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Here, we study the large-time limit of viscosity solutions of the Cauchy problem for second-order Hamilton–Jacobi–Bellman equations with convex Hamiltonians in the torus. This large-time limit solves the corresponding stationary problem, sometimes called the ergodic problem. This problem, however, has multiple viscosity solutions and, thus, a key question is which of these solutions is selected by the limit. Here, we provide a representation for the viscosity solution to the Cauchy problem in terms of generalized holonomic measures. Then, we use this representation to characterize the large-time limit in terms of the initial data and generalized Mather measures. In addition, we establish various results on generalized Mather measures and duality theorems that are of independent interest.

We revisit the Riemann-Hilbert problem determined by Donaldson-Thomas invariants for the resolved conifold and for other small crepant resolutions. While this problem can be recast as a system of TBA-type equations in the conformal limit, solutions are ill-defined due to divergences in the sum over infinite trajectories in the spectrum of D2-D0-brane bound states. We explore various prescriptions to make the sum well-defined, show that one of them reproduces the existing solution in the literature, and identify an alternative solution which is better behaved in a certain limit. Furthermore, we show that a suitable asymptotic expansion of the $\tau$ function reproduces the genus expansion of the topological string partition function for any small crepant resolution. As a by-product, we conjecture new integral representations for the triple sine function, similar to Woronowicz' integral representation for Faddeev's quantum dilogarithm., Comment: 25+14 pages; added proof for integral representation of the double sine function and a similar conjecture for the triple sine; version accepted for publication in Annales Henri Poincar\'e

We study higher order expansions both in the Berry-Ess\'een estimate (Edgeworth expansions) and in the local limit theorems for Birkhoff sums of chaotic probability preserving dynamical systems. We establish general results under technical assumptions, discuss the verification of these assumptions and illustrate our results by different examples (subshifts of finite type, Young towers, Sinai billiards, random matrix products), including situations of unbounded observables with integrability order arbitrarily close to the optimal moment condition required in the i.i.d. setting., Comment: 66 pages, 2 figures

In this paper, we investigate the solvability, regularity and the vanishing dissipation limit of solutions to the three-dimensional viscous magnetohydrodynamic (MHD) equations in bounded domains. On the boundary, the velocity field fulfills a Navier-slip condition, while the magnetic field satisfies the insulating condition. It is shown that the initial boundary value problem has a global weak solution for a general smooth domain. More importantly, for a flat domain, we establish the uniform local well-posedness of the strong solution with higher-order uniform regularity and the asymptotic convergence with a rate to the solution of the ideal MHD equation as the dissipations tend to zero.

I analyze the role of infinite idealizations used in the renormalization group (RG hereafter) method in explaining universality across microscopically different physical systems in critical phenomena. I argue that despite the reference to infinite limit systems such as systems with infinite correlation lengths during the RG process, the key to explaining universality in critical phenomena need not involve infinite limit systems. I develop my argument by introducing what I regard as the explanatorily relevant property in RG explanations: linearization* property; I then motivate and prove a proposition about the linearization* property in support of my view. As a result, infinite limit systems in RG explanations are dispensable.

We investigate strong stability preserving (SSP) implicit-explicit (IMEX) methods for partitioned systems of differential equations with stiff and nonstiff subsystems. Conditions for order p and stage order $$q=p$$ q = p are derived, and characterization of SSP IMEX methods is provided following the recent work by Spijker. Stability properties of these methods with respect to the decoupled linear system with a complex parameter, and a coupled linear system with real parameters are also investigated. Examples of methods up to the order $$p=4$$ p = 4 and stage order $$q=p$$ q = p are provided. Numerical examples on six partitioned test systems confirm that the derived methods achieve the expected order of convergence for large range of stepsizes of integration, and they are also suitable for preserving the accuracy in the stiff limit or preserving the positivity of the numerical solution for large stepsizes.

The logic $${\mathbb{L}}_\theta ^1$$ was introduced in [She12]; it is the maximal logic below $${{\mathbb{L}}_{\theta, \theta}}$$ in which a well ordering is not definable. We investigate it for θ a compact cardinal. We prove that it satisfies several parallels of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, two models are $${\mathbb{L}}_\theta ^1$$ -equivalent iff for some ω-sequence of θ-complete ultrafilters, the iterated ultrapowers by it of those two models are isomorphic. Also for strong limit λ>θ of cofinality $${\aleph _0}$$ , every complete $${\mathbb{L}}_\theta ^1$$ -theory has a so-called special model of cardinality λ, a parallel of saturated. For first order theory T and singular strong limit cardinal λ, T has a so-called special model of cardinality λ. Using “special” in our context is justified by: it is unique (fixing T and λ), all reducts of a special model are special too, so we have another proof of interpolation in this case.

