Continuous Dependence Research Articles

Structure and dimension of invariant subsets of expanding Markov maps and joint invariance

A long-standing question is what invariant subsets can be shared by two maps acting on the same space. A similar question stands for invariant measures. A particular interesting case are expanding Markov maps of the circle. If the two involved maps are commuting the answer is almost complete. However very little is known in the non-commutative case. A first step is to analyze the structure of the invariant subsets of a single map. For a mapping of the circle of class , , we study the topological structure of the set consisting of all compact invariant subsets. Furthermore for a fixed such mapping we examine locally, in the category sense, how big is the set of all maps that have at least one non trivial joint invariant compact subset. Lastly we show the strong dimensional relation between the maximal invariant subset of a given Markov map contained in a subinterval and the set of all right endpoints of its invariant subsets that are contained in the same subinterval, , as well as the continuous dependence of the dimension on the endpoints of the subinterval .

Quantum-well states at the surface of a heavy-fermion superconductor

Two-dimensional electronic states at surfaces are often observed in simple wide-band metals such as Cu or Ag (refs. 1–4). Confinement by closed geometries at the nanometre scale, such as surface terraces, leads to quantized energy levels formed from the surface band, in stark contrast to the continuous energy dependence of bulk electron bands2,5–10. Their energy-level separation is typically hundreds of meV (refs. 3,6,11). In a distinct class of materials, strong electronic correlations lead to so-called heavy fermions with a strongly reduced bandwidth and exotic bulk ground states12,13. Quantum-well states in two-dimensional heavy fermions (2DHFs) remain, however, notoriously difficult to observe because of their tiny energy separation. Here we use millikelvin scanning tunnelling microscopy (STM) to study atomically flat terraces on U-terminated surfaces of the heavy-fermion superconductor URu2Si2, which exhibits a mysterious hidden-order (HO) state below 17.5 K (ref. 14). We observe 2DHFs made of 5f electrons with an effective mass 17 times the free electron mass. The 2DHFs form quantized states separated by a fraction of a meV and their level width is set by the interaction with correlated bulk states. Edge states on steps between terraces appear along one of the two in-plane directions, suggesting electronic symmetry breaking at the surface. Our results propose a new route to realize quantum-well states in strongly correlated quantum materials and to explore how these connect to the electronic environment.

Well-posedness for a stochastic Camassa–Holm type equation with higher order nonlinearities

This paper aims at studying a generalized Camassa–Holm equation under random perturbation. We establish a local well-posedness result in the sense of Hadamard, i.e., existence, uniqueness and continuous dependence on initial data, as well as blow-up criteria for pathwise solutions in the Sobolev spaces Hs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^s$$\\end{document} with s>3/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s>3/2$$\\end{document} for x∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x\\in \\mathbb R$$\\end{document}. The analysis on continuous dependence on initial data for nonlinear stochastic partial differential equations has gained less attention in the literature so far. In this work, we first show that the solution map is continuous. Then we introduce a notion of stability of exiting time. We provide an example showing that one cannot improve the stability of the exiting time and simultaneously improve the continuity of the dependence on initial data. Finally, we analyze the regularization effect of nonlinear noise in preventing blow-up. Precisely, we demonstrate that global existence holds true almost surely provided that the noise is strong enough.

Analysis of sampling methods for imaging a periodic layer and its defects

We revisit the differential sampling method introduced in (Haddar and Nguyen 2017 Comput. Math. Appl. 74 2831–55) for the identification of a periodic domain and some local perturbation. We provide a theoretical justification of the method that avoids assuming that the local perturbation is also periodic. Our theoretical framework uses functional spaces with continuous dependence with respect to the Floquet–Bloch variable. The corner stone of the analysis is the justification of the generalized linear sampling method in this setting for a single Floquet–Bloch mode.

Well-posedness of a Navier–Stokes–Cahn–Hilliard system for incompressible two-phase flows with surfactant

We investigate a diffuse-interface model that describes the dynamics of viscous incompressible two-phase flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn–Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn–Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier–Stokes system for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. We first prove the existence of a global weak solution, which turns out to be unique in two dimensions. Stronger regularity assumptions on the initial data allow us to prove the existence of a unique global (respectively, local) strong solution in two (respectively, three) dimensions. In the two-dimensional case, we derive a continuous dependence estimate with respect to the norms controlled by the total energy. Moreover, we establish instantaneous regularization properties of global weak solutions for [Formula: see text]. In particular, we show that the surfactant concentration stays uniformly away from the pure states 0 and 1 after some positive time.

