Order Terms Research Articles

Stability estimates of Nyström discretizations of Helmholtz decomposition boundary integral equation formulations for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions

Abstract Helmholtz decompositions of elastic fields is a common approach for the solution of Navier scattering problems. Used in the context of boundary integral equations (BIE), this approach affords solutions of Navier problems via the simpler Helmholtz boundary integral operators (BIOs). Approximations of Helmholtz Dirichlet-to-Neumann (DtN) can be employed within a regularizing combined field strategy to deliver BIE formulations of the second kind for the solution of Navier scattering problems in two dimensions with Dirichlet boundary conditions, at least in the case of smooth boundaries. Unlike the case of scattering and transmission Helmholtz problems, the approximations of the DtN maps we use in the Helmholtz decomposition BIE in the Navier case require incorporation of lower order terms in their pseudodifferential asymptotic expansions. The presence of these lower order terms in the Navier regularized BIE formulations complicates the stability analysis of their Nyström discretizations in the framework of global trigonometric interpolation and the Kussmaul–Martensen kernel singularity splitting strategy. The main difficulty stems from compositions of pseudodifferential operators of opposite orders, whose Nyström discretization must be performed with care via pseudodifferential expansions beyond the principal symbol. The error analysis is significantly simpler in the case of arclength boundary parametrizations and considerably more involved in the case of general smooth parametrizations that are typically encountered in the description of one-dimensional closed curves.

An elliptic problem in dimension N with a varying drift term bounded in L N

The present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in L N ( Ω ), with N the dimension of the space. It is known that there exists a unique solution for each of these problems in the Sobolev space H 0 1 ( Ω ). However, because the operators are not coercive, there is no uniform estimate of the solutions in this space. We use some estimates in (J. Differential Equations 258 (2015) 2290–2314), and a regularization obtained by adding a small nonlinear first order term, to pass to the limit in these problems.

Effect of lower order terms on the well-posedness of Majda-Biello systems

Effect of lower order terms on the well-posedness of Majda-Biello systems

Wilson Lines in the Abelian Lattice Higgs Model

AbstractLattice gauge theories are lattice approximations of the Yang–Mills theory in physics. The abelian lattice Higgs model is one of the simplest examples of a lattice gauge theory interacting with an external field. In a previous paper (Forsström et al. in Math Phys 4(2):257–329, 2023), we calculated the leading order term of the expected value of Wilson loop observables in the low-temperature regime of the abelian lattice Higgs model on $${\mathbb {Z}}^4,$$ Z 4 , with structure group $$G = {\mathbb {Z}}_n$$ G = Z n for some $$n \ge 2.$$ n ≥ 2 . In the absence of a Higgs field, these are important observables since they exhibit a phase transition which can be interpreted as distinguishing between regions with and without quark confinement. However, in the presence of a Higgs field, this is no longer the case, and a more relevant family of observables are so-called open Wilson lines. In this paper, we extend and refine the ideas introduced in Forsström et al. (Math Phys 4(2):257–329, 2023) to calculate the leading order term of the expected value of the more general Wilson line observables. Using our main result, we then calculate the leading order term of several natural ratios of expected values and confirm the behavior predicted by physicists.

Comparative analysis of accelerated gradient algorithms for convex optimization: high and super resolution ODE approach

We investigate convex differentiable optimization and explore the temporal discretization of damped inertial dynamics driven by the gradient of the objective function. This leads to three accelerated gradient algorithms: Nesterov Accelerated Gradient (NAG), Ravine Accelerated Gradient (RAG), and (IGAHD). The latter was introduced by discretizing inertial dynamics with Hessian-driven damping to attenuate inherent oscillations in inertial methods. By analysing the high-resolution ODEs of order p = 0,1,2 for these algorithms, we gain insights into their similarities and differences. All three algorithms share the same low-resolution ODE of order 0, which is the dynamic proposed as a continuous surrogate for (NAG). To differentiate Nesterov from Ravine, we refine the comparison and demonstrate distinct high-resolution ODEs of order 2 in h (termed super-resolution). The corresponding Taylor expansions in h reveal matching terms of order 1 but differing terms of order 2. To the best of our knowledge, this result is completely new and emphasizes the need to avoid confusion between the Ravine and Nesterov methods in the literature. We present numerical experiments to illustrate our theoretical results. Performance profiles, measuring the number of iterations, indicate that (IGAHD) outperforms both (NAG) and (RAG) methods. (RAG) exhibits a slight advantage over (NAG) in terms of the average number of iterations. When considering CPU-time, both (RAG) and (NAG) outperform (IGAHD). All three algorithms exhibit similar behaviour when evaluating based on gradient norms.

