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This brief investigates the problem of state estimation of quaternion-valued neural networks (QVNNs) with time-varying delays. First, by extending the Jensen inequality to quaternion domain, an extended Jensen inequality with quaternion term is derived. Second, a class of proportional-integral state estimator (PISE) with exponential decay term is proposed. Then, by constructing a suitable Lyapunov-Krasovskii functional (LKF), some sufficient conditions are obtained to ensure the existence of the designed PISE and the gain matrices of the designed PISE can be directly computed. Simulations are given to illustrate the advantage of the proposed method.

This article mainly explores the local input-to-state stability (LISS) property of a class of nonlinear systems via a saturated control strategy, where both the external disturbance and impulsive disturbance being fully considered. In terms of the Lyapunov method and inequality techniques, some sufficient conditions under which the system can be made LISS are proposed, and the elastic constraint relationship among saturated control gain, rate coefficients, external disturbance, and domain of initial value is revealed. Moreover, the optimization design procedures are provided with the hope of obtaining the estimates of admissible external disturbance and domain of initial value as large as possible, where the corresponding saturated control law can be designed by solving LMI-based conditions. In the absence of an external disturbance, the locally exponential stability (LES) property can also be presented with a set of more relaxed conditions. Finally, two examples are presented to reveal the validity of the obtained results.

The adaptive hinging hyperplane (AHH) model is a popular piecewise linear representation with a generalized tree structure and has been successfully applied in dynamic system identification. In this article, we aim to construct the deep AHH (DAHH) model to extend and generalize the networking of AHH model for high-dimensional problems. The network structure of DAHH is determined through a forward growth, in which the activity ratio is introduced to select effective neurons and no connecting weights are involved between the layers. Then, all neurons in the DAHH network can be flexibly connected to the output in a skip-layer format, and only the corresponding weights are the parameters to optimize. With such a network framework, the backpropagation algorithm can be implemented in DAHH to efficiently tackle large-scale problems and the gradient vanishing problem is not encountered in the training of DAHH. In fact, the optimization problem of DAHH can maintain convexity with convex loss in the output layer, which brings natural advantages in optimization. Different from the existing neural networks, DAHH is easier to interpret, where neurons are connected sparsely and analysis of variance (ANOVA) decomposition can be applied, facilitating to revealing the interactions between variables. A theoretical analysis toward universal approximation ability and explicit domain partitions are also derived. Numerical experiments verify the effectiveness of the proposed DAHH. ispartof: Ieee Transactions On Neural Networks And Learning Systems vol:33 issue:11 pages:6373-6387 ispartof: location:United States status: published

Investigated in the present paper is a fifth-order nonlinear evolution (FONLE) equation, known as a nonlinear water wave (NLWW) equation, with applications in the applied sciences. More precisely, a traveling wave hypothesis is firstly applied that reduces the FONLE equation to a 1D domain. The Kudryashov methods (KMs) are then adopted as leading techniques to construct specific wave structures of the governing model which are classified as W -shaped and other solitons. In the end, the effect of changing the coefficients of nonlinear terms on the dynamical features of W -shaped and other solitons is investigated in detail for diverse groups of the involved parameters.

A novel approximate analytical solution to the linear damped Kawahara equation using a suitable hypothesis is reported for the first time. Based on the exact solutions (such as solitary waves, cnoidal waves, etc.) of the undamped Kawahara equation, the dissipative nonlinear structures like dissipative solitons and cnoidal waves are investigated. The obtained solution is considered a general solution, i.e., it can be applied for studying the properties of all dissipative traveling waves described by the linear damped Kawahara equation. Our technique is not limited to solve the linear damped Kawahara equation only, but it can be used for solving a large number of non-integrable evolution equations related to the realistic natural phenomena. Moreover, the maximum global residual error in the whole space-time domain is estimated for checking the accuracy of the obtained solutions. The obtained solutions can help many researchers in explaining the ambiguities about the mechanisms of propagation of nonlinear waves in complex systems such as seas, oceans, plasma physics, and much more.

