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Newton-step approximations to pseudo maximum likelihood estimates of spatial autoregressive models with a large number of parameters are examined, in the sense that the parameter space grows slowly as a function of sample size. These have the same asymptotic efficiency properties as maximum likelihood under Gaussianity but are of closed form. Hence they are computationally simple and free from compactness assumptions, thereby avoiding two notorious pitfalls of implicitly defined estimates of large spatial autoregressions. For an initial least squares estimate, the Newton step can also lead to weaker regularity conditions for a central limit theorem than those extant in the literature. A simulation study demonstrates excellent finite sample gains from Newton iterations, especially in large multiparameter models for which grid search is costly. A small empirical illustration shows improvements in estimation precision with real data., 36 pages

Given a dataset $\cal S$ composed of g subsets with each data items belonging to one of them, $\textit{multiset membership lookup}$ takes an item e as input and outputs a binary answer whether $e\in{\cal S}$ and, in case of yes, the ID of the subset to which e belongs. Overlaid upon while more sophisticated than the canonical membership lookup, multiset membership lookup emerges as a pivotal functionality in many computing and networking paradigms. The quest to achieve high-speed, high-accuracy lookup with limited memory cost makes lookup algorithm design a challenging task, particularly when the data items arrive as a stream. In this paper, we devise compact data structures and lookup algorithms that are amendable for hardware implementation, while guaranteeing high lookup accuracy and supporting interactive query processing. We first propose $\textit{multi-hash color table}$ , a variant of Bloom filter, to encode subset IDs compactly and map the ID of an item to its subset ID. We further construct a more balanced data structure called $\textit{balanced multi-hash color table}$ to improve the compactness by integrating the state-of-the-art load balancing technique. We complete our work by addressing the case of $\textit{batch arrivals}$ and design a batched recording algorithm optimizing the memory efficiency.

This article proposes an adaptive neural network (NN) control method for an n -link constrained robotic manipulator. Driven by actual demands, manipulator and actuator dynamics, state and input constraints, and unknown time-varying delays are taken into account simultaneously. NNs are employed to approximate unknown nonlinearities. Time-varying barrier Lyapunov functions are utilized to cope with full-state constraints. By resorting to saturation function and Lyapunov-Krasovskii functionals, the effects of actuator saturation and time delays are eliminated. It is proved that all the closed-loop signals are semiglobally uniformly ultimately bounded, full-state constraints and actuator saturation are not violated, and error signals remain within compact sets around zero. Simulation studies are given to demonstrate the validity and advantages of this control scheme.

Fix a prime $p$ and a positive integer $R$. We study the property testing of functions $\mathbb F_p^n\to[R]$. We say that a property is testable if there exists an oblivious tester for this property with one-sided error and constant query complexity. Furthermore, a property is proximity oblivious-testable (PO-testable) if the test is also independent of the proximity parameter $\epsilon$. It is known that a number of natural properties such as linearity and being a low degree polynomial are PO-testable. These properties are examples of linear-invariant properties, meaning that they are preserved under linear automorphisms of the domain. Following work of Kaufman and Sudan, the study of linear-invariant properties has been an important problem in arithmetic property testing. A central conjecture in this field, proposed by Bhattacharyya, Grigorescu, and Shapira, is that a linear-invariant property is testable if and only if it is semi subspace-hereditary. We prove two results, the first resolves this conjecture and the second classifies PO-testable properties. (1) A linear-invariant property is testable if and only if it is semi subspace-hereditary. (2) A linear-invariant property is PO-testable if and only if it is locally characterized. Our innovations are two-fold. We give a more powerful version of the compactness argument first introduced by Alon and Shapira. This relies on a new strong arithmetic regularity lemma in which one mixes different levels of Gowers uniformity. This allows us to extend the work of Bhattacharyya, Fischer, Hatami, Hatami, and Lovett by removing the bounded complexity restriction in their work. Our second innovation is a novel recoloring technique called patching. This Ramsey-theoretic technique is critical for working in the linear-invariant setting and allows us to remove the translation-invariant restriction present in previous work., Comment: 40 pages; updated with significantly improved main result