Existence results for p(x)-Laplacian equations with singular lower order terms and irregular data

Existence results for p(x)-Laplacian equations with singular lower order terms and irregular data

Fractional Caputo Operator and Takagi–Sugeno Fuzzy Modeling to Diabetes Analysis

Diabetes is becoming more and more dangerous, and the effects continue to grow due to the population’s ignorance of the seriousness of this phenomenon. The studies that have been carried out have not been able to follow the phenomenon more precisely, which has led to the use of the fractional derivative tool, which has a very great capability to study real problems and phenomena but is somewhat limited on nonlinear models. In this work, we will develop a new fractional derivative model of a diabetic population, the Takagi–Sugeno fractional fuzzy model, which will enable us to study the phenomenon with these nonlinear terms in order to obtain greater precision in the results. We will study the existence and uniqueness of the solution using the Lipschizian theorem and then turn to the new fuzzy model, which leads us to four dynamical systems. The interpretation results show the quality of fuzzy membership in tracking the malleable phenomena of nonlinear terms existing in the system.

Theoretical analysis and predictions for the two-neutrino double electron capture of 124Xe

Abstract We provide a complete theoretical description of the two-neutrino electron capture in 124Xe improving both the nuclear and the atomic structure calculations. We improve the general formalism through the use of the Taylor expansion method, leading to higher order terms in the decay rate of the process. The nuclear part is treated with pn-QRPA and interacting shell model (ISM) methods. The nuclear matrix elements (NMEs) are calculated with the pn-QRPA method with isospin restoration by fixing the input parameters so that the experimental decay rate is reproduced, resulting in values significantly lower than in previous calculations. The validity of the pn-QRPA NMEs is tested by showing their values to be comparable with the ones for double-beta decay with emission of two electrons of 128,130Te, which have similar pairing features. Within the ISM, we reproduce the total experimental half-life within a factor of two and predict the capture fraction to the KK channel of about 74%. We also predict the capture fractions to other decay channels and show that for the cumulative decay to the KL1-KO1 channels, a capture fraction of about 24% could be observed experimentally. On the atomic side, calculations are improved by accounting for the Pauli blocking of the decay of innermost nucleon states and by considering all s-wave electrons available for capture, expanding beyond the K and L1 orbitals considered in previous studies. We also provide improved atomic relaxation energies of the final atomic states of 124Te, which may be used as input for background modeling in liquid Xenon experiments.

A first order in time wave equation modeling nonlinear acoustics

In this paper we focus on a small amplitude approximation of a Navier-Stokes-Fourier system modeling nonlinear acoustics. Omitting all third and higher order terms with respect to certain small parameters, we obtain a first order in time system containing linear and quadratic pressure and velocity terms. Subsequently, the well-posedness of the derived system is shown using the classical method of Galerkin approximation in combination with a fixed point argument. We first prove the well-posedness of a linearized equation using energy estimates and then the well-posedness of the nonlinear system using a Newton-Kantorovich type argument. Based on this, we also obtain global in time well-posedness for small enough data and exponential decay. This is in line with semigroup results for a linear part of the system that we provide as well.

A new error estimate of a finite difference scheme for a fractional transport-advection equation with zero order term

In this work, we propose a finite difference scheme for Caputo’s fractional derivative transport equation in time and space, with a zero-order term. A new error estimation of the approximate solution has been demonstrated. By introducing an approximation of the Caputo derivative, we proved that the convergence is of order 2-α in time, and 2-β in space, for 0<α;β<1. Results of conditional stability and convergence of the numerical method are discussed. Finally a numerical implementation was presented to show the conformity between the theoretical and numerical approach of the finite difference scheme.