A biased assimilation model of opinion dynamics is a nonlinear model, in which opinions exchanged in a social network are multiplied by a state-dependent term having the bias as exponent and expressing the bias of the agents toward their own opinions. The aim of this article is to extend the bias assimilation model to signed social networks. We show that while for structurally balanced networks, polarization to an extreme value of the opinion domain (the unit hypercube) always occurs regardless of the value of the bias, for structurally unbalanced networks, a stable state of indecision (corresponding to the centroid of the opinion domain) also appears, at least for small values of the bias. When the bias grows and passes a critical threshold, which depends on the amount of "disorder" encoded in the signed graph, then a bifurcation occurs and opinions become again polarized. Funding Agencies|National Natural Science Foundation of China [61733018]; Australian Research Council [DP190103615]; Swedish Research Council [2020-03701]

Conditional distributivity (also called restricted distributivity) is a form of relaxed distributivity on the restricted domain. There are three options for conditional distributivity in literature. In this paper, we focus on type-III conditional distributivity equation involving nullnorms over uninorms. Firstly, we show that it can be transformed into the other two types of conditional distributivity equations involving t-norms/t-conorms over uninorms. Secondly, type-I and type-II conditional distributivity equations involving t-norms/t-conorms over uninorms that are related to our study and remain unknown are investigated. Finally, we show that this new perspective results in not only all known solutions, but new solutions as well.

We consider a boundary value problem in a bounded domain involving a degenerate operator of the form $$L(u)=-\textrm{div} (a(x)\nabla u)$$ and a suitable nonlinearity $f$. The function $a$ vanishes on smooth 1-codimensional submanifolds of $\Omega$ where it is not allowed to be $C^{2}$. By using weighted Sobolev spaces we are still able to find existence of solutions which vanish, in the trace sense, on the set where $a$ vanishes.

This paper outlines a novel approach for simulating general nonlinear circuits in the time-domain. The proposed approach can be considered as the generalization of the numerical inversion Laplace transform (NILT) which has been used for circuits with only linear elements. The new approach enables the well-known advantages of NILT such the guaranteed numerical stability and the high-order approximation, to be carried to the domain of nonlinear circuit. A numerical example is given to demonstrate the validity of the proposed method.

Recently, a symmetric differential operator (SDO) is attracted to studying in the field of mathematical analysis. A new formal of SDO is presented to generalize some well known differential operators in a complex domain. According to this formulation, we shall exam the boundedness and compactness of this operator in complex spaces, such as Hilbert space and Sobolev space. For this purpose, we suggest new norms to solve the fractional Beltrami equation in the open unit disk. This operator has ability to depict the analytic geometric representation of the solution of second order differential equation utilizing the concept of Schwarzian derivative in the open unit disk. Applications in imagings are given the sequel.

We study fluid-saturated porous materials that undergo poro-elastic deformations in thin domains. The mechanics in such materials are described using a biphasic model based on the theory of porous media (TPM) and consisting of a system of differential equations for material’s displacement and fluid’s pressure. These equations are in general strongly coupled and nonlinear, such that exact solutions are hard to obtain and numerical solutions are computationally expensive. This paper reduces the complexity of the biphasic model in thin domains with a scale separation between domain’s width and length. Based on standard asymptotic analysis, we derive a reduced model that combines two sub-models. Firstly, a limit model consists of averaged equations that describe the fluid pore pressure and displacement in the longitudinal direction of the domain. Secondly, a corrector model re-captures the mechanics in the transverse direction. The validity of the reduced model is finally tested using a set of numerical examples. These demonstrate the computational efficiency of the reduced model, while maintaining reliable solutions in comparison with original biphasic TPM model in thin domain., Deutsche Forschungsgemeinschaft, Projekt DEAL

This paper analyzes a Susceptible-Infected-Susceptible (SIS) model of epidemic propagation over hypergraphs and, motivated by an important special case, we refer to the model as to the simplicial SIS model. Classically, the multi-group SIS model has assumed pairwise interactions of contagion across groups and thus has been vastly studied in the literature. It is only recently that a renewed special attention has been drawn to the study of contagion dynamics over higher-order interactions and over more general graph structures, like simplexes. Previous work on mean-field approximation scalar models of the simplicial SIS model has indicated that a new dynamical behavior domain, compared to the classical SIS model, appears due to the newly introduced higher order interaction terms: both a disease-free equilibrium and an endemic equilibrium co-exist and are both locally asymptotically stable. This paper formally establishes that bistability (as a new epidemiological behavior) also appears in the multi-group simplicial SIS model. We give sufficient conditions over the model's parameters for the appearance of this and the other behavioral domains present in the classical

This paper addresses the problem of estimating the domain of attraction of positive systems under a saturated linear feedback. Compared with the analysis of general saturated linear systems, one remarkable difference is that only the domain of attraction inside the first orthant is of interest. Hence, we tackle this problem by exploiting different shapes of invariant sets confined in the first orthant, including hyper-pyramids and hyper-rectangles, which are, respectively, the level sets of sum-separable and max-separable Lyapunov functions. For an open-loop positive system, a sufficient condition is given to ensure that a hyper-pyramid (or a hyper-rectangle) is contractively invariant for the closed loop. This condition is non-conservative for the single-input case. Moreover, for a system without open-loop positivity, we establish conditions such that the closed loop has a hyper-rectangular invariant set. Examples show the superiority of the results, in comparison with those existing ones using invariant ellipsoids.

