance of the conventional unidirectional architecture by reducing the total number of sorting cycles via bidirectional sorting along with two auxiliary methods. Bidirectional sorting allows the sorting tasks to be conducted concurrently in the high- and low-index parts of our architecture. The first auxiliary method is boundary finding, which shortens the range for index searching by finding the boundaries of the range. The second auxiliary method is queue storing, which stores each useful index in a queue in advance to reduce the number of miss cycles during index searching. The performance of our architecture highly depends on the distribution of input data. For each set of input data to be sorted, five Gaussian distributions of the input data and four standard derivations for each distribution were adopted in our experiments. The results show that at the expense of some additional area cost, the number of sorting cycles and the energy consumption are significantly reduced by our method.

The paper is concerned with the numerical solution of two-dimensional potential problems, through a mesh-free boundary model in a multi-objective optimization framework that automatically generates Pareto-optimal mesh-free discretization arrangements. This robust new strategy of analysis allows for simultaneously improving the solution accuracy, the conditioning of the numerical solver, the stability and efficiency of the mesh-free analysis. The boundary mesh-free model (BMFM) is built on the boundary integral equation of the Laplace potential, with a moving least squares (MLS) approximation of variables. The model considers independent MLS approximations in each boundary segment and performs integration with standard numerical quadrature. The main novelty of the paper is the automatic generation of Pareto-optimal nodal arrangements and corresponding compact supports of the mesh-free boundary model, by means of an evolutionary multi-objective optimization process, based on genetic algorithms, which uses reliable very efficient objective functions. A benchmark problem is presented to assess the accuracy and efficiency of the modeling strategy. The remarkably accurate results obtained, in perfect agreement with those of analytical solutions, make very reliable this robust new strategy of automatic mesh-free boundary analysis in a multi-objective optimization framework.

The method of fundamental solutions is a more and more popular meshless method for solving boundary or initial–boundary value problems. The most important issue in this method is the determination of the positions of the source points. The accuracy of the method depends strongly on the distribution of the source points. In this paper placement of these points for transient heat conduction problems is studied. The problems are initial–boundary value problems and they are considered in a time-space domain. Because of that, the placement of the source points differs from the classical distribution of the source points for boundary values problems. In the paper, four different possible sources distributions are considered for 1D, 2D and 3D transient heat conduction problems. The results show very good accuracy in case of the source points placed in a space much bigger than the considered region, additionally with the negative time coordinate.

Traditional observation methods of the boundary layer are susceptible to external factors. In particular, detection data are relatively scarce in special areas such as plateaus and oceans. However, occultation observation data have the unique advantage of global coverage, and the boundary layer information of the occultation profile is rich. Based on the machine learning algorithm, this paper combines GPS occultation technology to improve the algorithm and constructs the boundary layer height simulation model based on the actual situation and analyzes its functional structure one by one. Moreover, in order to verify the validity of the simulation model, a simulation analysis is performed in combination with actual data. In addition, the performance of the model is verified from multiple aspects, and the results of the research are calculated by mathematical statistics, and the corresponding statistical diagram is drawn. Through experimental analysis, it can be known that the model constructed in this paper can effectively simulate the boundary height and has certain practical effects.

The paper presents the problem of modeling the uncertainty of the boundary shape in boundary problems (described by Laplace’s equation) and proposes a method for solving so-defined problems. The uncertainty of the boundary shape is modeled using interval numbers. The interval coordinates of control points, defined using classical and directed interval arithmetic, are considered in this paper. However, because of some disadvantages of well-known interval arithmetic, the authors propose a modification of directed interval arithmetic. This arithmetic is also used in the proposed modification of the traditional parametric integral equation system (PIES) method (previously used for exactly defined problems) to solve so-defined problems. Control points of the appropriate curves, necessary to define the boundary shape, are directly included in the mathematical formalism of the mentioned method. The developed interval method is tested on problems of various shapes, modeled with linear and curvilinear segments. The correctness of the obtained solutions is verified using proposed alternative methods. The obtained solutions indicate a very high potential of the proposed method in solving problems with an uncertainly defined boundary shape.