This study focuses on an adaptive fault-tolerant boundary control (BC) for a flexible string (FS) in the presence of unknown external disturbances, dead zone, and actuator fault. To tackle these issues, by employing some transformations, a part of the unknown dead zone and external disturbance can be regarded as a composite disturbance. Subsequently, an adaptive fault-tolerant BC is developed by utilizing strict formula derivations to compensate for unknown composite disturbance, dead zone, and actuator fault in the FS system. Under the proposed control strategy, the closed-loop system proves to be uniformly ultimately bounded, and the vibration amplitude is guaranteed to converge ultimately to a small compact set by choosing suitable design parameters. Finally, a numerical simulation is performed to demonstrate the control performance of the proposed scheme.

In this article, the adaptive fuzzy tracking control problem for a class of uncertain strict-feedback systems with unknown nonlinearities is investigated with particular emphasis on global stability. The proposed control scheme is designed by integrating the barrier Lyapunov functions (BLFs) with the techniques of fuzzy approximation and backstepping. The novel integral BLFs (iBLFs) are introduced to overcome the design difficulties induced by the virtual control coefficients and determine a priori the compact set for guaranteeing the validity of fuzzy approximation. Compared with existing approximation-based control results, the developed controller not only guarantees global stability without requiring prior information of system nonlinearities and assumptions on the time derivatives of virtual control coefficients, but also prevents the ``explosion of complexity'' issue without attaching additional filters. The simulation results further confirm the effectiveness of the theoretical findings.

In this article, the problem of adaptive tracking control is tackled for a class of high-order nonlinear systems. In contrast to existing results, the considered system contains not only unknown nonlinear functions but also unknown rational powers. By utilizing the fuzzy approximation approach together with the barrier Lyapunov functions (BLFs), we present a new adaptive tracking control strategy. Remarkably, the BLFs are employed to determine a priori the compact set for maintaining the validity of fuzzy approximation. The primary advantage of this article is that the developed controller is independent of the powers and can be capable of ensuring global stability. Finally, two illustrative examples are given to verify the effectiveness of the theoretical findings.

In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages

This paper investigates solvability of fully coupled systems of forward–backward stochastic differential equations (FBSDEs) with irregular coefficients. In particular, we assume that the coefficients of the FBSDEs are merely measurable and bounded in the forward process. We crucially use compactness results from the theory of Malliavin calculus to construct strong solutions. Despite the irregularity of the coefficients, the solutions turn out to be differentiable, at least in the Malliavin sense and, as functions of the initial variable, in the Sobolev sense.

Unsupervised domain adaptation (UDA) for the semantic segmentation of remote sensing images is challenging since the same class of objects may have different spectra while the different class of objects may have the same spectrum. To address this issue, we propose a class-aware generative adversarial network (CaGAN) for UDA semantic segmentation of multisource remote sensing images, which explicitly models the discrepancies of intraclass and the interclass between the source domain images with labels and the target domain images without labels. Specifically, first, to enhance the global domain alignment (GDA), we propose a transferable attention alignment (TAA) procedure to add more fine-grained features into the adversarial learning framework. Then, we propose a novel class-aware domain alignment (CDA) approach in semantic segmentation. CDA mainly includes two parts: the first one is adaptive category selection, which is to alleviate the class imbalance and select the reliable per-category centers in the source and target domains; the second one is adaptive category alignment, which is to model the intraclass compactness and interclass separability from source-only, target-only, and joint source and target images. Finally, the CDA plays as a penalty of GDA to train GaGAN in an alternating and iterative manner. Experiments on domain adaptation of space to space, spectrum to spectrum, both space-to-space and spectrum-to-spectrum data sets demonstrate that CaGAN outperforms the current state-of-the-art methods, which may serve as a starting point and baseline for the comprehensive applications of semantic segmentation in cross-space and cross-spectrum remote sensing images.