In this present paper, we propose and analyze a C 1 -conforming virtual element method to solve the so-called one-layer stationary quasi-geostrophic equations (QGE) with applications in the large scale wind-driven ocean circulation, formulated in terms of the stream-function. This problem corresponds to a nonlinear fourth order partial differential equation. The C 1 virtual space and the discrete scheme are built in a straightforward way due to the flexibility of the virtual approach. Under the assumption of small data, we prove well-posedness of the discrete problem by using a fixed-point strategy and under standard assumptions on the computational domain, we establish error estimates in H 2 -norm for the stream-function. Finally, we report four numerical experiments that illustrate the behavior of the proposed scheme and confirm our theoretical results on different families of polygonal meshes.

In this article we derive the expression of \textit{renormalized energies} for unit-valued harmonic maps defined on a smooth bounded domain in \(\mathbb{R}^2\) whose boundary has several connected components. The notion of renormalized energies was introduced by Bethuel-Brezis-H\'elein in order to describe the position of limiting Ginzburg-Landau vortices in simply connected domains. We show here, how a non-trivial topology of the domain modifies the expression of the renormalized energies. We treat the case of Dirichlet boundary conditions and Neumann boundary conditions as well.

A portmanteau-type statistic based on sample covariance is proposed to test for the existence of the unit-root of an autoregressive model under a heavy-tailed linear noise, where the noise ɛ t = ∑ j = 0 ∞ d j η t − j and { η t } is a sequence of i.i.d. innovations belonging to the domain of attraction of an α -stable distribution for some α ∈ ( 0 , 2 ) . The proposed procedure is nonparametric and easy to implement. Under certain regular assumptions, the limit distribution of the statistics is shown to be a functional of a standard stable distribution. Further, the good finite sample studies show that the new test outperforms PP, DF and ADF tests in size and power. Applications to the daily world crude oil price and 3-month AA financial commercial paper rate are also given to illustrate the performance of the new test.

In this paper, the generalized Nash equilibrium problem over a fuzzy domain is considered. An existence result of the solution is given. Then, on the basis of the variational inequality approach, an algorithm is developed to solve the fuzzy Nash equilibrium problem. In addition, a stability analysis of the solution set is provided while the membership function is perturbed by a parameter. Finally, the fuzzy Nash equilibrium formulation is applied to the study of the conflicts faced by the fashion and textile industry: production and pollution.

In this paper, we introduce crisp utility functions with range in the class of fuzzy numbers. These utility functions can be employed to prove a generalization of the classical result of Cantor using partially ordered sets as domain of the utility functions. In particular, for every partially ordered countable set, we prove that it can be represented by an order preserving function (utility).

Motivated by applications to congested traffic problems, we establish higher integrability results for the gradient of local weak solutions to the strongly degenerate or singular elliptic PDE − div ( ( | ∇ u | − 1 ) + q − 1 ∇ u | ∇ u | ) = f , in Ω , where Ω is a bounded domain in R n for n ≥ 2 , 1 q ∞ and ( ⋅ ) + stands for the positive part. We assume that the datum f belongs to a suitable Sobolev or Besov space. The main novelty here is that we deal with the case of subquadratic growth, i.e. 1 q 2 , which has so far been neglected. In the latter case, we also prove the higher fractional differentiability of the solution to a variational problem, which is characterized by the above equation. For the sake of completeness, we finally give a Besov regularity result also in the case q ≥ 2 .