Singular stochastic Volterra integral equations with Mittag–Leffler kernels: well-posedness and strong convergence of θ-Maruyama method

This paper focusses on the singular stochastic Volterra integral equations with Mittag–Leffler kernels. Some qualitative properties of the solution are given under local Lipschitz condition, which include uniqueness and existence, boundedness of pth moments, Hölder continuity and continuous dependence on the initial value. The θ-Maruyama method is proposed for solving the equations. The strong convergence results of this method are obtained under global Lipschitz condition and local Lipschitz condition, respectively. Some numerical examples are given to verify the theoretical results.

Cauchy Problem for Incompressible Neo-Hookean Materials

In this paper we consider the Cauchy problem for neo-Hookean incompressible elasticity in spatial dimension $$d \geqq 2$$ . The Cauchy problem can be formulated in terms of maps $$x(t,\cdot )$$ with domain a reference space $${\mathbb {R}}^d_\xi $$ , and with values in space $${\mathbb {R}}^d_x$$ . Initial data consists of initial deformation $$\phi (\xi ) = x(0,\xi )$$ and velocity $$\psi (\xi ) = \partial x(t,\xi )/\partial t |_{t=0}$$ . We consider the initial deformations of the form $$x(0, \xi ) = A \xi + \varphi (\xi )$$ , where A is a constant $$SL(d, {\mathbb {R}})$$ matrix. We assume that $$\varphi $$ and $$\psi $$ are in Sobolev spaces $$(\varphi , \psi ) \in H^{s+1}({\mathbb {R}}^d)\times H^{s}({\mathbb {R}}^d)$$ . If $$s>s_{crit}= d/2+1$$ , well-posedness is well-known. We are here interested primarily in the low regularity case, $$s \le s_{crit}$$ . For $$d = 2, 3$$ , we prove existence and uniqueness for $$s_0 < s\le s_{crit}$$ , and we can prove the well-posedness, but for a smaller range, $$s_1 < s \le s_{crit}$$ , where, if $$d = 2$$ , $$s_0 = 7/4$$ and $$s_1 = 7/4 + (\sqrt{65}-7)/8$$ , and if $$d=3$$ , then $$s_0=2$$ and $$s_1 = 1 + \sqrt{3/2}$$ . For the full range (in s) results, as indicated above, we need additional Hölder regularity assumptions on certain combinations of second order derivatives of $$\varphi $$ . A key observation in the proof is that the equations of evolution for the vorticities decompose into a first-order hyperbolic system, for which a Strichartz estimate holds, and a coupled transport system. This allows one to set up a bootstrap argument to prove local existence and uniqueness. Continuous dependence on initial data is proved using an argument inspired by Bona and Smith, and Kato and Lai, with a modification based on new estimates for Riesz potentials. The results of this paper should be compared to what is known for the ideal fluid equations, where, as shown by Bourgain and Li, the requirement $$s > s_{crit}$$ is necessary.

Continuous Dependence in a Problem of Convergence of Random Iteration

We show that an invariant measure of a Markov operator that is contracting in the Hutchison distance, acting on a space of Borel probability measures on a Polish space, depends continuously on the given operator. In addition, we establish an estimate for a distance between invariant measures. Some applications to the weak limit of iterates of random-valued functions (in particular, the so-called random affine maps, occuring, e.g., in perpetuities analysis) are also given.

Positive periodic solutions for a delay model of erythropoiesis with iterative terms

In the present paper, we mainly focus on an iterative erythropoiesis model with a nonlinear harvesting term involving a time-varying delay. Under certain conditions and by the Schauder fixed point theorem, we show that the existence of positive periodic solutions can be guaranteed and based on the Banach fixed point theorem, we establish the existence as well as the continuous dependence on parameters theorems of the unique solution. Moreover, two examples are provided to validate the correctness of our outcomes. Our theoretical findings extend and improve several related ones in the existing literature.

A higher order implicit adaptive finite point method for the Burgers' equation

ABSTRACT The method proposed in this manuscript is constructed for the classical non-linear Burgers' equation on the basis of the tailored finite point method. Attempts have been made in the direction to enhance the tailored finite point method with appealing features like cost-effectiveness and rapidity through the usage of a minimum machinery algorithm, ability to handle shocks and boundary layers without mesh refinement, and continuous dependence on the initial conditions throughout the algorithm. Also, the foundation of the construction of the algorithm is completely based on the local properties of the solution which thereby brings in the influence of localized behaviours even when the kinematic viscosity is infinitesimal. Moreover, the usage of a 4-point stair stencil makes it an implicit scheme, and the adaptation of adequate linearization techniques reduces the usage of higher-order machinery to handle non-linearity. The conditional stability, consistency, and convergence of the method have been deliberated with appropriate theories. The method is second-order convergent in space and first-order convergent in time. The credibility of the method is tested through diverse test examples by comparing it with the exact solutions and other methods in the literature and the results are found to be at par excellence with the classical methods.

