Stability In Norm Research Articles

A fully discrete GL-ADI scheme for 2D time-fractional reaction-subdiffusion equation

Alternating direction implicit (ADI) difference method for solving a 2D reaction-subdiffusion equation whose solution behaves a weak singularity at t=0 is studied in this paper. A Grünwald-Letnikov (GL) approximation is used for the discretization of Caputo fractional derivative (of order α, with 0<α<1) on a uniform mesh. Stability and convergence of the fully discrete ADI scheme are rigorously established. With the help of a discrete fractional Gronwall inequality, we get the sharp error estimate. The stability in L2 norm and the convergence of the GL-ADI scheme are strictly proved, where the convergent order is O(τtsα−1+τ2α+h12+h22). Numerical experiments are given to verify the theoretical analysis.

Simultaneous compensation of input delay and state/input quantization for linear systems via switched predictor feedback

We develop a switched predictor-feedback law, which achieves global asymptotic stabilization of linear systems with input delay and with the plant and actuator states available only in (almost) quantized form. The control design relies on a quantized version of the nominal predictor-feedback law for linear systems, in which quantized measurements of the plant and actuator states enter the predictor state formula. A switching strategy is constructed to dynamically adjust the tunable parameter of the quantizer (in a piecewise constant manner), in order to initially increase the range and subsequently decrease the error of the quantizers. The key element in the proof of global asymptotic stability in the supremum norm of the actuator state is derivation of solutions’ estimates combining a backstepping transformation with small-gain and input-to-state stability arguments, for addressing the error due to quantization. We extend this result to the input quantization case and illustrate our theory with a numerical example.

Unconditionally energy stable high-order BDF schemes for the molecular beam epitaxial model without slope selection

In this paper, we consider a class of k-order (3≤k≤5) backward differentiation formulas (BDF-k) for the molecular beam epitaxial (MBE) model without slope selection. Convex splitting technique along with k-th order Douglas-Dupont regularization term τnk(−Δ)kD_kϕn (D_k represents a truncated BDF-k formula) is added to the numerical schemes to ensure unconditional energy stability. The stabilized convex splitting BDF-k (3≤k≤5) methods are unique solvable unconditionally. Then the modified discrete energy dissipation laws are established by using the discrete gradient structures of BDF-k (3≤k≤5) formulas and processing k-th order explicit extrapolations of the concave term. In addition, based on the discrete energy technique, the L2 norm stability and convergence of the stabilized BDF-k (3≤k≤5) schemes are obtained by means of the discrete orthogonal convolution kernels and the convolution type Young inequalities. Numerical results are carried out to verify our theory and illustrate the validity of the proposed schemes.

Computational evolution of social norms in well-mixed and group-structured populations

Models of indirect reciprocity study how social norms promote cooperation. In these models, cooperative individuals build up a positive reputation, which in turn helps them in their future interactions. The exact reputational benefits of cooperation depend on the norm in place, which may change over time. Previous research focused on the stability of social norms. Much less is known about how social norms initially evolve when competing with many others. A comprehensive evolutionary analysis, however, has been difficult. Even among the comparably simple space of so-called third-order norms, there are thousands of possibilities, each one inducing its own reputation dynamics. To address this challenge, we use large-scale computer simulations. We study the reputation dynamics of each third-order norm and all evolutionary transitions between them. In contrast to established work with only a handful of norms, we find that cooperation is hard to maintain in well-mixed populations. However, within group-structured populations, cooperation can emerge. The most successful norm in our simulations is particularly simple. It regards cooperation as universally positive, and defection as usually negative-unless defection takes the form of justified punishment. This research sheds light on the complex interplay of social norms, their induced reputation dynamics, and population structure.

Time two-grid fitted scheme for the nonlinear time fractional Schrödinger equation with nonsmooth solutions

In this paper, we delve into the regularity properties and the development of an efficient numerical strategy for addressing the nonlinear time-fractional Schrödinger equation. Initially, we embark on an examination of the solution’s regularity, followed by an investigation into regularity enhancement through the application of the Laplace and finite Fourier sine transforms. Subsequently, after the decomposition of u-u0, we introduce an innovative time two-grid fitted scheme designed for the equation’s resolution, which demonstrates unconditional stability and convergence in the L2 norm, achieving the global convergence order of O(τF2+τC4+h2). Here, τF and τC denote the temporal step sizes on the fine and coarse grids, respectively, while h represents the spatial step size. Notably, this is the inaugural application of such an approach for solving the nonlinear time fractional Schrödinger equation. Ultimately, our numerical experiments validate that this method not only retains computational efficiency but also significantly enhances convergence accuracy.