We address the problem of catching all speed $1$ geodesics of a Riemannian manifold with a moving ball: given a compact Riemannian manifold $(M,g)$ and small parameters $\varepsilon>0$ and $v>0$, is it possible to find $T>0$ and an absolutely continuous map $x:[0,T]\rightarrow M, t\mapsto x(t)$ satisfying $\|\dot{x}\|_{\infty}\leq v$ and such that any geodesic of $(M,g)$ traveled at speed $1$ meets the open ball $B_g(x(t),\varepsilon)\subset M$ within time $T$? Our main motivation comes from the control of the wave equation: our results show that the controllability of the wave equation can sometimes be improved by allowing the domain of control to move adequately, even very slowly. We first prove that, in any Riemannian manifold $(M,g)$ satisfying a geodesic recurrence condition (GRC), our problem has a positive answer for any $\varepsilon>0$ and $v>0$, and we give examples of Riemannian manifolds $(M,g)$ for which (GRC) is satisfied. Then, we build an explicit example of a domain $X\subset\mathbb{R}^2$ (with flat metric) containing convex obstacles, not satisfying (GRC), for which our problem has a negative answer if $��$ and $v$ are small enough, i.e., no sufficiently small ball moving sufficiently slowly can catch all geodesics of $X$., Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore In press

In many engineering applications such as structural mechanics, heat transfer and fluid dynamics numerical methods are employed to solve the governing partial differential equations. The numerical data is available at discrete points in the computational domain. It is then necessary to use numerical differentiation to evaluate or estimate derivatives. Usually finite difference approximations are used to evaluate derivatives. Forward, backward and divided differences dealt with in section 5.3are in fact related to approximations of derivatives of a given function. We look at numerical differentiation in greater detail in this chapter.

Let us consider the quasilinear problem \[ (P_\varepsilon) \ \ \left\{ \begin{array}{ll} - \varepsilon^p \Delta _{p}u + u^{p-1} = f(u) & \hbox{in} \ \Omega \newline u>0 & \hbox{in} \ \Omega \newline u=0 & \hbox{on} \ \partial \Omega \end{array} \right. \] where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary, $N\geq 2$, $1< p < 2$, $\varepsilon >0$ is a parameter and $f: \mathbb{R} \to \mathbb{R}$ is a continuous function with $f(0)=0$, having a subcritical growth. We prove that there exists $\varepsilon^* >0$ such that, for every $\varepsilon \in (0, \varepsilon^*)$, $(P_\varepsilon)$ has at least $2{\mathcal P}_1(\Omega)-1$ solutions, possibly counted with their multiplicities, where ${\mathcal P}_t(\Omega)$ is the Poincar\'e polynomial of $\Omega$. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on $\Omega$, approximating $(P_\varepsilon)$., Comment: to be published in "Communications in Contemporary Mathematics"

This paper is concerned with the mode-dependent sliding mode control (SMC) for uncertain discrete-time semi-Markov switching systems (S-MSSs). The switching is governed by a semi-Markov chain with a discrete-time semi-Markov kernel (SMK). First, a sliding mode surface (SMS) is constructed to be related to the current switching mode. Second, by means of the upper bound of the sojourn time (ST), some feasible conditions are derived to guarantee σ-error mean square stability for the corresponding S-MSSs. Third, an appropriate SMC law is established to drive the state onto the specified sliding domain. Finally, an F-404 aircraft model validates the proposed control algorithm.

This paper addresses sampled-data control of 2D Kuramoto-Sivashinsky equation over a rectangular domain. We suggest to divide the 2D rectangular into N sub-domains, where sensors provide spatially averaged or point state measurements to be transmitted through communication network to the controller. Note that differently from 2D heat equation, here we manage with sampled-data control under point measurements. We design a regionally stabilizing controller applied through distributed in space characteristic functions. Sufficient conditions ensuring regional stability of the closed-loop system are established in terms of linear matrix inequalities (LMIs). By solving these LMIs, we find an estimate on the set of initial conditions starting from which the state trajectories of the system are exponentially converging to zero. A numerical example demonstrates the efficiency of the results.

This paper is concerned with the recovery of (approximate) solutions to parabolic problems from incomplete and possibly inconsistent observational data, given on a time-space cylinder that is a strict subset of the computational domain under consideration. Unlike previous approaches to this and related problems our starting point is a regularized least squares formulation in a continuous infinite-dimensional setting that is based on stable variational time-space formulations of the parabolic partial differential equation. This allows us to derive a priori as well as a posteriori error bounds for the recovered states with respect to a certain reference solution. In these bounds the regularization parameter is disentangled from the underlying discretization. An important ingredient for the derivation of a posteriori bounds is the construction of suitable Fortin operators which allow us to control oscillation errors stemming from the discretization of dual norms. Moreover, the variational framework allows us to contrive preconditioners for the discrete problems whose application can be performed in linear time, and for which the condition numbers of the preconditioned systems are uniformly proportional to that of the regularized continuous problem. In particular, we provide suitable stopping criteria for the iterative solvers based on the a posteriori error bounds. The presented numerical experiments quantify the theoretical findings and demonstrate the performance of the numerical scheme in relation with the underlying discretization and regularization.