In this paper we study the notion of Salem set from the point of view of descriptive set theory. We first work in the hyperspace $\mathbf{K}([0,1])$ of compact subsets of $[0,1]$ and show that the closed Salem sets form a $\boldsymbol{\Pi}^0_3$-complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in $\mathbf{K}([0,1]^d)$. We then generalize the results by relaxing the compactness of the ambient space, and show that the closed Salem sets are still $\boldsymbol{\Pi}^0_3$-complete when we endow the hyperspace of all closed subsets of $\mathbb{R}^d$ with the Fell topology. A similar result holds also for the Vietoris topology., Comment: Extended Lemma 3.1, fixed Lemma 5.3 and improved the presentation of the results. To appear in Fundamenta Mathematicae

This paper is concerned with a class of nonlocal reaction-diffusion equations with time-delay and degenerate diffusion. Affected by the degeneracy of diffusion, it is proved that, the Cauchy problem of the equation possesses the Holder-continuous solution. Furthermore, the non-critical traveling waves are proved to be globally L 1 -stable, which is the first frame work on L 1 -wavefront-stability for the degenerate diffusion equations. The time-exponential convergence rate is also derived. The adopted approach for the proof is the technical L 1 -weighted energy estimates combining the compactness analysis, but with some new development.

In this letter, we study the problem of non-blockingness verification by tapping into the basis reachability graph (BRG). Non-blockingness is a property that ensures that all pre-specified tasks can be completed, which is a mandatory requirement during the system design stage. We develop a condition of transition partition of a given net such that the corresponding conflict-increase BRG contains sufficient information on verifying non-blockingness of its corresponding Petri net. Thanks to the compactness of the BRG, our approach possesses practical efficiency since the exhaustive enumeration of the state space can be avoided. In particular, our method does not require that the net is deadlock-free.

IEEE Keyword search problem has been widely studied to retrieve relevant substructures from graphs. However, existing approaches aim at finding compact trees/subgraphs containing the keywords, and ignore density to evaluate how strongly and stablely the keyword nodes are connected. In this paper, we study the problem of finding cohesive subgraph containing query keywords with high density and compactness, and formulate it as minimal dense truss search problem based on k-truss model. However, unlike ${k}$-truss based community search that can be efficiently done by local search from a given set of nodes, this problem is nontrivial as the keyword nodes to be included in the retrieved substructure is previously unknown. To tackle this problem, we first design a novel hybrid KT-Index that keeps the keyword and truss information compactly to support efficient search of dense truss with the maximum trussness ${G_{den}}$. Then, we develop a novel refinement approach to extract minimal dense truss from ${G_{den}}$, by checking each node at most once based on the anti-monotonicity property of k-truss, together with several optimization strategies including batch based deletion, early-stop based deletion, and local exploration. Extensive experimental studies on real-world networks validated the effectiveness and efficiency of our approaches.

A century after the first attempts of Wentworth to characterise the shape of cobbles, our understanding of particle morphology is still expanding. A plethora of shape indices has been proposed in the literature to characterise the morphology of individual particles. This study aims to shed light on the merits and limitations of the indices currently used to characterise particle elongation, flatness and compactness, adopting a unified classification framework. Second, new indices are proposed to address the identified shortcomings. Third, a new particle classification system derived from the proposed indices is illustrated. It is shown the new system overcomes the misclassification of a range of particles that are incorrectly classified as bladed in the Zingg system.

The set-theoretical convergence of models of continuous and discrete synthesis processes in the compositions of metals and non-metals at the action of ray sources is shown. The non-metal density function obtained by the IDS thermal impulses on the temperature axis projection has a periodic view with fi xed stationary temperatures. They coincide with the boundaries of diff erent areas of change in the speeds of metal heating. This convergence of models to the general periodic operator is possible under four conditions of beam scanning: separation, compactness, property of states, stability. Then the current reciprocity ratios of the system (metals/non-metals) allow the use of the temperature of stationary non-metals as support for metals and serves to determine the parameters of ray sources when synthesizing structures in the compositions of the volume of the material.