The (2 + 1)-dimensional generalized Bogoyavlensky-Konopelchenko equation is a significant physical model. By using the long wave limit method and confining the conjugation conditions on the interrelated solitons, the general M-lump, high-order breather, and localized interaction hybrid solutions are investigated, respectively. Then we implement the numerical simulations to research their dynamical behaviors, which indicate that different parameters have very different dynamic properties and propagation modes of the waves. The method involved can be validly employed to get high-order waves and study their propagation phenomena of many nonlinear equations.

This study aims to propose state-of-the-art techniques in predicting and controlling slope stability in open-pit mines based on limit equilibrium analysis, artificial neural networks, and optimization algorithms. Accordingly, the simplified Bishop method was used to analyze the slope stability of an open-pit coal mine through the limit equilibrium analysis method. Various rock mass properties and the geometrical parameters of the slopes were considered, such as bench height, slope angle, unit weight, cohesion, and friction angle. Finally, 495 cases were analyzed to compute the factor of safety (FOS). Subsequently, the radial basis function neural network (RBFNN) model was applied to predict FOS. In order to optimize the RBFNN model, the brainstorm optimization (BSO) algorithm was applied to train the RBFNN model, named as BSO-RBFNN model. The genetic algorithm (GA)-RBFNN, RBFNN (without optimization), and multiple layers perceptron (MLP) neural network were also developed to predict FOS and compared with the proposed BSO-RBFNN model as part of the study. The results revealed that the optimization of the BSO algorithm and RBFNN model provided a state-of-the-art technique (i.e., BSO-RBFNN) for predicting and controlling slope stability with high accuracy (i.e., mean absolute error (MAE) = 0.047, root-mean-squared error (RMSE) = 0.057, determination coefficient (R2) = 0.929, variance accounted for (VAF) = 92.948), and reliability (i.e., absolute error of 5.89% for 80% of cases in practice). Comparisons also indicated that the proposed BSO-RBFNN model is the most dominant model for predicting slope stability in this study (i.e., MAEGA-RBFNN = 0.048, RMSEGA-RBFNN = 0.060, R2GA-RBFNN = 0.927, VAFGA-RBFNN = 92.534; MAERBFNN = 0.064, RMSERBFNN = 0.081, R2RBFNN = 0.925, VAFRBFNN = 89.189; MAEMLP = 0.065, RMSEMLP = 0.081, R2MLP = 0.873, VAFMLP = 85.724). Furthermore, the slope angle and bench height should be taken into account to control slope stability in practical engineering based on the proposed BSO-RBFNN model.

The paper deals with the limit behavior of a compound Poisson process with switching between a finite number of sequences of i.i.d. random variables. The switching is provided by Bernoulli’s random variables. Under suitable normalization, the limit process is a Brownian motion with switching variance.

We introduce notions of generalized $$\varepsilon $$ -quasi solutions to approximate set type solutions of set optimization problems. We study their properties, consistency and limit behavior as approximations to efficient and strict weak efficient solutions. Moreover, we prove an existence result for such solutions and a bound for their asymptotic cone. Finally, we obtain optimality conditions for them.

We prove a limit theorem on the convergence of mathematical expectations of functionals of sums of independent random variables to the Cauchy problem solution for the nonstationary Schr¨odinger equation with a symmetric fractional derivative operator of order $$ \upalpha\ \upepsilon \bigcup_{m=3}^{\infty}\left(m-1,m\right) $$ in the righthand side.

The paper is concerned with the two-machine flow shop, where each job requires an additional resource (referred to as storage space) from the start of its first operation till the end of its second operation. The storage requirement of a job is determined by the processing time of its first operation. At any point in time, the total consumption of this additional resource cannot exceed a given limit (referred to as the storage capacity). The goal is to minimise the makespan, i.e. to minimise the time needed for the completion of all jobs. This problem is NP-hard in the strong sense. The paper analyses how the parameter - a lower bound on the storage capacity specified in terms of the processing times, affects the computational complexity.