On the stability of norms and norm-following propensity: a cross-cultural panel study with adolescents.

The online version contains supplementary material available at 10.1007/s10683-024-09821-5.

Can Asymmetric Punishment Explain Norm Changes?

Abstract: We test a general theory of norm changes based on evidence that people will punish less hygienic others more strongly than more hygienic others. The theory concludes that such asymmetric punishment would result in hygiene norms becoming ever stricter. We argue that, because complaints about one’s behavior might lead to protest, norms might not always change because of such complaints. We conducted an online experiment ( N = 1,023 Swiss adults) using handwashing as the target behavior. We replicated the asymmetry in punishment intensities and found that the intensity of protests against complaints about one’s too unhygienic behavior approached the intensity of these complaints. We conclude that, while asymmetric punishment may drive norm change, protests may lead to norm stability.

«Правила словацкого правописания» (1940) и стабилизация норм словацкого литературного языка

The article deals with the reasons behind the appearance of the Slovak Spelling Rules of 1940, the most important changes introduced in the codification of the Slovak literary language, and their significance in the development of the language. The need to develop new spelling rules arose immediately after the publication of the Slovak Spelling Rules of 1931, which were not accepted by the majority of Slovak intelligentsia, who perceived them as the first stage on the way to the absorption of the Slovak language by the Czech one. The new rules were prepared mainly by the leader of the purist movement in Slovakia, H. Bartek, who completed his work in 1939. That version, however, was not approved by the Ministry of Education of the Slovak Republic and was sent for revision. The Rules were published the following year in a modified form and affected the morphology, lexical composition, and orthography of the Slovak literary language. Among the most important changes were the unification of the -l participles in plural forms, a significant reduction of doublet forms by abolishing forms coinciding with the Czech language, the written marking of actually pronounced long vowels in words of foreign origin, the removal from the literary language of a number of lexemes common with Czech, and others. Practically all the changes made contributed to the stabilization of the Slovak literary language norms and were confirmed in the subsequent Slovak Spelling Rules of 1953. Moreover, in 1953 even the proposals of H. Bartek rejected in 1940 were adopted.

LEGAL EFFECTS OF THE UKRAINIAN NATIONAL POLICE IN THE FIELD OF ENSURING THE COMPLIANCE WITH ECONOMIC RIGHTS AND FREEDOMS OF HUMAN AND CITIZEN

The aim is to analyse problematic issues related to the legal effect of the activities of the National Police of Ukraine in the sphere of ensuring the observance of economic rights and freedoms of a person and a citizen. The authors used the following methods: analysis (study of the subject of the article as a complex theoretical and legal category), logic (sequential and structured analysis of the topics related to the subject of the paper), interpretation (clarification of the legal content of legislative acts), systemic and structural (combination of the information obtained on the subject of research and its sequential presentation), etc. Results. The authors have established a stable relationship between the activities of the National Police of Ukraine and the mechanism for ensuring compliance with the economic rights and freedoms of a person and a citizen in a number of forms and segments that make up the national economic system. It was noted that the most important thing in the matter of ensuring the functioning of the specified mechanism is the stability and universality of legal norms determining the standards and limits of social and legal interaction in this sphere. This is because, as the example of the ongoing war unleashed by Russia on the territory of Ukraine shows, this destructive factor significantly endangers the functioning sustainability of all, without exception, state and social institutions. Practical significance. The corresponding practical component of the study is the possibility of using the above-mentioned developments in the legislative and regulatory activities, which are systematically implemented taking into account the dynamics of social and legal relations in the conditions of the active phase of hostilities on the territory of Ukraine. Value/originality. The authors have proved that the modern algorithm of the National Police of Ukraine functioning in the field of ensuring the observance of economic rights and freedoms of a man and a citizen is stable, although it is not without a number of shortcomings systematically affecting the relevant institution, which is confirmed by statistical data. The role and place of individual police units, which in turn directly ensure the functioning of the mechanism for ensuring the observance of economic rights and freedoms of a man and a citizen, have been defined.