DOMESTIC VIOLENCE PROTECTION ORDER APPLICATION GAPS AS A VIOLATION OF HUMAN RIGHTS

The article examines the main gaps of the implementation of the new version of the Law on Protection Against Domestic Violence, which entered into force in Lithuania on the 1st of July 2023, and its accompanying legislation. The topic analysed in the article is of great relevance today, because the number of criminal offences related to domestic violence in Lithuania is increasing. Domestic violence is defined as any form of physical, emotional, sexual or psychological abuse, it is a violation of human rights. The new version of the Law on Protection Against Domestic Violence introduced a new concept called the Domestic Violence Protection Order, which is a preventive protection measure designed to protect a victim of domestic violence, and which obliges abuser to move temporarily from the place of residence. This is an important step towards creating a secured society, however, there is a lack of information addressed to the problems of the implementation of Domestic Violence Protection Order in Lithuania in the context of the protection of human rights. It is therefore reasonable to investigate what problems are faced by police officers when issuing a domestic violence order and by the courts when dealing with complaints about compliance with the terms of the order issued. Thus, the object of the investigation is the Domestic Violence Protection Order. The aim of the article is to investigate the Domestic Violence Protection Order application gaps as a violation of human rights. The Objectives of the study are 1) To explore the concept of domestic violence and its legal regulation; 2) To carry out a case study of Lithuanian court practice of the Domestic Violence Protection Order application problems. The methods used for the investigation: analysis of literature and legislation and a case study of a court decision. To fulfil the purpose of the research, a specific case of court practice was analysed. In this case even several violations committed by police officers when issuing a domestic violence order for a person and checking the person's compliance with the terms of the domestic violence order were recorded and analysed in the court decision. It was found that despite police officers play an important role in ensuring the safety of the public, in some cases they decide without analysing and investigating the factual circumstances, without giving reasons and legal arguments. This creates several problems in terms of human rights protection. Thus, police officers in some cases violate the rights of the violent person and this highlights the problems faced by police officers when issuing a Domestic Violence Protection Order and by the courts when dealing with complaints about compliance with the terms of the order issued. A special attention of Lithuanian legislators should be given to these gaps of Domestic Violence Protection Order application as they can lead to the violation of human rights.

Nonlinear Elliptic Equations with Variable Exponents Anisotropic Sobolev Weights and Natural Growth Terms

Abstract The purpose of our paper is to prove the existence of the distributional solutions for anisotropic nonlinear elliptic equations with variable exponents, which contain lower order terms dependent on the gradient of the solution and on the solution itself. The terms are weighted, and the main results rely on the possibility of comparing the weights with each other, where the right-hand side is a sum of the natural growth term and the datum f ∈ L 1(Ω). Furthermore the weight function θ(·) is in W̊ 1,p→(·) (Ω), with θ(·) &gt; 0 and connected with the coefficient b(·) ∈ L 1(Ω) of the lower order term.

Rethinking the effective field theory formulation of gravity

General relativity is highly successful in explaining a wide range of gravitational phenomena including the gravitational waves emitted by binary systems and the shadows cast by supermassive black holes. From a modern perspective the theory is not fundamental though, but constitutes the lowest order term in an effective field theory description of the gravitational force. As a consequence, the gravitational dynamics should receive corrections by higher-derivative terms. This essay discusses structural aspects associated with these corrections and summarizes their imprint on static, spherically symmetric geometries. Along these lines, we critically reassess the common practice of using local field redefinitions in order to simplify the dynamics at the danger of shifting physics effects into sectors which are beyond the approximation under consideration.

Theoretical Analysis of Viscoelastic Friction System Characteristics of Robotic Arm Brake Based on Fractional Differential Theory

In engineering practice, the nonlinear vibration effect can easily lead to chaos in the system, which will not only reduce the performance of the system but also lead to premature fatigue of components, control failure, and increased safety risks. In view of the core position of the robotic arm in modern industry, this study relies on the robotic arm brake system to explore the theoretical basis of integrated viscoelastic materials as a vibration isolation layer. By analyzing the dynamic characteristics of the friction braking system with fractional differential terms, it aims to provide a new perspective for understanding and controlling the chaotic phenomena of a class of nonlinear friction systems. Firstly, we construct a model of a friction system and analyze its dynamic characteristics in detail. The self-excited vibration of the system under disturbance is studied. The relationship between amplitude and frequency is calculated by a nonlinear approximate analytical algorithm, and the accuracy of this relationship is verified by a numerical algorithm. Then, we compare the differences between non-fractional systems and fractional systems. It is found that with the increase in the fractional order term, the vibration amplitude of the system decreases significantly, which helps to reduce the nonlinear characteristics generated by the friction system and narrow the range of unstable solutions. Secondly, we also study the influence of parameter coefficients on the amplitude–frequency characteristics and analyze the local static bifurcation characteristics through singularity theory. Finally, we study the dynamic bifurcation behavior under different parameter perturbations and find that the change in system parameters will lead to the alternation of periodic motion and chaotic motion.

Dangerous dichotomies and misunderstandings in L2 research

This article draws attention to the obstacles created by imprecise definitions and misleading dichotomies. In this case the focus is on second language and multilingualism research literature. Despite the obvious benefits of reformulating otherwise complex ideas and approaches in simpler terms in order to distill their essential meaning, misunderstandings and misrepresentations can too easily arise. These can then proliferate within and beyond these research fields and hinder productive debates between proponents of different theoretical approaches. A selection of classic examples relating to issues of language cognition are discussed in this article. They include definitions of terms like cognitive, nativist, innatist, interactionist, usage-based, and dynamic. Simple contrasts introduced to improve readability and to introduce a longer discussion can become highly misleading where the context fails to include clarification. Incorrect inferences may otherwise be drawn by less well-informed readers. In other cases, dichotomies can be just wrong. We discuss the choice and phrasing of various terms and distinctions and argue that more care is needed by all involved. Discussions can still be carried out in a combative style if discussants so wish but not to the point of introducing conceptual confusion in debates that should serve to advance understanding in second language acquisition research.