We present a unified approach to obtain sharp mean-squared and multiplicative inequalities of Hardy-Littlewood-Polya and Taikov types for multiple closed operators acting on Hilbert space. We apply our results to establish new sharp inequalities for the norms of powers of the Laplace-Beltrami operators on compact Riemannian manifolds and derive the well-known Taikov and Hardy-Littlewood-Polya inequalities for functions defined on the d-dimensional space in the limit case. Other applications include the best approximation of unbounded operators by linear bounded ones and the best approximation of one class by elements of another class. In addition, we establish sharp Solyar type inequalities for unbounded closed operators with closed range.

In this paper, meant as a companion to arXiv:2006.04458, we consider a class of non-integrable $2D$ Ising models in cylindrical domains, and we discuss two key aspects of the multiscale construction of their scaling limit. In particular, we provide a detailed derivation of the Grassmann representation of the model, including a self-contained presentation of the exact solution of the nearest neighbor model in the cylinder. Moreover, we prove precise asymptotic estimates of the fermionic Green's function in the cylinder, required for the multiscale analysis of the model. We also review the multiscale construction of the effective potentials in the infinite volume limit, in a form suitable for the generalization to finite cylinders. Compared to previous works, we introduce a few important simplifications in the localization procedure and in the iterative bounds on the kernels of the effective potentials, which are crucial for the adaptation of the construction to domains with boundaries., 69 pages. Minor revisions. Version accepted by Annales Henri Poincar\'e. arXiv admin note: substantial text overlap with arXiv:2006.04458

We consider a model whereby a given response variable Y following a transformation $${{\mathcal {Y}}}:=\mathcal {T}(Y)$$ , satisfies some classical regression equation. In this transformation model the form of the transformation is specified analytically but incorporates an unknown transformation parameter. We develop testing procedures for the null hypothesis of homoskedasticity for versions of this model where the regression function is considered either known or unknown. The test statistics are formulated on the basis of Fourier-type conditional contrasts of a variance computed under the null hypothesis against the same quantity computed under alternatives. The limit null distribution of the test statistic is studied, as well as the behaviour of the test criterion under alternatives. Since the limit null distribution is complicated, a bootstrap version is suggested in order to actually carry out the test procedures. Monte Carlo results are included that illustrate the finite-sample properties of the new method. The applicability of the new tests on real data is also illustrated.

Predictions of future water resources are essential for the strategic plans of any country especially in arid regions such as Saudi Arabia (SA). This paper presents a modeling study of the temporal annual rainfall variability over SA, evaluating the best model for future rainfall predictions and mapping spatiotemporal rainfall over SA. Common time series models of orders p and q such as autoregressive, AR(p), moving average, MA(q), and the combined autoregressive-moving average, ARMA(p,q), models are utilized. The models are applied to 28 metrological stations distributed over SA. Spatiotemporal statistical analysis of rainfall data is performed over a period between 1970 and 2012. The minimum and maximum fitted parameters of the models are φ1 = − 0.55, 0.46 for ARMA (1,0), θ1 = − 0.66, 0.17 for ARMA (0,1) and φ1 = − 0.84, 0.94, θ1 = − 0.87, 0.78 for ARMA (1,1), respectively. It has been shown that ARMA (1,0) is the best to model the temporal variability based on the Akaike information criterion (AIC), the correlation coefficient (R), and the root mean square error (RMSE). The Monte Carlo method is used to make future predictions (100 realizations) with the confidence levels (CIs) based on ARMA (1,0). Spatial distribution of the ensemble predictions and their CIs are presented graphically at the upper limit of 95%, 97.5%, and 99% and the lower limit of 5%, 2.5%, and 1% confidence, respectively, for the year 2030 to help decision-makers for future water resources planning of the country. Abha city has the highest annual rainfall prediction in 2030 (221 mm) with upper confidences (436, 533, and 643 mm) for 95%, 97.5%, and 99%, respectively. The prediction results indicate that the high mountainous areas (Asir and Taif) are expected to have more rainfall in the future than the rest of the regions in SA. The use of non-traditional water resources is the solution to future challenges.