We concern $C^2$-compactness of the solution set of the boundary Yamabe problem on smooth compact Riemannian manifolds with boundary provided that their dimensions are $4$, $5$ or $6$. By conducting a quantitative analysis of a linear equation associated with the problem, we prove that the trace-free second fundamental form must vanish at possible blow-up points of a sequence of blowing-up solutions. Applying this result and the positive mass theorem, we deduce the $C^2$-compactness for all $4$-manifolds (which may be non-umbilic). For the $5$-dimensional case, we also establish that a sum of the second-order derivatives of the trace-free second fundamental form is non-negative at possible blow-up points. We essentially use this fact to obtain the $C^2$-compactness for all $5$-manifolds. Finally, we show that the $C^2$-compactness on $6$-manifolds is true if the trace-free second fundamental form on the boundary never vanishes., Comment: 29 pages, This version treats general 5-manifolds as well, Comments are welcome

Good image segmentation produces objects that are internally homogeneous and distinct from their neighbors. Several approaches are used to assess the quality of segmentation parameters including visual observation of multiple segmentation results and techniques, which calculate measures of intra segment homogeneity and inter segment heterogeneity. Some of these techniques have been reported without considering under and over segmentation; occurrences and others exhibit some mathematical formulation instabilities when dealing with very heterogeneous areas. Moreover, the majority of segmentation assessment techniques focus on the evaluation of the scale parameter and give little attention to the influence of other parameters, such as compactness on the performance of the scale parameter. To the best of our knowledge, no existing approach can predict the optimal compactness thresholds that would enhance the performance of the scale parameters, or predict which scale parameter will perform best under certain compactness constraints. This paper proposes three spatial autocorrelation models, which identify the optimal parameters for single scale and multilevel segmentations of urban environments. Tested using a GeoEye satellite image of the City of Cape Town (South Africa), results reflect that the best combination of parameters to optimally segment the area and those that would poorly perform were successfully identified.

We present a search for transient radio sources on timescales of 2 to 9 yr at 150 MHz. This search is conducted by comparing the first Alternative Data Release of the TIFR GMRT Sky Survey (TGSS ADR1) and the second data release of the LOFAR Two-metre Sky Survey (LoTSS DR2). The overlapping survey area covers 5570 $\rm{deg}^2$ on the sky, or 14 per cent of the total sky. We introduce a method to compare the source catalogues that involves a pair match of sources, a flux density cutoff to meet the survey completeness limit and a newly developed compactness criterion. This method is used to identify both transient candidates in the TGSS source catalogue that have no counterpart in the LoTSS catalogue and transient candidates in LoTSS without a counterpart in TGSS. We find that imaging artefacts and uncertainties and variations in the flux density scales complicate the transient search. Our method to search for transients by comparing two different surveys, while taking into account imaging artefacts around bright sources and misaligned flux scales between surveys, is universally applicable to future radio transient searches. No transient sources were identified, but we are able to place an upper limit on the transient surface density of $, 14 pages, 11 figures

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.

Let $\alpha\in (0, 1]$, $\beta\in [0, n)$ and $T_{\Omega,\beta}$ be a singular or fractional integral operator with homogeneous kernel $\Omega$. In this article, a CMO type space ${\rm CMO}_\alpha(\mathbb R^n)$ is introduced and studied. In particular, the relationship between ${\rm CMO}_\alpha(\mathbb R^n)$ and the Lipchitz space $Lip_\alpha(\mathbb R^n)$ is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator $(T_{\Omega,\beta})^m_b$ on weighted Lebesgue spaces via functions in $Lip_\alpha(\mathbb R^n)$, and an equivalent characterization of the compactness for $(T_{\Omega,\beta})^m_b$ via functions in ${\rm CMO}_\alpha(\mathbb R^n)$ are obtained. Some results are new even in the unweighted setting for the first order commutators., Comment: arXiv admin note: text overlap with arXiv:1712.08292

We provide parameter dependent version of the Browder–Minty Theorem in case when the solution is unique utilizing different types of monotonicity and compactness assumptions related to condition (S)2. Potential equations and the convergence of their Euler action functionals is also investigated. Applications towards the dependence on parameters for both potential and non-potenial nonlinear Dirichlet boundary problems are given.