A strain-based criterion for predicting size-dependent fracture resistance of quasi-brittle materials under mixed mode loading

In the present study, a strain-based approach called the modified maximum tangential strain criterion has been developed for evaluating size effect on the mixed-mode fracture resistance of quasi-brittle materials. This approach relies on the generalized maximum tangential strain criterion, which considers the singular (K) and the first non-singular (T) terms of Williams series expansion. Furthermore, as an important and size-dependent parameter in the proposed criterion, a new formulation is presented to calculate the critical distance (rc). The predictions of the new criterion are compared not only with the available experimental data, but also with the results estimated by another size effect criterion named the modified maximum tangential stress criterion. It is shown that the proposed strain-based criterion is highly capable of estimating the size-dependent mixed-mode fracture behavior of quasi-brittle materials without requiring to compute the other higher order terms.

Moments of Artin–Schreier L-functions

Abstract We compute moments of L-functions associated to the polynomial family of Artin–Schreier covers over $\mathbb{F}_q$, where q is a power of a prime p &amp;gt; 2, when the size of the finite field is fixed and the genus of the family goes to infinity. More specifically, we compute the $k{\text{th}}$ moment for a large range of values of k, depending on the sizes of p and q. We also compute the second moment in absolute value of the polynomial family, obtaining an exact formula with a lower order term, and confirming the unitary symmetry type of the family.

Inventing and Re-inventing Populism to Protect Europe: The Case of the Italian Partito Democratico, 2007-2022

ABSTRACT Populism has been used by its critics to support European integration. The case of the Italian Partito Democratico (PD) highlights that, as a relatively empty signifier, ‘populism’ affords parties scope to adapt to different circumstances, as evidenced by the evolving use of the term in relation to shifts in policymaking paradigms at the European level. In the period between the Eurozone crisis and the Covid pandemic, the PD changed the connotation of populism in relation to key debates at the European level. While initially the term was used mainly to contrast unproductive uses of spending by the Berlusconi government and then by the M5S-League coalition, the definition of populism narrowed as European support for austerity measures faded. More recently, the PD has started using the term almost exclusively as a synonym for right-wing extremism. The Italian case suggests that critics of populism continuously shift their understanding of the term in order to protect European integration.

Hierarchical three-body problem at high eccentricities = simple pendulum I: octupole

ABSTRACT The gradual evolution of the restricted hierarchical three body problem is analysed analytically, focusing on conditions of Kozai–Lidov cycles that may lead to orbital flips from prograde to retrograde motion due to the octupole (third order) term which are associated with extremely high eccentricities. We revisit the approach described by Katz, Dong and Malhotra (2011) and show that for most initial conditions, to an excellent approximation, the analytical derivation can be greatly simplified and reduces to a simple pendulum model allowing an explicit flip criterion. The resulting flip criterion is much simpler than the previous one but the latter is still needed in a small fraction of phase space. We identify a logical error in the earlier derivation but clarify why it does not affect the final results.

Characterization of the aerosol contribution to the top-of-atmosphere radiance for satellite ocean color retrievals

A method is proposed for characterization of the aerosol contribution to the top-of-atmosphere (TOA) radiance. The method is based on solving the problem of radiative transfer in the atmosphere-ocean system and expanding the solution in powers of the aerosol optical thickness τ a . We show that the linear term of the expansion is analytically expressed in terms of the bidirectional transmittance/reflectance of the aerosol-free Rayleigh atmosphere. A procedure is also proposed for successively extracting the terms of higher order in τ a from the data of the TOA radiance computation with the DISORT code. As analysis shows, the radiance expansion in τ a is not purely polynomial. Beginning from the quadratic term, the coefficients of the series expansion in powers of τ a become dependent logarithmically on τ a . The approach proposed enables us to reproduce analytically the τ a -dependence of the TOA radiance with controlled accuracy. We determine the expansion coefficients up to the cubic term inclusive and validate our results on the aerosol model embedded in NASA’s SeaDAS algorithm for aerosol loadings, representative for the Barents and Kara seas. In the visible and near-infrared spectral ranges, accounting for the terms up to a quadratic one is found to be sufficient for the atmospheric correction of satellite ocean color data typical for the Arctic region.