Nonlocal Case Research Articles

Free vibration response of micromorphic Timoshenko beams

In this paper the authors investigate the free vibration of a two-length-scale nonlocal micromorphic Timoshenko beam, which is shown to overlap with the nonlocal strain gradient Timoshenko beam under certain conditions. Hamilton’s principle is utilized to obtain a system of two coupled fourth-order equations of motion governing the eigen-deflection and the eigen-rotation of the beam. Uncoupling both equations leads to two eight-order differential equations. Using Ferrari’s method, exact solutions are derived for the eigenfrequencies for various boundary conditions, including simply supported, clamped-clamped, clamped-free, and clamped-hinged boundary conditions. The obtained results are compared with those published in the literature using similar nonlocal strain gradient cases. A detailed parametric study is then performed to emphasize the role of the variationally-derived higher-order boundary conditions (natural higher-order boundary conditions or mixed higher-order boundary conditions). It is noted that when the difference in length-scales is large, the effect of the slenderness of the beam on the frequencies is amplified. Finally, the hardening or the softening effect of the beam model can be achieved through a choice of the ratio between the two length-scales.

Activation of Einstein–Podolsky–Rosen steering sharing with unsharp nonlocal measurements

Einstein–Podolsky–Rosen (EPR) steering is commonly shared among multiple observers by utilizing unsharp measurements. Nevertheless, their usage is restricted to local measurements and does not encompass all nonlocal measurement-based cases. In this work, a method for finding beneficial local measurement settings has been expanded to include nonlocal measurement cases. This method is applicable for any bipartite state and offers benefits even in scenarios with a high number of measurement settings. Using the Greenberger–Horne–Zeilinger state as an illustration, we show that employing unsharp nonlocal measurements can activate the phenomenon of steering sharing in contrast to using local measurements. Furthermore, our findings demonstrate that nonlocal measurements with unequal strength possess a greater activation capability compared to those with equal strength. Our activation method generates fresh concepts for conservation and recycling quantum resources.

Renormalisable Non-Local Quark–Gluon Interaction: Mass Gap, Chiral Symmetry Breaking and Scale Invariance

We derive a Nambu–Jona-Lasinio (NJL) model from a non-local gauge theory and show that it has confining properties at low energies. In particular, we present an extended approach to non-local QCD and a complete revision of the technique of Bender, Milton and Savage applied to non-local theories, providing a set of Dyson–Schwinger equations in differential form. In the local case, we obtain closed-form solutions in the simplest case of the scalar field and extend it to the Yang–Mills field. In general, for non-local theories, we use a perturbative technique and a Fourier series and show how higher-order harmonics are heavily damped due to the presence of the non-local factor. The spectrum of the theory is analysed for the non-local Yang–Mills sector and found to be in agreement with the local results on the lattice in the limit of the non-locality mass parameter running to infinity. In the non-local case, we confine ourselves to a non-locality mass that is sufficiently large compared to the mass scale arising from the integration of the Dyson–Schwinger equations. Such a choice results in good agreement, in the proper limit, with the spectrum of the local theory. We derive a gap equation for the fermions in the theory that gives some indication of quark confinement in the non-local NJL case as well. Confinement seems to be a rather ubiquitous effect that removes some degrees of freedom in the original action, favouring the appearance of new observable states, as seen, e.g., for quantum chromodynamics at lower energies.

Unraveling the Photoluminescent Properties of Sb-Doped Cd-Based Inorganic Halides: A First-Principles Study.

Sb-doped Cd-based inorganic halides, with varying connections of CdCl6 octahedra ranging from 0D to 3D, exhibit a variety of photoluminescent properties. Single-band emission is observed in Sb-doped Rb4CdCl6 (0D) and Cs2CdCl4 (2D), while dual-band emission is seen in Sb-doped RbCdCl3 (1D) and CsCdCl3 (3D). Density-functional-based first-principles calculations were conducted. The results reveal that cation vacancies, acting as charge compensators, influence the luminescence properties of dopant centers. In CsCdCl3, the local cation vacancy VCd″ for Sb3+ at the Cd2+ site ([Sb□Cl9]6-) significantly modifies the photoluminescence property, accounting for the observed dual-band emission alongside the nonlocal compensation case. This effect is insignificant in Sb-doped Rb4CdCl6, RbCdCl3, and Cs2CdCl4, due to the large distances or high formation energies of Cd vacancies in these hosts. However, in Sb-doped RbCdCl3, two different potential energy minima, one that involves typical structure relaxation and the other that is off-center, lead to the observed dual-band emission. Furthermore, the shift of the charge transition level illustrates the different temperature dependences of the dual-band emission caused by the charge-compensating point defects. These insights not only enhance our understanding of luminescent materials based on halides containing ns2 dopants but also provide valuable guidance for the design and optimization of luminescent materials.