We extend a result on existence of Walrasian equilibria in He and Yannelis (Econ Theory 61:497–513, 2016) by replacing the compactness assumption on consumption sets made there by the standard assumption that these sets are closed and bounded from below. This provides a positive answer to a question explicitly raised in He and Yannelis (Econ Theory 61:497–513, 2016). Our new equilibrium existence theorem generalizes many results in the literature as we do not require any transitivity or completeness or continuity assumption on preferences, initial endowments need not be in the interior of the consumption sets, preferences may be interdependent and price-dependent, and no monotonicity or local non satiation is needed for any of the agents.

We show that the prescribed Gaussian curvature equation in $\mathbb {R}^{2}$ $$ -\varDelta u= (1-|x|^{p}) e^{2u}, $$ has solutions with prescribed total curvature equal to ${{\varLambda }}:={\int \limits }_{\mathbb {R}^{2}}(1-|x|^{p})e^{2u}dx\in \mathbb {R}$ , if and only if $$ p\in(0,2) \qquad \text{and} \qquad (2+p)\pi\le{\varLambda}

We consider stochastic two-dimensional constrained Navier-Stokes equations driven by Levy noise in the Marcus canonical form. The aim of this work is two-fold. At first, we prove the existence of a martingale solution based on the construction relying on classical Faedo-Galerkin approximations, compactness method and the Jakubowski's version of Skorokhod representation theorem for non-metric spaces. We further prove that the martingale solution is pathwise unique and deduces the existence of a strong solution. In the second part of the paper, we establish a Wentzell-Freidlin type large deviations principle for the small noise asymptotic of solutions using weak convergence method.

In this paper we are concerned with some abstract results regarding to fractional Orlicz-Sobolev spaces. Precisely, we ensure the compactness embedding for the weighted fractional Orlicz-Sobolev space into the Orlicz spaces, provided the weight is unbounded. We also obtain a version of Lions' “vanishing” Lemma for fractional Orlicz-Sobolev spaces, by introducing new techniques to overcome the lack of a suitable interpolation law. Finally, as a product of the abstract results, we use a minimization method over the Nehari manifold to prove the existence of ground state solutions for a class of nonlinear Schrodinger equations, taking into account unbounded or bounded potentials.

We study Mean Field Games (MFGs) driven by a large class of nonlocal, fractional and anomalous diffusions in the whole space. These non-Gaussian diffusions are pure jump L\'evy processes with some $\sigma$-stable like behaviour. Included are $\sigma$-stable processes and fractional Laplace diffusion operators $(-\Delta)^{\frac{\sigma}2}$, tempered nonsymmetric processes in Finance, spectrally one-sided processes, and sums of subelliptic operators of different orders. Our main results are existence and uniqueness of classical solutions of MFG systems with nondegenerate diffusion operators of order $\sigma\in(1,2)$. We consider parabolic equations in the whole space with both local and nonlocal couplings. Our proofs uses pure PDE-methods and build on ideas of Lions et al. The new ingredients are fractional heat kernel estimates, regularity results for fractional Bellman, Fokker-Planck and coupled Mean Field Game equations, and a priori bounds and compactness of (very) weak solutions of fractional Fokker-Planck equations in the whole space. Our techniques requires no moment assumptions and uses a weaker topology than Wasserstein., Comment: Major update. A much larger class of anomalous diffusions, local coupling results are now complete, problems are posed in the whole space and not the compact torus. New equicontinuity, $L^1$, and $L^\infty$-results for the Fokker-planck equation are needed and proved by using pure PDE arguments. We now work in a more general metrics than Wasserstein