In this paper, the two-zone sound field reproduction problem is investigated with a newly proposed method. The aim of the two-zone reproduction problem is to achieve sound field reproduction in the bright zone while keeping the sound field energy in the dark zone as low as possible. The main difference between the proposed method and the conventional method is that the sound field control in the dark zone is not limited to a small number of measurement points on the region boundary but to measurement points throughout the whole dark zone. For practical reasons, the whole region is discretized into a grid, and the grid selection method is also provided in this paper. Simulations are performed to compare the performances of the proposed and conventional methods with different virtual source directions, and a comparison of the robustness to the regularization parameter shows that the proposed method is less sensitive to the selection of the regularization parameter. Finally, an experiment is performed with a circular loudspeaker array in a real listening room. A quieter dark zone is achieved by the proposed method while keeping the reproduction error in the bright zone close to that of the conventional method, similar to the simulation results. These outcomes verify that the proposed method performs well in both free-field and reverberation environments.

In this paper we give the necessary and sufficient conditions for the Discrete Maximum Principle (DMP) to hold. We prove the convergence of the nonlinear finite element method applied to the logistic equation by using that the Jacobian matrix evaluated in the supersolution, provided by the a priori bound, is a non-singular M-matrix, which is proved in a fast way using both, the positiveness of its principal eigenvalue and the DMP. Meanwhile a positive subsolution provides the coercivity constant. The numerical simulations show that the nonlinear finite element approximate solutions do not oscillate if the DMP is fulfilled. The characterization of the DMP and the mesh sizes guaranteeing the existence of positive sub- and supersolutions of the nonlinear finite element approximate problem, in the case of variable coefficients and all types of boundary conditions are some of the novelties of this paper. The excellent performance of the method is tested in two examples with boundary layers caused by very small diffusion.

In this paper, the one-side physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the physical vacuum boundary, the sound speed is C 1 / 2 -Holder continuous. The coefficient of the time-dependent damping is given by μ ( 1 + t ) λ , ( 0 λ , 0 μ ) which decays by order −λ in time. First we give an one-side physical vacuum background solution whose density and velocity have a growing order with respect to time. Then the main purpose of this paper is to prove the stability of this background solution under the assumption that 0 λ 1 , 0 μ or λ = 1 , 2 μ . The pointwise convergence rate of the density, velocity and the expanding rate of the physical vacuum boundary are also given. The proof is based on the space-time weighted energy estimates, elliptic estimates and the Hardy inequality in the Lagrangian coordinates.

In this paper, I consider the Cauchy problem for the incompressible Navier-Stokes equations in ℝ + n for n ≥ 3 with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial data. This paper is a continuation of my work in my previous papers, where the initial data are considered in T n and ℝ n respectively. In this paper, because of the nonempty boundary in our domain of interest, the details in obtaining the desired result are significantly different and more challenging than the work of my previous papers. This challenges arise due to the possible noncommutativity nature of the Leray projector with the derivatives in the direction of normal to the boundary of the domain of interest. Therefore, we only consider one derivative of the velocity field in that direction.

This paper proposes a new multiobjective optimization with elliptical constraints approach for nonlinear models implemented in a cladding process of ABNT 1020 carbon steel plate using austenitic ABNT 316L stainless steel cored wire. Stainless steel stands out among the cladding materials as it allows obtaining surfaces with determined desirable characteristics from lower cost materials. This kind of process may be controlled by a relatively small number of input variables, i.e., the wire feed rate (WF), voltage (V), welding speed (WS), and the distance from the contact tip to the workpiece (N). Besides that, many outputs can be evaluated and simultaneously optimized. In the present paper, dilution (D), yield (Y), convexity index (CI), and penetration index (PI) were investigated. In order to consider the problem’s multivariate nature, techniques such as factor analysis and Bonferroni’s multivariate intervals were applied combined with elliptical constraints. The response variables were mathematically modeled using Poisson regression, and the obtained results were satisfactory since accurate models were achieved. The normal boundary intersection (NBI) method produced a set of viable configurations for the input variables that allows the experimenter to encounter the best system setup regarding the importance level of each response. Feasible and non-dominated solutions were found which means that the best possible scenario considering all the constraints was reached.