Dynamic analysis of wave scenarios based on enhanced numerical models for the good Boussinesq equation

The good Boussinesq equation, a modification of the Boussinesq equation, aims to enhance predictions about shallow water wave behavior. This paper introduces two finite difference schemes for solving the good Boussinesq equation including linear and nonlinear implicit finite difference methods. Both schemes utilize the pseudo-compact difference approach, delivering second-order precision with an additional term to boost numerical simulation accuracy while maintaining the grid points of the standard scheme. These schemes rigorously preserve the critical physical characteristics of the good Boussinesq equation, ensuring more precise representation. We establish the existence of solutions with discrete differences and demonstrate, through the discrete energy method, their uniqueness, stability, and second-order convergence in the maximum norm. Furthermore, we propose an iterative algorithm tailored for the nonlinear implicit finite difference scheme, resulting in significant reductions in computational costs compared to the linear scheme. The results of our numerical experiments demonstrate that our methods are competitive and efficient when compared to difference schemes and previously used methods, while maintaining crucial physical qualities. Furthermore, we run relevant numerical simulations to demonstrate the accuracy of the current methods using evidence from the solitary wave interaction with the initial amplitudes of the wave. It is also suggested that the issue has a critical initial wave amplitude for the interaction of two solitary waves, where blow up occurs in a finite amount of time for initial wave amplitudes greater than the new blow-up criteria value.

The stability of Sobolev norms for the linear wave equation with unbounded perturbations

In this paper, we prove that the Sobolev norms of solutions for the linear wave equation with unbounded perturbations of order one remain bounded for all time. The main proof is based on the KAM reducibility of the linear wave equation. To the best of our knowledge, this is the first reducibility result for the linear wave equation with general quasi-periodic unbounded perturbations on the one-dimensional torus.

A norm stability condition of neutral-type Cohen-Grossberg neural networks with multiple time delays

By constructing an appropriate Lyapunov functional, this paper obtains a novel delay-independent stability criterion of neutral-type Cohen-Grossberg neural networks possessing multiple time delays. Although this type of system cannot be represented as vector–matrix form due to the presence of multiple delays, our stability conclusion is fully defined by the infinite norm of parametric matrices and the network parameters first time. Due to the feasibility and simplicity of method proposed, our stability conclusion reducing the computational complexity while also reducing the conservatism compared with several onetime literature. Two concrete neural network models are applied to confirm the effectiveness and superiority of our conclusion.

On stability in the maximum norm of difference scheme for nonlinear parabolic equation with nonlocal condition

We construct and analyze the backward Euler method for one nonlinear one-dimensional parabolic equation with nonlocal boundary condition. The main objective of this article is to investigate the stability and convergence of the difference scheme in the maximum norm. For this purpose, we use the M-matrices theory. We describe some new approach for the estimation of the error of solution and construct the majorant for it. Some conclusions and discussion of our approach are presented.

The efficient ADI Galerkin finite element methods for the three-dimensional nonlocal evolution problem arising in viscoelastic mechanics

In this article, we investigate and analyze new methods for the numerical solution of the three-dimensional nonlocal evolution problem arising in viscoelastic mechanics. Then these methods combine Galerkin finite element methods (FEMs) for the spatial discretization with corresponding alternating direction implicit (ADI) algorithms, based on the backward Euler (BE) method and Crank-Nicolson (CN) method, respectively, from which, the Riemann-Liouville (R-L) integral term is approximated by relevant convolution quadrature rules. The $ L^2 $-norm stability and convergence of two ADI Galerkin schemes are proved by the energy argument. Numerical results confirm the predicted space-time convergence orders.