The point of this paper is to analyze and investigate the analytic-approximate solutions for fractional system of Volterra integro-differential equations in framework of Caputo-Fabrizio operator. The methodology relies on creating the reproducing kernel functions to gain analytical solutions in a uniform form of a rapidly convergent series in the Hilbert space. Using the Gram-Schmidt orthonomalization process, the orthonormal basis system is constructed in a dense compact domain to encompass the Fourier series expansion in view of reproducing kernel properties. Besides, convergence and error analysis of the proposed technique are discussed. For this purpose, several numerical examples are tested to demonstrate the great feasibility and efficiency of the present method and to support theoretical aspect as well. From a numerical point of view, the acquired solutions simulation indicates that the methodology used is sound, straightforward, and appropriate to deal with many physical issues in light of Caputo-Fabrizio derivatives.

Traditional proper orthogonal decomposition (POD)-Galerkin projection for reduced-order models (ROMs) of fluid flows is based on a fixed domain. The current method removes this limitation by consid...

We propose a novel three-level implicit half-step method of order two in time and four in space using unequal mesh for the solution of three-space dimensional quasi-linear hyperbolic equations on an irrational domain. The proposed method is directly applicable to 3D hyperbolic equation with singular coefficients, which is the key charisma of our scheme and we do not need any fictitious points for computation. The stability analysis of the model problem for unequal mesh has been discussed and it has been shown that the proposed method for Telegraphic equation is unconditionally stable. Alternating direction implicit (ADI) method by the help of a tri-diagonal solver is used to solve 3D linear schemes in three sweeps. The proposed method is scrutinized on several physical problems on dissimilar domain to exhibit the accuracy and effectiveness of the proposed method.

This paper studies the iterative learning control (ILC) algorithm for first-order hyperbolic systems. Unlike most of the ILC literature of distributed parameter systems, in the iteration domain, that require identical desired trajectories. Here the desired trajectories are iteratively varying and described by a high-order internal model (HOIM). The HOIM-based P-type ILC design is firstly introduced in this paper to the first-order hyperbolic systems, which enable the systems to achieve the perfect tracking for the iteration-varying desired trajectories on L 2 space. Meanwhile, the convergence theorem of the proposed algorithm is established for first-order time-delay hyperbolic systems. Finally, simulation results testify the validity of the algorithm.

In this paper, we provide rigorous justification of the hydrostatic approximation and the derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in the anisotropic horizontal viscosity regime. Setting e > 0 to be the small aspect ratio of the vertical to the horizontal scales of the domain, we investigate the case when the horizontal and vertical viscosities in the incompressible three-dimensional Navier-Stokes equations are of orders O ( 1 ) and O ( e α ) , respectively, with α > 2 , for which the limiting system is the primitive equations with only horizontal viscosity as e tends to zero. In particular we show that for “well prepared” initial data the solutions of the scaled incompressible three-dimensional Navier-Stokes equations converge strongly, in any finite interval of time, to the corresponding solutions of the anisotropic primitive equations with only horizontal viscosities, as e tends to zero, and that the convergence rate is of order O ( e β 2 ) , where β = min ⁡ { α − 2 , 2 } . Note that this result is different from the case α = 2 studied in Li and Titi (2019) [38] , where the limiting system is the primitive equations with full viscosities and the convergence is globally in time and its rate of order O ( e ) .

In this paper, we establish the global existence of strong solution to the chemotaxis-shallow water system with vacuum in a bounded domain Ω ⊂ R 2 provided that the initial data is partially small. We first get a new blow-up criterion and then we use bootstrap argument to obtain the global existence of strong solution.

In this paper, the fictitious stress method (FSM) as a meshless technique is extended to solve problems with inhomogeneities. The Eshelby's equivalent inclusion theory is coupled with the FSM as a set of particular solutions. There is no domain discretization as analytical solutions for inclusions are employed. The coupling procedure is presented, and the solution is carried out either in a direct or in an iterative approach. Numerical examples are presented to verify accuracy of the developed formulations.