We analyze the stability of elastic systems composed of the shells of revolution with variable curvature and complex structures in the field of conservative axisymmetric loads of different nature. Within the framework of classical and refined theories of shells, we determine the limit and bifurcation critical values of the acting loads based on the geometrically nonlinear statement of the problem and a criterion of dynamic stability. To solve the corresponding nonlinear and eigenvalue problems, we propose to use a numerical-analytic approach based on their rational reduction to one-dimensional linear boundaryvalue problems in the meridional coordinate and their numerical solution by the discrete-orthogonalization method. We present test examples that confirm the applicability of the proposed procedure to the analyzed class of problems. The limit and bifurcation values of the critical loads in the shell system are analyzed depending on its geometric parameters.

In this work we show how it is possible to derive a new set of nudging equations, a tool still used in many data assimilation problems, starting from statistical physics considerations and availing ourselves of stochastic parameterizations that take into account unresolved interactions. The fluctuations used are thought of as Gaussian white noise with zero mean. The derivation is based on the conditioned Langevin dynamics technique. Exploiting the relation between the Fokker–Planck and the Langevin equations, the nudging equations are derived for a maximally observed system that converges towards the observations in finite time. The new nudging term found is the analog of the so called quantum potential of the Bohmian mechanics. In order to make the new nudging equations feasible for practical computations, two approximations are developed and used as bases from which extending this tool to non-perfectly observed systems. By means of a physical framework, in the zero noise limit, all the physical nudging parameters are fixed by the model under study and there is no need to tune other free ad-hoc variables. The limit of zero noise shows that also for the classical nudging equations it is necessary to use dynamical information to correct the typical relaxation term. A comparison of these approximations with a 3DVar scheme, that use a conjugate gradient minimization, is then shown in a series of four twin experiments that exploit low order chaotic models.

A partial-limit method is developed to understand solutions related to real eigenvalues for the Fokas–Lenells equation. By applying a partial-limit procedure to soliton solutions of the Fokas–Lenells equation, new multiple-pole solutions related to real repeated eigenvalues are obtained. For the envelop $$|u|^2$$ , the simplest solution corresponds to a real double eigenvalue, showing a solitary wave with algebraic decay. Two such solitons allow elastic scattering but asymptotically with zero phase shift. Single eigenvalue with higher multiplicity gives rise to rational solutions which contain an intrinsic parameter, live on a zero background, and have slowly changing amplitudes. The partial-limit procedure may be extended to other integrable systems and generate new solutions.

We consider a nonlinear nonautonomous system of two ordinary differential equations with a stable fixed point and assume that the non-Hamiltonian part of the system tends to zero at infinity. We examine the asymptotic behavior of a two-parameter family of solutions that start from a neighborhood of the stable equilibrium. The proposed construction of asymptotic solutions is based on the averaging method and the transition in the original system to new dependent variables, one of which is the angle of the limit Hamiltonian system, and the other is the Lyapunov function for the complete system.

To reduce the adverse vibrations of buildings, tuned mass dampers (TMDs), which are the most representative passive control devices, have been widely used and studied in aerospace, machinery, civil engineering, and other fields for many years. Most scholars used to treat the TMD as a linear damper, but they show some nonlinear characteristics owing to the use of limit devices and large displacements. It is necessary to consider the nonlinear coefficient of the TMD when designing its parameters. In this study, the mass ratio of the TMD was optimized with considering the nonlinear coefficient of the TMD. The complex variable average method and multiscale method were used for analysis. A mass ratio interval was found on the “ε- N2” curve in which modulation response can occur, and then an analytical method for obtaining the optimal mass ratio of TMD was derived based on this phenomenon. The numerical results showed that taking the midpoint of this mass ratio interval as the optimal mass ratio can yield a better damping effect and robustness than using the traditional linear design method.