Local discontinuous Galerkin schemes for an ultrasonic propagation equation with fractional attenuation

The goal of this article is to develop local discontinuous Galerkin (LDG) schemes for solving a time fractional equation describing the ultrasonic wave in a homogeneous isotropic porous material. Two novel semi-discrete LDG schemes are designed for the considered model. The semi-discrete LDG schemes are constructed by splitting the original model into a coupled system. The first semi-discrete scheme follows the traditional LDG method by splitting second-order space derivative. The second one splits the original model for both time and space derivatives. The discontinuous Galerkin is used for the spatial discretization. Two kinds of fully discrete LDG schemes are presented by using the Grünwald-Letnikov and L1 approximation formulas for the time fractional derivatives. The $ L^2 $ norm stability and convergence analysis are carried out for both semi-discrete and fully discrete LDG schemes. The stability analysis reveals that the numerical schemes are unconditionally stable in $ L^2 $ norm and convergence with optimal convergence rate. Finally, numerical examples are presented to test the effectiveness of the proposed schemes and the correctness of the theoretical analysis.

An energy stable linear BDF2 scheme with variable time-steps for the molecular beam epitaxial model without slope selection

An implicit–explicit linear BDF2 scheme with variable steps is proposed for the molecular beam epitaxial model without slope selection. The fourth-order surface diffusion term is treated implicitly and the nonlinear term is approximated by a second order explicit extrapolation. A new type regularization term Aτ2Δh2D2ϕn (D2ϕn represents the variable-step BDF2 formula) is designed to ensure unconditional energy stability in long-time simulations. Under the time-step ratio constraint 0<τk/τk−1<4.864, the linear BDF2 method preserves the modified discrete energy dissipation law unconditionally if a proper large constant A is chosen. Furthermore, with the help of discrete orthogonal convolution kernels and the corresponding convolution Young inequalities, the L2 norm stability and convergence analysis of the linear BDF2 scheme for the MBE model are established. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed scheme.

Global exponential p-norm stability of BAM neural networks with unbounded time-varying delays: A method based on the representation of solutions

This paper studies the global exponential [Formula: see text]-norm stability of bidirectional associative memory (BAM) neural networks with unbounded time-varying delays. A novel method based on the representation of solutions is put forward to deduce a global exponential [Formula: see text]-norm stability criterion. This method does not need to set up any Lyapunov–Krasovskii functionals (LKF), which can greatly reduce a large amount of computations and is simpler than the existing methods. In the end, representative numerical examples are given to illustrate the availability of the method.

Secrecy and Norm Emergence in Cyber-Space. The US, China and Russia Interaction and the Governance of Cyber-Espionage

ABSTRACT When and how does state interaction in cyber-space result in norm development? In this article, we contend that governments take on roles vis-á-vis both domestic and foreign audiences, often resulting in conflicts between crosscutting internal and external role expectations. To alleviate these role conflicts, governments use various secret instruments of statecraft that, in turn, shape international norm development. We theorize the nexus between domestic and foreign role play and secrecy, thereby extending the understanding of role taking in international relations to the cyber-space. We argue that whereas the role conceptions of autocratic powers China and Russia have been geared toward cyber-sovereignty of the regime vis-à-vis internal and external others, the United States, as a democratic power, has sought cyber-security for both state and non-state actors in the international realm. Trying to hide some of their cyber-operations, the interaction between China, Russia and the US has resulted in a distinct pattern of cyber-proxy use and state-based surveillance interaction that has facilitated the stabilization of illiberal cyber-espionage norms and the spread of diverging notions of information sovereignty.

The shadow banning controversy: perceived governance and algorithmic folklore

In this paper, I approach platform governance through algorithmic folklore, consisting of beliefs and narratives about moderation systems that are passed on informally and can exist in tension with official accounts. More specifically, I analyse user discussions on ‘shadow banning’, a controversial, potentially non-existing form of content moderation on popular social media platforms. I argue that discursive mobilisations of the term can act as a methodological entry point to studying the shifting grounds and emerging logics of algorithmic governance, not necessarily in terms of the actual practices themselves, but in terms of its experiential dimension that, in turn, indicates broader modalities and relationalities of control. Based on my analysis of the user discussions, I argue that the constitutive logics of social media platforms increasingly seem to run counter to the values of good governance, such as clarity and stability of norms, and consistency of enforcement. This is reflected in how users struggle, desperately, to form expectations about system operation and police themselves according to perceived rules, yet are left in a state of dependency and frustration, unable to take hold of their digital futures.

On the Stability of Norms and Norm-Following Propensity: A Cross-Cultural Panel Study with Adolescents

On the Stability of Norms and Norm-Following Propensity: A Cross-Cultural Panel Study with Adolescents