Analysis of SIRVI model with time dependent coefficients and the effect of vaccination on the transmission rate and COVID-19 epidemic waves.

COVID-19 epidemic models with constant transmission rate cannot capture the patterns of the infection data in the presence of pharmaceutical and non-pharmaceutical interventions during a pandemic. Because of this, a new modification of SIR model that contain the vaccination compartment with time dependent coefficients and weak/loss-immunity is explored. Literature review confirms that the effect of vaccination on the time dependent transmission rate is still an open problem. This study answers this open problem. In this study, we first prove the well-posedness and investigate the model dynamics to show their continuous dependence on the model parameters. We then provide an algorithm to derive the time-dependent transmission function for the epidemiologic model and the data of the infected cases. The derived coupled nonlinear differential equations show the effect of vaccination on the transmission rate. Unlike previous studies, we first filter the published data and solve the nonlinear coupled differential equations using the finite difference technique, where the coefficient of the coupled nonlinear differential equations is a function of given data. We then show that time-dependent transmission function can be represented by linear combinations of Gaussian radial base function. We then validate the prediction of our models using numerical simulations, where we used the published data of COVID-19 confirmed cases by the Ministries of Health in Saudi Arabia and Poland. Finally, the numerical solutions of a SIRVI model with time dependent transmission rate show that the waves for currently active cases are in good agreement with the data of Saudi Arabia and Poland.

TÍNH CHẤT NGHIỆM CHO MỘT LỚP CÁC BẤT ĐẲNG THỨC HEMI-BIẾN PHÂN KIỂU PARABOLIC

In this paper, we study a class of parabolic hemivariational inequalities in infinite dimensional spaces. We prove the solvability and the continuous dependence on initial conditions for this problem based on surjective lemma, monotone theory and estimates.

Mixed cumulative probit: a multivariate generalization of transition analysis that accommodates variation in the shape, spread and structure of data.

Biological data are frequently nonlinear, heteroscedastic and conditionally dependent, and often researchers deal with missing data. To account for characteristics common in biological data in one algorithm, we developed the mixed cumulative probit (MCP), a novel latent trait model that is a formal generalization of the cumulative probit model usually used in transition analysis. Specifically, the MCP accommodates heteroscedasticity, mixtures of ordinal and continuous variables, missing values, conditional dependence and alternative specifications of the mean response and noise response. Cross-validation selects the best model parameters (mean response and the noise response for simple models, as well as conditional dependence for multivariate models), and the Kullback-Leibler divergence evaluates information gain during posterior inference to quantify mis-specified models (conditionally dependent versus conditionally independent). Two continuous and four ordinal skeletal and dental variables collected from 1296 individuals (aged birth to 22 years) from the Subadult Virtual Anthropology Database are used to introduce and demonstrate the algorithm. In addition to describing the features of the MCP, we provide material to help fit novel datasets using the MCP. The flexible, general formulation with model selection provides a process to robustly identify the modelling assumptions that are best suited for the data at hand.

Qualitative Behaviour of Stochastic Integro-differential Equations with Random Impulses

In this paper, we study the existence and some stability results of mild solutions for random impulsive stochastic integro-differential equations (RISIDEs) with noncompact semigroups in Hilbert spaces via resolvent operators. Initially, we prove the existence of mild solution for the system is established by using Mönch fixed point theorem and contemplating Hausdorff measures of noncompactness. Then, the stability results includes continuous dependence of solutions on initial conditions, exponential stability and Hyers–Ulam stability for the aforementioned system are investigated. Finally, an example is proposed to validate the obtained results.

Reading Machiavelli and La Boétie with Lefort

Abstract This article will explore the historical account and political actualisation of Machiavelli and La Boétie in the work of Claude Lefort. In the 1970s, Lefort renewed the interpretation of Machiavelli and La Boétie by underlining their common ‘radical humanism’. The long-overlooked insights into desire and social division of the two Renaissance thinkers underline the subversive potential of humanism against its common ideological and oligarchic uses. But the history of radical humanism cannot be separated from its topicality, as it is one of the germs of the democratic revolution. This radicalism echoes and inspires Lefort's agonistic theory of democracy. After Machiavelli and La Boétie, his work grapples with the continued dependency of the subjects on visible and invisible masters. Indeed, according to Lefort, contemporary societies long for certainty and remain haunted by servitude in the form of ideology (be it bourgeois, totalitarian or invisible).