In this work, we use the constrained Gaussian realization technique to study the early growth of supermassive black holes (SMBHs) in cosmological hydrodynamic simulations, exploring its relationship with features of the initial density peaks on large scales, ~1 Mpc/h. Our constrained simulations of volume (20 Mpc/h)^3 successfully reconstruct the large-scale structure as well as the black hole growth for the hosts of the rare 10^9 Msun SMBHs found in the BlueTides simulation at z~7. We run a set of simulations with constrained initial conditions by imposing a 5 \sigma_0(R_G) peak on scale of R_G = 1 Mpc/h varying different peak features, such as the shape and compactness as well as the tidal field surrounding the peak. We find that initial density peaks with high compactness and low tidal field induce the most rapid BH growth at early epochs. This is because compact density peaks with a more spherical large scale matter distribution lead to the formation of high density gas clumps in the centers of halos, and thus boost early BH accretion. Moreover, such initially compact density peaks in low tidal field regions also lead to a more compact BH host galaxy morphology. This can explain the tight correlation between BH growth and host galaxy compactness seen in observations., Comment: matches MNRAS accepted version

We consider the fully nonlinear equation with variable-exponent double phase type degeneracies $$ \big[|Du|^{p(x)}+a(x)|Du|^{q(x)}\big]F(D^2u)=f(x). $$ Under some appropriate assumptions, by making use of geometric tangential methods and combing a refined improvement-of-flatness approach with compactness and scaling techniques we obtain the sharp local $C^{1,\alpha}$ regularity of viscosity solutions to such equations.

Given a domain $D$ in $\mathbb{C}^n$ and $K$ a compact subset of $D$, the set $\mathcal{A}_K^D$ of all restrictions of functions holomorphic on $D$ the modulus of which is bounded by $1$ is a compact subset of the Banach space $C(K)$ of continuous functions on $K$. The sequence $(d_m(\mathcal{A}_K^D))_{m\in \mathbb{N}}$ of Kolmogorov $m$-widths of $\mathcal{A}_K^D$ provides a measure of the degree of compactness of the set $\mathcal{A}_K^D$ in $C(K)$ and the study of its asymptotics has a long history, essentially going back to Kolmogorov's work on $\epsilon$-entropy of compact sets in the 1950s. In the 1980s Zakharyuta showed that for suitable $D$ and $K$ the asymptotics \begin{equation*} \lim_{m\to \infty}\frac{- \log d_m(\mathcal{A}_K^D)}{m^{1/n}} = 2\pi \left ( \frac{n!}{C(K,D)}\right ) ^{1/n}\,, \end{equation*} where $C(K,D)$ is the Bedford-Taylor relative capacity of $K$ in $D$ is implied by a conjecture, now known as Zakharyuta's Conjecture, concerning the approximability of the regularised relative extremal function of $K$ and $D$ by certain pluricomplex Green functions. Zakharyuta's Conjecture was proved by Nivoche in 2004 thus settling the asymptotics above at the same time. We shall give a new proof of the asymptotics above for $D$ strictly hyperconvex and $K$ non-pluripolar which does not rely on Zakharyuta's Conjecture. Instead we proceed more directly by a two-pronged approach establishing sharp upper and lower bounds for the Kolmogorov widths. The lower bounds follow from concentration results of independent interest for the eigenvalues of a certain family of Toeplitz operators, while the upper bounds follow from an application of the Bergman-Weil formula together with an exhaustion procedure by special holomorphic polyhedra., Comment: 34 pages; strengthened result: compact $K$ now only assumed to be non-pluripolar

This paper has two goals: to present some new results that are necessary for further study and applications of quasi-linear functionals, and, by combining known and new results, to serve as a convenient single source for anyone interested in quasi-linear functionals on locally compact non-compact spaces or on compact spaces. We study signed and positive quasi-linear functionals paying close attention to singly generated subalgebras. The paper gives representation theorems for quasi-linear functionals on $C_c(X)$ and for bounded quasi-linear functionals on $C_0(X)$ on a locally compact space, and for quasi-linear functionals on $C(X)$ on a compact space. There is an order-preserving bijection between quasi-linear functionals and compact-finite topological measures, which is also "isometric" when topological measures are finite. Finally, we further study properties of quasi-linear functionals and give an explicit example of a quasi-linear functional., Comment: 30 pages