There has been much research about regularizing optimal portfolio selections through $$\ell _1$$ norm and/or $$\ell _2$$ -norm squared. The common consensuses are (i) $$\ell _1$$ leads to sparse portfolios and there exists a theoretical bound that limits extreme shorting of assets; (ii) $$\ell _2$$ (norm-squared) stabilizes the computation by improving the condition number of the problem resulting in strong out-of-sample performance; and (iii) there exist efficient numerical algorithms for those regularized portfolios with closed-form solutions each step. When combined such as in the well-known elastic net regularization, theoretical bounds are difficult to derive so as to limit extreme shorting of assets. In this paper, we propose a minimum variance portfolio with the regularization of $$\ell _1$$ and $$\ell _2$$ norm combined (namely $$\ell _{1, 2}$$ -norm). The new regularization enjoys the best of the two regularizations of $$\ell _1$$ norm and $$\ell _2$$ -norm squared. In particular, we derive a theoretical bound that limits short-sells and develop a closed-form formula for the proximal term of the $$\ell _{1,2}$$ norm. A fast proximal augmented Lagrange method is applied to solve the $$\ell _{1,2}$$ -norm regularized problem. Extensive numerical experiments confirm that the new model often results in high Sharpe ratio, low turnover and small amount of short sells when compared with several existing models on six datasets.

The lump solutions and interaction solutions are mainly investigated for the (2 + 1)-dimensional KPI equation. According to relations of the undetermined parameters of the test functions, the N-soliton solutions are showed by computations of the Maple using the Hirota bilinear form for (2 + 1)-dimensional KPI equation. One type of the lump solutions for (2 + 1)-dimensional KPI equation has been deduced by the limit method of the N-soliton solutions. In addition, the interaction solutions between the lump and N-soliton solutions of it are studied by the undetermined interaction functions. The sufficient conditions for the existence of the interaction solutions are obtained. Furthermore, the new breather solutions for the (2 + 1)-dimensional KPI equation are considered by the homoclinic test method via new test functions including more parameters than common test functions.

We consider a mathematical model with five compartments relevant to depict the feature of a certain type of epidemic transmission. We aim to identify some system parameters by means of a minimization problem for a functional involving available measurements for observable compartments, which we treat by an optimal control technique with a state constraint imposed by realistic considerations. The proof of the maximum principle is done by passing to the limit in the conditions of optimality for an appropriate approximating problem. The proof of the estimates for the dual approximating system requires a more challenging treatment since the trajectories are not absolutely continuous, but only with bounded variation. These allow to pass to the limit to obtain the conditions of optimality for the primal problem and lead to a singular dual backward system with a generalized solution in the sense of measure. As far as we know, this approach developed here for the identification of parameters in an epidemic model considering a state constraint related to the actions undertaken for the disease containment was not addressed in the literature and represents a novel issue in this paper.

In this paper, our discussion mainly focuses on equations with energy supercritical nonlinearities. We establish probabilistic global well-posedness (GWP) results for the cubic Schr\"odinger equation with any fractional power of the Laplacian in all dimensions. We consider both low and high regularities in the radial setting, in dimension $\geq 2$. In the high regularity result, an {\it Inviscid - Infinite dimensional (IID) limit} is employed while in the low regularity global well-posedness result, we make use of the Skorokhod representation theorem. The IID limit is presented in details as an independent approach that applies to a wide range of Hamiltonian PDEs. Moreover we discuss the adaptation to the periodic settings, in any dimension, for smooth regularities., Comment: 39 pages

We study the influence of decaying perturbations on autonomous oscillatory systems in a plane under the assumption that the perturbations preserve the equilibrium state of the limit system, oscillate with asymptotically constant frequency, and satisfy the nonresonance condition. We discuss the long-term behavior of the perturbed trajectories in a neighborhood of the equilibrium state. We describe conditions on the perturbation parameters that guarantee preservation or loss of stability of the equilibrium. The results are illustrated by an example of decaying perturbations of the Duffing oscillator.

We investigate a particle system with mean field interaction living in a random environment characterized by a regime-switching process. The switching process is allowed to be dependent on the particle system. The well-posedness and various properties of the limit conditional McKean-Vlasov SDEs are studied, and the conditional propagation of chaos is established with explicit estimate of the convergence rate.

We study the effect of the unfolding of a reducible double confluent Heun equation from the point of view of the Stokes phenomenon. We introduce a small complex parameter e that splits together the non-resonant singular points x = 0 and $x=\infty $ into four different Fuchsian singularities $x_{L}=-\sqrt {\varepsilon }, x_{R}=\sqrt {\varepsilon }$ , and $x_{LL}=-1/\sqrt {\varepsilon }, x_{RR}=1/\sqrt {\varepsilon }$ , respectively. The perturbed equation is a symmetric general Heun equation and its general solution depends analytically on $\sqrt {\varepsilon }$ . Then we prove that when the perturbed equation has exactly two resonant singularities of different type, all the Stokes matrices of the initial double confluent Heun equation are realized as a limit of the upper-triangular parts of the monodromy matrices of the perturbed equation when $\sqrt {\varepsilon } \rightarrow 0$ . To establish this result we combine a direct computation with a theoretical approach.