The 2.5D approach is an efficient tool for the dynamic analysis of half-space with longitudinally invariant properties. In this paper, a newly formulated 2.5D version of the NURBS-based isogeometric analysis (IGA) and the scaled boundary IGA (SBIGA) is proposed for analyzing the bounded and unbounded domains, respectively, of a soil-tunnel system subjected to moving railway loads. The bounded domain is discretized by IGA, since the NURBS used can exactly represent the variations in geometry and materials with negligible errors. The unbounded domain is discretized by SBIGA, which automatically satisfies the Sommerfeld radiation condition and can be easily connected with the bounded IGA domain. The 2.5D approach differs from the 2D approach in that both the in-plane and out-of-plane displacements of the profile of the soil-tunnel system are included in analysis. By applying the Fourier transformation, the IGA and SBIGA equations of the 2.5D version in the frequency-wavenumber domain are derived individually and then assembled at the interface of the bounded and unbounded domains. Two examples are prepared for the ground and underground trains moving at subsonic and transonic speeds. Both single and multi moving loads are considered. It was demonstrated that the wave propagation characteristics including the Mach cones at transonic speeds can be precisely predicted. The results in time and frequency domains have been compared with existing ones. It was shown that the proposed method is accurate and reliable for application to a wide range of soil-tunnel systems.

In this letter, a new full-wave inversion (FWI) scheme is proposed to reconstruct multiple dielectric parameters of 3-D arbitrary anisotropic objects buried in layered media. Three inverse solvers, including the isotropic one, biaxial anisotropic one, and the arbitrary anisotropic one, are cascaded sequentially. The dielectric parameters obtained by the first solver are used as the initial values of the next solver. Meanwhile, the inversion domain is synchronously downsized on the basis of discrepancies between the inverted dielectric parameters and the background ones. Numerical simulations show that, compared with the direct arbitrary anisotropic inverse solver, the cascading inversion scheme not only can produce more reliable reconstructed profiles but also significantly lowers the computational cost. In addition, the antinoise ability of the cascaded solvers is also tested.

In this paper, the space-time fractional advection-diffusion equation (STFADE) is considered in the finite domain that the time and space derivatives are the Caputo fractional derivative. At first, a quadratic interpolation with convergence order O ( τ 3 - α ) is applied to obtain the semi-discrete in time variable. Then, the Chebyshev collocation method of the fourth kind has been used to approximate the spatial fractional derivative. In addition, the energy method has been employed to show the unconditional stability and gained convergence order of the time-discrete scheme. Finally, the accuracy of the numerical method is analyzed and showed that our method is much more accurate than existing techniques.

This paper presents an H ∞ event-triggered state-feedback controller design for continuous-time nonlinear systems via convex optimization techniques. The proposal is based on an exact discretization and its quasi-linear parameter varying representation. Thus, two sets of design conditions in terms of linear matrix inequalities by means of both parameter-dependent and quadratic Lyapunov functions are proposed. The proposed conditions also provide an estimate to system’s domain of attraction and an extra set of conditions is presented for a guaranteed minimum time between events. Well-known examples are employed to illustrate the effectiveness of the proposal.

It is known from the literature that solutions of homogeneous linear stable difference equations may experience large deviations, or peaks, from the nonzero initial conditions at finite time instants. While the problem has been studied from a deterministic standpoint, not much is known about the probability of occurrence of such event when both the initial conditions and the coefficients of the equation have random nature. In this paper, by exploiting results on the volume of the Schur domain, we are able to compute the probability for deviations to occur. This turns out to be very close to unity, even for equations of low degree. Hence, we claim that “solutions of stable difference equations probably experience peak”. Then, we make use of tools from statistical learning to address other issues such as evaluation of the mean magnitude and maximum value of peak.

This letter introduces the notion of linear Discrete-time Output Negative Imaginary (D-ONI) systems. The D-ONI class is defined in the $z$ -domain and it includes the systems having poles on the unit circle. The proposed definition involves a real parameter $\delta \geq 0$ , which indicates the strictness properties. $\delta > 0$ specifies the strict subset, Discrete-time Output Strictly Negative Imaginary (D-OSNI), within the stable D-ONI class. Interestingly, the new D-ONI class captures the existing D-NI systems while restricted to discrete-time LTI systems having a real, rational and proper transfer function. However, the D-OSNI systems are not identical to the existing strictly D-NI (D-SNI) subset. Instead, these two subsets intersect each other. An LMI-based state-space characterisation is derived to check the strict/non-strict D-ONI properties of a given system relying on the value of $\delta $ . The paper also establishes the connections between the discrete-time Passive and discrete-time NI systems. Finally, a closed-loop stability result is proposed for a positive feedback interconnection of two D-ONI systems without poles at $z = -1$ and $z = +1$ .