Magnetic fractional Poincaré inequality in punctured domains

We study Poincaré-Wirtinger type inequalities in the framework of magnetic fractional Sobolev spaces. In the local case, Lieb et al. (2003) [19] showed that, if a bounded domain Ω is the union of two disjoint sets Γ and Λ, then the Lp-norm of a function calculated on Ω is dominated by the sum of magnetic seminorms of the function, calculated on Γ and Λ separately. We show that the straightforward generalisation of their result to nonlocal setup does not hold true in general. We provide an alternative formulation of the problem for the nonlocal case. As an auxiliary result, we also show that the set of eigenvalues of the magnetic fractional Laplacian is discrete.

Isoperimetric inequalities in nonlocal diffusion problems with integrable kernel

We deduce isoperimetric estimates for solutions of linear stationary and evolution problems. Our main result establishes the comparison in norm between the solution of a problem and its symmetric version when nonlocal diffusion defined through integrable kernels is replacing the usual local diffusion defined by a second order differential operator. Since an appropriate kernel rescaling allows to define a sequence of solutions of the nonlocal diffusion problems converging to their local diffusion counterparts, we also find the corresponding isoperimetric inequalities for the latter, i.e. we prove the classical Talenti's theorem. The novelty of our approach is that we replace the measure geometric tools employed in Talenti's proof, such as the geometric isoperimetric inequality or the coarea formula, by the Riesz's rearrangement inequality. Thus, in addition to providing a proof for the nonlocal diffusion case, our technique also introduces an alternative proof to Talenti's theorem.

Traveling wave fronts in a single species model with cannibalism and strongly nonlocal effect

<p>In this paper we studied traveling front solutions of a single species model with cannibalism and nonlocal effect. For a particular class of kernels, the existence of traveling front solutions connecting the extinction state with the positive equilibrium was established for the strongly nonlocal effect case. Our approach was to reformulate it as a singular perturbed problem, and then tackle this problem by using dynamical systems techniques, in particular, geometric singular perturbation theory and Fenichel's invariant manifold theory.</p>

К вопросу о пространственных отношениях в финском языке

The study presents a thorough review, conducted from a linguocultural perspective, of the primary principles used in expressing spatial relations in Finnish through linguistic categories and means. It covers the genesis and changes of cases in the Finnish language, paying special attention to those that conveyed spatial meaning at a particular stage of the language development. The issue is particularly interesting as there is an ongoing debate about the application of certain cases owing to the significant influence of dialects in Finnish, which is characterized by an egocentric spatial orientation. The study aimed to explore the instances of diverging spatial representations regarding the direction of actions in different linguocultures, with Finnish and Russian linguocultures taken as examples. The research applied a quantitative analysis approach to examine competing forms, with a corpus of YLE newspaper texts from 2011 to 2020 as the research material. The study analyzed instances of differing depictions through verb-noun pairs and verbs related to sensual and tangible perception (vaikutelmaverbit). Additionally, we investigated the variations in verb usage of the latter and the verb ‘käydä’ (‘to go, to visit’), which is generally presented as ‘static’, but expresses its dynamic qualities in different collocations. The analysis indicates that the system of spatial relations in Finnish is characterized by linguocultural specificity and is expressed grammatically through local and non-local case forms, prepositions, postpositions, and verb-noun syntactic structures. Furthermore, one and the same name in the internal and external case forms may convey not only distinct grammatical but also different lexical meanings. The use of local cases with certain verbs can lend expressions and verbs a figurative meaning that is only loosely connected to position in space.

Diversification, vertical integration and economic resilience: evidence from intercity truck flows during COVID-19 in China

AbstractThis article examines economic resilience by combining high-frequency truck flows and the lockdown policy shock during COVID-19 in China. We discover that the truck flows in regions with higher levels of diversification and vertical integration see a smaller decrease in response to the COVID-19 shock. Dynamically, such moderating effects of diversification and vertical integration get smaller with the recovery of interregional economic linkages. Diversification and integration also mitigate the negative impact from nonlocal infection cases. The association between the industrial structure attributes and economic resilience is more prominent in regions with lower centrality in the nationwide intercity truck flow network.

Nonautonomous dynamics of local and nonlocal Fokas–Lenells models

We investigate the dynamics of solutions to the local and nonlocal nonautonomous Fokas–Lenells (FL) models. Solutions to the local and nonlocal reductions are presented using the Wronskian technique by imposing constraints on the double Wronskian solutions to the unreduced nonautonomous FL system. Classifications of the soliton solutions are shown in relation to the distribution of the eigenvalues. Moreover, dynamics of these solutions are presented, along with the impacts of the variable coefficient . Notably, the coefficient influences the top traces and velocities of the found solutions in both the local and nonlocal cases, enriching this study.