In the present paper, the enumeration of a certain class of directed lattice paths is based on the analysis of dynamical correlation functions of the exactly solvable XX0 model. This model is the zero anisotropy limit of one of the basic models of the theory of integrable systems, the XXZ Heisenberg magnet. It is demonstrated that the considered correlation functions under different boundary conditions are the exponential generating functions of various types of paths, in particular, Dyck and Motzkin paths.

The six-vertex model on a finite square lattice with the so-called partial domain wall boundary conditions is considered. For the case of rational Boltzmann weights, the polarization on the free boundary of the lattice is computed. For the finite lattice the result is given in terms of a ratio of determinants. In the limit, where the side of the lattice with free boundary tends to infinity (the limit of a semi-infinite lattice), they are simplified and can be evaluated in a closed form.

There are a number of situations in which rescaled interacting particle systems have been shown to converge to a reaction diffusion equation (RDE) with a bistable reaction term, see e.g., Cox et al. (Asterisque 349:1–127, 2013), Durrett (Ann Appl Prob 19:477–496, 2009, Electron J Probab 19:1–64, 2014), Durrett and Neuhauser (Ann Probab 22:289–333, 1994). These RDEs have traveling wave solutions. When the speed of the wave is nonzero, block constructions have been used to prove the existence or nonexistence of nontrivial stationary distributions. Here, we follow the approach in a paper by Etheridge et al. (Electron J Probab 22:1–40, 2017) to show that in a wide variety of examples when the RDE limit has a bistable reaction term and traveling waves have speed 0, one can run time faster and further rescale space to obtain convergence to motion by mean curvature. This opens up the possibility of proving that the sexual reproduction model with fast stirring has a discontinuous phase transition, and that in Region 2 of the phase diagram for the nonlinear voter model studied by Molofsky et al. (Theoret Pop Biol 55(1999):270–282, 1999) there were two nontrivial stationary distributions.

The existence of a smooth Lyapunov function for any smooth planar dynamical system with one limit cycle is proved, which is based on a novel decomposition of the dynamical system from the perspective of mechanics and some definitions and known theorems. By considering the smooth simple closed curve in the complex plane corresponding to the limit cycle of a smooth planar dynamical system and the definition of Morse decomposition, we first prove a theorem in which the limit cycle is diffeomorphic to the unit circle for any smooth planar dynamical system with one limit cycle, and we deduce another theorem, that any two smooth planar systems with one limit cycle are diffeomorphic (or smoothly equivalent). Next, through the definition of the potential function, the explicit construction of a smooth Lyapunov function for a smooth planar dynamical system with one circular limit cycle is given. Then, according to these results, we obtain the following theorem: there always exists a smooth Lyapunov function for any smooth planar dynamical system with one limit cycle. Additionally, two examples are given. Finally, with respect to the coexistence of the limit cycle and Lyapunov function, we discuss two criteria related to system dissipation (divergence and dissipative power) in an example, find that they are not consistent, and explain the meaning of dissipation in infinitely repeated motion in the limit cycle. This result may provide a deeper understanding of the existence of a Lyapunov function for systems with limit cycles.

For the plain Polya urn with two colors, black and white, we prove a functional central limit theorem for the number of white balls, assuming that the initial number of black balls is large. Depending on the initial number of white balls, the limit is either a pure birth process or a diffusion.

We consider the Robinson-Schensted-Knuth algorithm applied to a random input and investigate the shape of the bumping route (in the vicinity of the $y$-axis) when a specified number is inserted into a large Plancherel-distributed tableau. We show that after a projective change of the coordinate system the bumping route converges in distribution to the Poisson process., Comment: version 2: 54 pages