In this paper, a two-grid mixed finite volume element (MFVE) algorithm is presented for the nonlinear time fractional reaction-diffusion equations, where the Caputo fractional derivative is approximated by the classical $ L1 $-formula. The coarse and fine grids (containing the primal and dual grids) are constructed for the space domain, then a nonlinear MFVE scheme on the coarse grid and a linearized MFVE scheme on the fine grid are given. By using the Browder fixed point theorem and the matrix theory, the existence and uniqueness for the nonlinear and linearized MFVE schemes are obtained, respectively. Furthermore, the stability results and optimal error estimates are derived in detailed. Finally, some numerical results are given to verify the feasibility and effectiveness of the proposed algorithm.

This work presents some improvements on related papers that investigate certain overdetermined problems associated to elliptic quasilinear operators. Our model operator is the $ p $-Laplacian. Under suitable structural conditions, and assuming that a solution exists, we show that the domain of the problem is a ball centered at the origin. Furthermore we discuss a convenient form of comparison principle for this kind of problems.

We propose seismic elastic full-waveform inversion (EFWI) using both the multiparametric approximate Hessian and the discrete cosine transform (DCT). EFWI is a promising technology for obtaining physical parameters. EFWI using the multiparametric approximate Hessian can suppress crosstalk artifacts for each physical parameter, but the computational burden increases as the number of physical parameters considered increases. In this study, to reduce the computational burden, DCT was used to compress the model parameters and accelerate EFWI in the truncated DCT domain. EFWI using DCT is possible to increase the calculation efficiency by reducing the total number of unknown variables. We implemented EFWI using monoparametric Hessian and multiparametric Hessian, which reduced the computational burden using DCT and conducted tests through numerical experiments using each Hessian. In numerical result, despite using a small number of DCT coefficients, we confirmed that contamination due to crosstalk artifacts was suppressed when EFWI was performed using the multiparametric approximate Hessian.

For any $\Omega\subset \mathbb{R}^N$ smooth and bounded domain, we prove uniqueness of positive solutions of free boundary problems arising in plasma physics on $\Omega$ in a neat interval depending only by the best constant of the Sobolev embedding $H^{1}_0(\Omega)\hookrightarrow L^{2p}(\Omega)$, $p\in [1,\frac{N}{N-2})$ and show that the boundary density and a suitably defined energy share a universal monotonic behavior. At least to our knowledge, for $p>1$, this is the first result about the uniqueness for a domain which is not a two-dimensional ball and in particular the very first result about the monotonicity of solutions, which seems to be new even for $p=1$. The threshold, which is sharp for $p=1$, yields a new condition which guarantees that there is no free boundary inside $\Omega$. As a corollary, in the same range, we solve a long-standing open problem (dating back to the work of Berestycki-Brezis in 1980) about the uniqueness of variational solutions. Moreover, on a two-dimensional ball we describe the full branch of positive solutions, that is, we prove the monotonicity along the curve of positive solutions until the boundary density vanishes., Comment: 29 pages

In this paper we study the Neumann problem for a type of fully nonlinear second order elliptic partial differential equations on domains in C n without any curvature assumptions on the domain.

Linearized algorithms based on the Born approximation are well-known and popular techniques for quantitative seismic imaging and inversion. However, linearization methods usually suffer from some significant problems, such as the computational cost for the required number of iterations, requirement for background models, and uncertain and unstable multiparameter extraction, which make the methods difficult to implement in practical applications. To avoid these problems, we have developed an angle-domain generalized Radon transform (AD-GRT) inversion in 2D elastic isotropic media. This AD-GRT is an approximate transform between the seismic data and an angle-domain model, which acts as a scattering function, and the seismic data can be reconstructed accurately, even when the background models are incorrect. The density and Lamé moduli perturbation parameters can be extracted stably from the inverted angle-domain scattering function. Deconvolution of the source wavelet is taken into account to remove the effect of the wavelet and improve the resolution and accuracy of the inversion results. The derived AD-GRT inversion is noniterative and is as efficient as the traditional elastic GRT method. The additional dimension of the angle domain has little effect on the computational cost of the AD-GRT, as opposed to other extended-domain inversion/migration methods. Our method also can be used to solve nonlinear Born inversion problems using iteration, which can significantly improve their convergence rate. Three numerical examples illustrate that the angle-domain scattering function inversion, data reconstruction, and multiparameter extraction using the presented AD-GRT inversion are effective.