Analytical Contribution to a Cubic Functional Integral Equation with Feedback Control on the Real Half Axis

Synthetic biology involves trying to create new approaches using design-based approaches. A controller is a biological system intended to regulate the performance of other biological processes. The design of such controllers can be based on the results of control theory, including strategies. Integrated feedback control is central to regulation, sensory adaptation, and long-term effects. In this work, we present a study of a cubic functional integral equation with a general and new constraint that may help in investigating some real problems. We discuss the existence of solutions for an equation that involves a control variable in the class of bounded continuous functions BC(R+), by applying the technique of measure of noncompactness on R+. Furthermore, we establish sufficient conditions for the continuous dependence of the state function on the control variable. Finally, some remarks and discussion are presented to demonstrate our results.

Continuous dependence of eigenvalues on potential functions for nonlocal Sturm–Liouville equations

In this paper, we discuss the continuous dependence of eigenvalues on the potential function for a nonlocal Sturm–Liouville equation with truncated Marchaud fractional derivative term by two‐parameter method. To this end, we first study the properties of eigenvalues for a two‐parameter nonlocal Sturm–Liouville eigenvalue problem. We obtain the two‐parameter nonlocal Sturm–Liouville eigenvalue problem that has countable number of simple real eigenvalues. Meanwhile, the asymptotic behavior of eigenvalues is studied by aid of the analytic perturbation theory.

Age-and sex-specific reference intervals of total cholesterol, LDL cholesterol, HDL cholesterol and non-HDL cholesterol. Comparison of two algorithms for the indirect estimation of reference intervals

Abstract Objectives Reference intervals of total cholesterol concentrations in plasma and of their fractions low-density lipoprotein (LDL)-, high-density lipoprotein (HDL)-and non-HDL concentrations are seldom studied with respect to the relevance of age and sex. Therefore, the effect of age and sex on the reference intervals was reinvestigated with 2 indirect procedures. Methods As an indirect approach, the truncated minimum chi-square method was applied. All analyses were performed by computer programs available. The script published on the homepage of the German Society of Clinical Chemistry and Laboratory Medicine (DGKL) allows to derive a continuous age dependency of reference intervals together with their confidence and equivalence limits. The results of this approach were compared with those obtained by an indirect method developed more recently, the refineR algorithm. Results In the present study, the upper reference limits of total cholesterol varied from 5.1 to 7.8 mmol/L (197–302 mg/dL) depending on various biological variables (as age, sex, inpatients versus outpatients). These upper limits increased with age. Differences between sexes can be neglected except for the age above 80 years. The pattern of reference limits of LDL cholesterol and non-HDL cholesterol paralleled those of total cholesterol. The reference limits of HDL cholesterol were higher in women than in men but were independent of age. Conclusions Reference limits for the concentrations of total cholesterol and their fractions LDL-, HDL-and non-HDL concentrations should be stratified for age and sex.

Exploring donor-driven skills development as a channel of continued aid dependency

This article examines donor-driven skills training programmes and projects at the technical and vocational level in post-war Sierra Leone, using dependency theory as an analytical framework. Based on qualitative data collected from fieldwork in 2017, it emerges that such programmes and projects are driven by donor strategies and largely detached from local market demands, are oftentimes based on convenience and historical relationships in regions/sectors rather than evolving needs of the country, promise employment but instead deliver informal self-employment and focus heavily on outputs rather than outcomes. From these empirical observations, it can be argued that donor-driven skills development, as it has manifested in Sierra Leone, is unlikely to create skills that can meaningfully contribute to national growth and development. Instead, these interventions may drive outcomes that lead to continued aid dependency. The article thus presents a renewed argument for skills for self-reliant development.

Existence and regularity for global weak solutions to the 𝜆-family water wave equations

In this paper, we prove the global existence of Hölder continuous solutions for the Cauchy problem of a family of partial differential equations, named as λ \lambda -family equations, where λ \lambda is the power of nonlinear wave speed. The λ \lambda -family equations include Camassa-Holm equation ( λ = 1 \lambda =1 ) and Novikov equation ( λ = 2 \lambda =2 ) modelling water waves, where solutions generically form finite time cusp singularities, or, in other words, show wave breaking phenomenon. The global energy conservative solution we construct is Hölder continuous with exponent 1 − 1 2 λ 1- \frac {1}{2\lambda } . The existence result also paves the way for the future study on uniqueness and Lipschitz continuous dependence.