This article settles the issue of adaptive fixed-time control of uncertain nonlinear quantized systems. Different from the traditional study about fixed-time control for uncertain nonlinear systems, quantitative control issue is considered in this paper, and the nonlinear term can be unknown. The new adaptive control tactic of fixed-time tracking control is proposed via fuzzy logic systems approaching unknown nonlinearity, which overcomes the existing limitation of the upper boundary of system settling time relies on the initial condition. The closed-loop system stability is guaranteed in a fixed time. At the end of this paper, the availability of the strategy is proved by a numerical simulation.

In this paper we introduce a cavity reconstructing algorithm for 3D surface scans (CRASS) developed for filling cavities in point clouds representing human body surfaces. The presented method uses Bezier patches to reconstruct missing data. The source of input data for the algorithm was an 8-directional structured light scanning system for the human body. Typical 3D scan representing human body consists of about 1 million points with average sampling density of 1 mm. The paper describes the complete scan processing pipeline: data pre-processing, boundary selection, cavity extraction and reconstruction, and a post-processing step to smooth and resample resulting geometry. The developed algorithm was tested on simulated and scanned 3D input data. Quality assessment was made based on simulated cavities, reconstructed using presented method and compared to original 3D geometry. Additionally, comparison to the state-of-the-art screened Poisson method is presented. Values’ ranges of parameters influencing result of described method were estimated for sample scans and comprehensively discussed. The results of the quantitative assessment of the reconstruction were lower than 0,5 of average sampling density.

The goals of this paper are twofold: selection of pseudo-boundaries for sources nodes in the method of fundamental solutions (MFS), and comparisons of the MFS, the method of particular solutions (MPS) and the MFS-QR of Antunes. To pursue better pseudo-boundaries, we provide new estimates of the condition number (Cond) by the MFS for arbitrary pseudo-boundaries, and propose a new sensitivity index of stability via accuracy. Numerical experiments and comparisons are carried out to verify the analysis made. For five-pedal-flower-like domains, numerical comparisons are made by the sensitivity index. Circular pseudo-boundaries are optimal for highly smooth solutions, but the pseudo-boundaries near the domain boundary may be better for singular solutions. In this paper the gap has been shortened between theoretical analysis and numerical computation of the MFS, to provide some guidance for users. This is the first goal of this paper. The second goal is to compare the MFS, the MPS and the MFS-QR. Characteristics of the MFS-QR are explored. The new basis functions of the MFS-QR are the very particular solutions (PS), and the MFS-QR may be regarded as a special MPS. The MFS-QR is not a variant of the MFS but a variant of the MPS. The MFS-QR also plays a role in bridging from the MFS to the MPS. Both the MFS and the MPS can also be recognized as twins via the MFS-QR in the Trefftz family. The comparisons in this paper are more comprehensive.

Traditional medical image segmentation methods have problems such as low segmentation accuracy and low adaptive ability. Therefore, many scholars have proposed a medical image segmentation method based on deep learning, which has achieved good results in the field of medical image segmentation. However, this type of method has the following problems in the application process: (1) Medical image segmentation target boundary positioning problem. Constrained by factors such as medical image contrast, heterogeneity, and boundary resolution, existing convolution models still cannot accurately locate boundaries. (2) Deep adaptability of deep learning network structure to medical images. Because medical images have more distinct and different feature information than natural images, the current deep learning-based medical segmentation methods have not fully considered this feature. In view of this, this paper proposes a multi-level boundary-aware RUNet segmentation model. The network structure consists of a U-Net-based segmentation network and a multi-level boundary detection network. It can solve the problem of boundary positioning. At the same time, in order to solve the problem of poor adaptability of deep learning network structures to medical images, this paper proposes to introduce a new interactive self-attention module into deep learning models. It can make the feature map get global information, and realize the effective extraction of medical image feature information. It solves the problem of weak matching between the deep learning network structure and medical images. Based on the above ideas, this paper proposes an image segmentation algorithm based on a multi-layer boundary perception-self-attention mechanism deep learning model. This method and other mainstream segmentation algorithms are used to perform experiments on related medical databases. The results show that the proposed method not only improves the segmentation effect significantly compared with traditional machine learning methods, but also improves it to a certain extent compared with other deep learning methods.