Time-dependent flow of a viscoelastic material through a wavy curved channel is studied. The flow traversing the wavy curved domain is modelled in the curvilinear coordinates. The viscoelastic fluid is described by the fluid relaxation and retardation times, and their variations are considered in relation to the flow characteristic time scale. The wavy walls of the curved channels are in sinusoidal form, with arbitrary phase difference. Using domain perturbation technique, the analytical solution of the model is obtained, including the velocity and the volumetric flow rate. The analytical solution is explained as a function of the flow domain structure, the viscoelastic fluid properties, and the flow generating force. Flow enhancement due to the flow domain is discussed; we have shown that the flow augmentation depends on the properties of the viscoelastic fluid. Furthermore, comparisons have been made with Newtonian fluid flow, and the prominent variations in the flow behaviours have been reported.

In survey domain estimation, a priori information can often be imposed in the form of linear inequality constraints on the domain estimators. Wu et al. (2016) formulated the isotonic domain mean estimator, for the simple order restriction, and methods for more general constraints were proposed in Oliva-Aviles et al. (2020). When the assumptions are valid, imposing restrictions on the estimators will ensure that the a priori information is respected, and in addition allows information to be pooled across domains, resulting in estimators with smaller variance. Here, we propose a method to further improve the estimation of the covariance matrix for these constrained domain estimators, using a mixture of possible covariance matrices obtained from the inequality constraints. We prove consistency of the improved variance estimator, and simulations demonstrate that the new estimator results in improved coverage probabilities for domain mean confidence intervals, while retaining the smaller confidence interval lengths.

Traveling-wave modulation is a form of space-time modulation which has been shown to enable unique electromagnetic phenomena such as non-reciprocity, beam-steering, frequency conversion, and amplification. In practice, traveling-wave modulation is achieved by applying a staggered time-modulation signal to a spatially-discrete array of unit cells. Therefore, the capability to accurately simulate spatially-discrete traveling-wave modulated structures is critical to design. However, simulating these structures is challenging due to the complex space-time dependence of the constituent unit cells. In this paper, a field relation (referred to as the interpath relation) is derived for spatially-discrete traveling-wave modulated structures. The interpath relation reveals that the field within a single time-modulated unit cell (rather than an entire spatial period) is sufficient to determine the field solution throughout space. It will be shown that the interpath relation can be incorporated into existing periodic method of moments solvers simply by modifying the source basis functions. As a result, the computational domain is reduced from an entire spatial period to a single time-modulated unit cell, dramatically reducing the number of unknowns. In the context of traveling-wave modulation, this enables researchers to efficiently simulate both complex structures with patterned unit cells in addition to continuous structures with infinitesimal unit cells., 15 pages

Recently, a polygonal boundary element method (PBEM) has been developed for solving heat conduction problems. In the PBEM, the radial integration method (RIM) is employed to shift the domain and surface integrals in the boundary-domain integral equation into equivalent line integrals, and all the radial integrals are computed by Gaussian quadrature, which may obtain an accurate solution. However, the computation costs much. Therefore, analytical expressions of radial integrals are derived in this paper, concerning four kinds of varying thermal conductivities and two kinds of heat generation functions, aiming at improving the efficiency. Then all the radial integrals in the boundary-domain integral equation can be analytically calculated. Three examples are employed to examine the analytically-integrated PBEM for solving steady heat conduction problems. The results show that the proposed method is accurate, and it is more efficient than traditional PBEM.

In this paper, we study a Susceptible-Vaccinated-Infected-Recovered (SVIR) epidemic model in a spatially heterogeneous environment under the Dirichlet boundary condition. We define the basic reproduction number ℜ 0 by the spectral radius of the next generation operator, and show that it is a threshold parameter. The disease extinction and persistence in the case of a bounded domain are considered. More precisely, we show that the disease-free equilibrium is globally asymptotically stable if ℜ 0 1 ; the system is uniformly persistent and an endemic equilibrium exists if ℜ 0 > 1 . To verify our theoretical results, we perform some numerical simulations, using the Fredholm discretization method to identify ℜ 0 .

We analyze a system of differential equations with state-dependent delay (SD-DDE) from cell biology, in which the delay is implicitly defined as the time when the solution of an ODE, parametrized by the SD-DDE state, meets a threshold. We show that the system is well-posed and that the solutions define a continuous semiflow on a state space of Lipschitz functions. Moreover we establish for an associated system a convex and compact set that is invariant under the time-t-map for a finite time. It is known that, due to the state dependence of the delay, necessary and sufficient conditions for well-posedness can be related to functionals being almost locally Lipschitz, which roughly means locally Lipschitz on the restriction of the domain to Lipschitz functions, and our methodology involves such conditions. To achieve transparency and wider applicability, we elaborate a general class of two component functional differential equation systems, that contains the SD-DDE from cell biology and formulate our results also for this class.