For parameters in a threshold autoregressive process, the paper proposes a sequential modification of the least squares estimates with a specific stopping rule for collecting the data for each parameter. In the case of normal residuals, these estimates are exactly normally distributed in a wide range of unknown parameters. On the base of these estimates, a fixed-size confidence ellipsoid covering true values of parameters with prescribed probability is constructed. In the i.i.d. case with unspecified error distributions, the sequential estimates are asymptotically normally distributed uniformly in parameters belonging to any compact set in the ergodicity parametric region. Small-sample behavior of the estimates is studied via simulation data.

This paper studies the synchronization of two different fractional-order chaotic systems through the fractional-order control method, which can ensure that the synchronization error converges to a sufficiently small compact set. Afterwards, the disturbance observer of the synchronization control scheme based on adaptive parameters is designed to predict unknown disturbances. The Lyapunov function method is used to verify the appropriateness of the disturbance observer design and the convergence of the synchronization error, and then the feasibility of the control scheme is obtained. Finally, our simulation studies verify and clarify the proposed method.

In the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019), the authors have used the Krasnoselskii fixed point theorem for showing the existence of mild solutions of an abstract class of conformable fractional differential equations of the form: $\frac{d^{\alpha }}{dt^{\alpha }}[\frac{d^{\alpha }x(t)}{dt^{\alpha }}]=Ax(t)+f(t,x(t))$ d α d t α [ d α x ( t ) d t α ] = A x ( t ) + f ( t , x ( t ) ) , $t\in [0,\tau ]$ t ∈ [ 0 , τ ] subject to the nonlocal conditions $x(0)=x_{0}+g(x)$ x ( 0 ) = x 0 + g ( x ) and $\frac{d^{\alpha }x(0)}{dt^{\alpha }}=x_{1}+h(x)$ d α x ( 0 ) d t α = x 1 + h ( x ) , where $\frac{d^{\alpha }(\cdot)}{dt^{\alpha }}$ d α ( ⋅ ) d t α is the conformable fractional derivative of order $\alpha \in\, ]0,1]$ α ∈ ] 0 , 1 ] and A is the infinitesimal generator of a cosine family $(\{C(t),S(t)\})_{t\in \mathbb{R}}$ ( { C ( t ) , S ( t ) } ) t ∈ R on a Banach space X. The elements $x_{0}$ x 0 and $x_{1}$ x 1 are two fixed vectors in X, and f, g, h are given functions. The present paper is a continuation of the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) in order to use the Darbo–Sadovskii fixed point theorem for proving the same existence result given in (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) [Theorem 3.1] without assuming the compactness of the family $(S(t))_{t>0}$ ( S ( t ) ) t > 0 and any Lipschitz conditions on the functions g and h.

We consider a structural-acoustic wall problem in three dimensions, in which the structural wall is modeled by a 2D Kirchhoff-Boussinesq plate and the acoustic medium is subject to boundary damping. For this model we study the existence of a continuous nonlinear semigroup associated with the model in the finite energy space. We show that strong/weak continuity of the semigroups depends on the support of the boundary damping. The complications are related to supercritical nonlinearity exhibited by the plate along with the compromised boundary regularity of the acoustic waves. Compensated compactness methods along with a hidden boundary regularity of hyperbolic traces are exploited in order to establish weak (resp. strong) generation of a nonlinear semigroup subjected to feedback forces placed on the boundary of the acoustic medium.

The aim of this paper is to study the controllability and stabilization for the Benjamin equation on a periodic domain T . We show that the Benjamin equation is globally exactly controllable and globally exponentially stabilizable in H p s ( T ) , with s ≥ 0 . The global exponential stabilizability corresponding to a natural feedback law is first established with the aid of certain properties of solution, viz., propagation of compactness and propagation of regularity in Bourgain's spaces. The global exponential stability of the system combined with a local controllability result yields the global controllability as well. Using a different feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrary large decay rate. The results obtained here extend the ones we proved for the linearized Benjamin equation in [32] .