This paper investigates the state bounding problem of fuzzy memristive neural networks (FMNNs) with bounded input disturbances. By using the characters of Metzler, Hurwitz and nonnegative matrices, this paper obtains the exact delay-independent and delay-dependent boundary ranges of the solution, which have less conservatism than the results in existing literatures. The validity of the results is verified by two numerical examples.

The Lattice Boltzmann Method (LBM) is a numerical method developed in recent decades. It has the characteristics of high parallel efficiency and simple boundary processing. The basic idea is to construct a simplified dynamic model so that the macroscopic behavior of the model is the same as the macroscopic equation. From the perspective of micro-dynamics, LBM treats macro-physical quantities as micro-quantities to obtain results by statistical averaging. The Finite-difference LBM (FDLBM) is a new numerical method developed based on LBM. The first finite-difference LBE (FDLBE) was perhaps due to Tamura and Akinori and was examined by Cao et al. in more detail. Finite-difference LBM was further extended to curvilinear coordinates with nonuniform grids by Mei and Shyy. By improving the FDLBE proposed by Mei and Shyy, a new finite difference LBM is obtained in the paper. In the model, the collision term is treated implicitly, just as done in the Mei-Shyy model. However, by introducing another distribution function based on the earlier distribution function， the implicitness of the discrete scheme is eliminated, and a simple explicit scheme is finally obtained, such as the standard LBE. Furthermore, this trick for the FDLBE can also be easily used to develop more efficient FVLBE and FELBE schemes. To verify the correctness and feasibility of this improved FDLBM model, which is used to calculate the square cavity model, and the calculated results are compared with the data of the classic square cavity model. The comparison result includes two items: the velocity on the centerline of the square cavity and the position of the vortex center in the square cavity. The simulation results of FDLBM are very consistent with the data in the literature. When Re=400, the velocity profiles of u and v on the centerline of the square cavity are consistent with the data results in Ghia's paper, and the vortex center position in the square cavity is also almost the same as the data results in Ghia's paper. Therefore, the verification of FDLBM is successful and FDLBM is feasible. This improved method can also serve as a reference for subsequent research.

Purpose The purpose of this paper is to implement the idea of the sphere on systems theory (type: Niklas Luhmann) and to test whether this theory can be used to gain a sharper approach to the complex of ideas of the sphere. The main theme is “the conceptual blurriness of the word sphere.” This essay analyzes the societal and historical semantics of the concept of sphere and nearby notions in the thought traditions in medieval society until the modern, i.e. the functionally differentiated society. Nearby notions as limitationality and boundary are discussed, as well as sphere as a “lexicalized metaphor.” At the end, the paper turns to the concept of sphere in modern society that is based on the distinction “public/private” and applies the discussion to the system of mass media. Design/methodology/approach This paper can be classified as a “conceptual essay” that draws on basic systems theoretical concepts. Findings The result of the theoretical considerations is that the word “sphere” represents nothing more than a “catchword” and can hardly lay the groundwork for empirical and theoretical research that would be instructive with regard to the new media and their massive influence on society. Originality/value This paper was researched and written solely by the author. All sources are clearly identified.