Recently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

The performance of the Scheduled Relaxation Jacobi method for levels as high as 25 and mesh size as large as $$4096\times 4096$$ is studied in the present work. The optimal values for the relaxation parameters have been obtained using a search procedure proposed in an earlier study by the same author after suitable modifications and improvements which allow it to be used for the present purpose. It is shown that for a given number of levels and mesh size, the theoretical spectral radius and the speed-up of the method improve as the upper limit for the value of the relaxation factor is increased, albeit at the cost of numerical stability. Since the values for the over-relaxation factors for the cases considered here are high and there are more than one under-relaxation factor, a proper sequencing of the over- and under-relaxed iterations is essential and a strategy for the same is proposed. The five level method with suitably determined optimal values is shown to perform better than a twenty five level method, when used for solving the Neumann–Laplace problem, despite the theoretical spectral radius of the former being less than that of the latter. The numerical calculations demonstrate that the SRJ method with optimal parameters can achieve a reduction in the initial error by four orders of magnitude within 3000 iterations even for a 4096 $$\times $$ 4096 mesh.

We consider Whittle estimation for the parameters of a stationary solution of a continuous-time linear state space model sampled at low frequencies. In our context, the driving process is a Levy process which allows flexible margins of the underlying model. The Levy process is supposed to have finite second moments. Then, the classes of stationary solutions of linear state space models and of multivariate CARMA processes coincide. We prove that the Whittle estimator, which is based on the periodogram, is strongly consistent and asymptotically normal. A comparison with ARMA models shows that in the continuous-time setting the limit covariance matrix of the estimator has an additional term for non-Gaussian models. Thereby, we investigate the asymptotic normality of the integrated periodogram. Furthermore, for univariate processes we introduce an adjusted version of the Whittle estimator and derive its asymptotic properties. The practical applicability of our estimators is demonstrated through a simulation study.

A new probability distribution, called the beta–epsilon distribution, is introduced in this study and shown to possess desirable statistical properties for application in reliability studies. Its hazard rate function is bathtub shaped, and exhibits differing shapes with increasing value of its upper support parameter, $$\delta $$ . The boundedness of the upper limit describes its potential to model finite lifetimes of systems, organisms and humans. Two real life applications in this study provided good fit to the data and further buttress the potential of the application of the new distribution.

For delayed neural networks with randomly occurring uncertainties (ROU), this paper uses an improved integral inequality to optimize the stability of the receding horizon. The ROU follows some uncorrelated Bernoulli distribution white noise sequence, which it can enter the neural network in a free and random manner. By using a suitable lemma, the ROU problem added in this paper is transformed into a linear matrix inequality. Based on the auxiliary function-based integral inequality method, the new cross terms matrix of linear matrix inequality in the improved Lyapunov-Krasovskii functional is processed. Therefore, some new matrix variables containing more information are generated, so that the results have more degrees of freedom. This paper has obtained the new condition of the end-weighting matrix of the receding horizon cost function, thereby reducing its conservativeness and increasing its upper limit of delay. Finally, the superiority of the method has be illustrated by giving some simulation numerical examples.

We prove a Berry-Esseen theorem, a local central limit theorem and (local) large and (global) moderate deviations principles for i.i.d. (uniformly) random non-uniformly expanding or hyperbolic maps with exponential first return times. Using existing results the problem is reduced to certain random (Young) tower extensions, which is the main focus of this paper. On the random towers we will obtain our results using contraction properties of random complex equivariant cones with respect to the complex Hilbert projective metric., Comment: 35 pages, accepted to Ann. Henri Poincar\'e

We study the models of multicomponent discrete dynamical conflict systems with attractive interaction characterized by a positive value called the attractor index. The existence of limit equilibrium states in these systems is proved and their description in terms of the attractor index is given. An explicit relationship between the limit state of the system and the attractor index is established. A series of concrete examples is presented. They illustrate the dynamics of the system for different attractor indices.

We study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; ‘slow’ rates are constant, and ‘fast’ rates are scaled as $$1/\epsilon $$ 1 / ϵ , and we prove the convergence in the fast-reaction limit $$\epsilon \rightarrow 0$$ ϵ → 0 . We establish a $$\Gamma $$ Γ -convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterises both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the $$\Gamma $$ Γ -convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.

Shaft boring machine (SBM) plays a significant role in mechanized construction of deep and large shafts. However, surrounding soil instability of SBM pose great threats to the shaft construction. Therefore, it is necessary to analyse the soil stability underlying SBM’s gripper. On the basis of the first SBM (MSJ5.8/1.6D) that independently developed in China, the interaction between the SBM’s gripper and surrounding soil is investigated. According to the failure mode of the shaft obtained by numerical simulation, the geometry of the failure surfaces is first derived, and the stresses distribution on the failure surfaces are clearly presented. Then, the ultimate bearing capacity formula of the surrounding soil underlying SBM’s gripper is derived based on plastic limit equilibrium theory with Mohr-Coulomb criterion. The change rules of bearing capacity factors and the influence rules of basic parameters in this formula are obtained. Finally, the theoretical values in typical soils are compared with those from the numerical simulation to illustrate the rationality of the formula. The results of this investigation not only provide an advanced method for evaluating the surrounding soil stability of SBM in shaft construction, but also provide guidelines for the design and construction risk control of SBM.

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).

We prove a distributional limit theorem conjectured in [Journal of Statistical Physics 174, No. 6, 1372-1403 (2019)] for partition functions defining models of directed polymers on diamond hierarchical graphs with disorder variables placed at the graphical edges. The limiting regime involves a joint scaling in which the number of hierarchical layers, $n\in \mathbb{N}$, of the graphs grows as the inverse temperature, $\beta\equiv \beta(n)$, vanishes with a fine-tuned dependence on $n$. The conjecture pertains to the marginally relevant disorder case of the model wherein the branching parameter $b \in \{2,3,\ldots\}$ and the segmenting parameter $s \in \{2,3,\ldots\}$ determining the hierarchical graphs are equal, which coincides with the diamond fractal embedding the graphs having Hausdorff dimension two. Unlike the analogous weak-disorder scaling limit for random polymer models on hierarchical graphs in the disorder relevant $b, Comment: 79 pages, 2 figures. Additional results have been added on the vertex-disorder version of the model (Theorem 3.1) and the rate of distributional convergence (Theorem 7.3). Various corrections and improvements to the presentation have been made in the text based on the suggestions given by three anonymous referees

For the smoke filling in enclosures with floor leaks and growing fires, a comprehensive governing equation has been set up by Copper. However, this governing equation can hardly be exactly solved analytically because it is inseparable. Up to now, the existing analytical solutions for it are all approximate, including the so-called SFPE solution, the Deli-2 solution, the Deli-3 solution, etc. Among these approximate solutions, the SFPE solution almost always overestimates the smoke-interface height, while the Deli-2 solution almost always underestimates the interface height. In general, the Deli-3 solution is the most accurate but it can generate considerable error when the smoke interface is near the floor level. In this note, a new approximate solution is derived, which is actually a correction of the SFPE solution. The basic idea of the new solution is to resume the effect of the expansion term that is neglected in the deduction of the SFPE solution. It is found that, while the smoke interface in not very close to the floor level, the accuracy of the new solution is generally of the same order of magnitude as that of the Deli-3 solution; while the smoke interface is near the floor level, the accuracy of the new solution can be one order of magnitude higher than that of the Deli-3 solution. Moreover, it is shown that the Deli-2 solution is the limit of the new solution as the non-dimensional heat release rate approaching to infinity.

Some limit theorems are derived for a class of controled Markov systems with small noises. The aim is to understand the effects of strategies of actions on the limiting behaviors of the systems, so that optimization for associated utilities can be considered. In deriving the limits, we apply a large deviation approach with a somewhat new technique of proof. An almost sure convergence theorem is then derived. To meet more realistic situations, the noises need not to be smooth or Lipschitz continuous even are allowed to be discontinuous in the states. Dependence of the limits on the strategies can be found, which involves a certain differential equation. Some illustrative examples are provided.

In this paper, we investigate the first Steklov–Dirichlet eigenvalue on eccentric annuli. The main geometric parameter is the distance t between the centers of the inner and outer boundaries of an annulus. We first show the differentiability of the eigenvalue in t and obtain an integral expression for the derivative value in two and higher dimensions. We then derive an upper bound of the eigenvalue for each t, in two dimensions, by the variational formulation. We also obtain a lower bound of the eigenvalue, given a restriction that the two boundaries of the annulus are sufficiently close. The key point of the proof of the lower bound is in analyzing the limit behavior of an infinite series expansion of the first eigenfunction in bipolar coordinates. We also derive a relation between the first eigenvalue and a sequence of eigenvalues obtained by a finite section method. Based on this relation, we also perform numerical experiments that exhibit the monotonicity for two dimensions.