The paper is concerned with the classical Flamant problem of the theory of elasticity. The classical solution of this problem obtained by A. Flamant in the nineteenth century specifies the stresses and the displacements induced in a half-plane by a concentrated force applied to the half-plane boundary in the normal direction. Being presented in numerous textbooks in the theory of elasticity, this solution provides results that can be qualified as paradoxical and which traditionally are left without proper comments in the literature. Particularly, the half-plane displacements are singular at the point of the force application and are infinitely increasing with a distance from this point. The displacement of the boundary in the boundary direction is not continuous and experiences a jump at the point of the force application. A boundary element under any force, irrespective of how low it is, rotates at the point of the force application by 90 $$^{\circ }$$ which does not correspond to basic assumptions of the linear theory of elasticity; hence, the classical solution cannot be qualified as consistent. The consistent solution of the problem is constructed in the paper on the basis of the generalized theory of elasticity the equations of which are obtained for the solid element that has small but not infinitesimal dimensions. As a result, these equations provide regular solutions of the problems that are singular in the classical theory of elasticity. The obtained generalized solution of the Flamant problem demonstrates regular behavior of the displacements which are not singular at the point of the force application. The solution is supported by experimental results obtained for the plate of silicon rubber simulating the half-plane.

This paper deals with the robust stability analysis problem of linear time-invariant systems with a state delay and polytopic uncertainty. The delay parameter is constant which does not limited by any upper bound. The system’s matrices linearly depend on uncertain parameters which belong to the unit simplex known as the uncertain space. An extended uncertain system is introduced based on the original delay model that contains the uncertain parameters and two slack parameters. The slack parameters belong to the boundary of the unit circle in two dimensional real space. Then, it has been proven that the delay system is robust stable if and only if the extended uncertain system is robust stable over the uncertain space and the unit circle’s boundary. Since the unit circle’s boundary is not convex, a containing polygonal space is developed that consists of a group of polygonal subspaces. This paper proposes a novel algorithm to check system’s robust stability via linear dependent Lyapunov functions that correspondingly defined for each polygonal subspace. The algorithm is able to establish robust stability of the model for all positive values of the delay parameter. There simulation examples are provided to clarify the notations and facts of this paper. First example presents the notation and results in an instance uncertain delay system. Two extensive examples compare feasibility performances of the proposed algorithm to some existing methods that reveal the superiority of the prosed algorithm.

We assume that $$n\ge 3$$ , $$u \in C^{2}( \mathbb {B}^{n},\mathbb {R}^{n}) \cap C(\overline{\mathbb {B}}^{n},\mathbb {R}^{n})$$ is a solution to the hyperbolic Poisson equation $$\Delta _{h}u=\psi $$ in $$\mathbb {B}^{n}$$ with the boundary condition $$u|_{\mathbb {S}^{n-1}}=\phi $$ , where $$\Delta _{h}$$ is the hyperbolic Laplace operator and $$\psi \in C( \mathbb {B}^{n},\mathbb {R}^{n})$$ . In Chen et al. (Calc Var Partial Differ Equ 57:32, 2018), the first, the second, and the last author of this paper, together with Rasila, studied expressions of u, and proved that $$u=P_{h}[\phi ]-G_{h}[\psi ]$$ , where $$P_{h}[\phi ]$$ and $$G_{h}[\psi ]$$ denote the Poisson integral of $$\phi $$ and the Green integral of $$\psi $$ with respect to $$\Delta _h$$ , respectively. With the assumption $$|\psi (x)|\le M(1-|x|^{2})$$ $$(M\ge 0$$ is a constant), the Lipschitz-type continuity of u was also investigated. As a continuation, in this paper, we first consider the existence of the solutions, and demonstrate that if $$\phi \in L^{\infty }(\mathbb {S}^{n-1}, \mathbb {R}^{n})$$ , $$\psi \in L^{\infty }(\mathbb {B}^{n}, \mathbb {R}^{n})$$ , $$\int _{\mathbb {B}^{n}} (1-|x|^2)^{n-1} |\psi (x)|\mathrm{d}\tau (x)