Effect of non-local transport of hot electrons on the laser-target ablation

The non-local heat transport of hot electrons during high-intensity laser interaction with plasmas can preheat the fuel and limit the heat flow in inertial confinement fusion. It increases the entropy of the fuel and decreases the final compression. In this paper, the non-local electron transport model that is based on the improved Schurtz–Nicolaï–Busquet (SNB) algorithm has been embedded into the radiation hydrodynamic code and is benchmarked with two classical non-local transport cases. Then, we studied a 2ω laser ablating a CH target by using the non-local module. It is found that the non-local effect becomes significant when the laser intensity is above 1×1014W/cm2. The mass ablation rate from the SNB model is increased compared to that of the flux-limited model due to the lower coronal plasma temperature. This non-local model has a better agreement with the experimental results compared to that of the flux-limited model. The non-local transport is strongly dependent on the laser frequency, and the thresholds that the non-local transport should be considered are obtained for lasers of different frequencies. The appropriate flux-limiters that should be employed in the flux-limited model for different lasers are also presented. The results here should have a good reference for the laser-target ablation applications.

Normalized Solutions to the Fractional Schrödinger Equation with Potential

This paper is concerned with the existence of normalized solutions to a class of Schrödinger equations driven by a fractional operator with a parametric potential term. We obtain minimization of energy functional associated with that equations assuming basic conditions for the potential. Our work offers a partial extension of previous results to the non-local case.

Positive solutions for Kirchhoff equation in exterior domains with small Sobolev critical perturbation

ABSTRACT The paper concerns with the existence of positive solutions for the following Kirchhoff problem { − ( a + b ∫ Ω | ∇u | 2 d x ) Δu + u = | u | p − 2 u + ε | u | 4 u in Ω , u ∈ H 0 1 ( Ω ) , where a, b>0, 4<p<6, 0 $ ]]> ε > 0 is a parameter and Ω ⊂ R 3 is an exterior domain, that is, Ω is an unbounded domain with R 3 ∖Ω nonempty and bounded. After showing the nonexistence of ground state solutions, we prove the existence of one positive solution with higher energy when R 3 ∖Ω is contained in a small ball and 0 $ ]]> ε > 0 is sufficiently small. The novelty of this paper is that we extend the result obtained in Alves and de Freitas [Existence of a positive solution for a class of elliptic problems in exterior domains involving critical growth. Milan J Math. 2017;85:309–330] to nonlocal case and generalize the subcritical nonlinearity discussed in Chen and Liu [Positive solutions for Kirchhoff equation in exterior domains. J Math Phys. 2021;62:Article ID 041510] to small critical perturbation.

On a Fractional Nirenberg Problem Involving the Square Root of the Laplacian on $${\mathbb {S}}^{3}$$

In this paper, we are devoted to establishing the compactness and existence results of the solutions to the fractional Nirenberg problem for $$n=3,$$ $$\sigma =1/2,$$ when the prescribing $$\sigma $$ -curvature function satisfies the $$(n-2\sigma )$$ -flatness condition near its critical points. The compactness results are new and optimal. In addition, we obtain a degree-counting formula of all solutions. From our results, we can know where blow up occur. Moreover, for any finite distinct points, the sequence of solutions that blow up precisely at these points can be constructed. We extend the results of Li (Commun Pure Appl Math 49:541–597, 1996) from the local problem to nonlocal cases.

Weak Harnack inequality for a mixed local and nonlocal parabolic equation

This article proves a weak Harnack inequality with a tail term for sign changing supersolutions of a mixed local and nonlocal parabolic equation. Our argument is purely analytic. It is based on energy estimates and the Moser iteration technique. Instead of the parabolic John-Nirenberg lemma, we adopt a lemma of Bombieri-Giusti to the mixed local and nonlocal parabolic case. To this end, we prove an appropriate reverse Hölder inequality and a logarithmic estimate for weak supersolutions.

Inverse scattering transforms for the nonlocal Hirota–Maxwell–Bloch system

The Hirota and Maxwell–Bloch (H-MB) system is a mathematical model that can be used to describe the wave propagation in an erbium-doped nonlinear fiber with higher-order dispersion. By extending the parameter ω in the H-MB system to the complex domain, all the nonlocal forms of the H-MB system, including the reverse-space-time, complex reverse-space, complex reverse-time, complex reverse-space-time H-MB systems, are found. Then the Riemann–Hilbert problem of the nonlocal H-MB system is established to analyze the corresponding inverse scattering problem and construct the corresponding soliton solutions. When N = 1, the singularity of the one-soliton solutions of each nonlocal H-MB system is analyzed. When N = 2, we take the nonlocal reverse-space-time H-MB system and the nonlocal complex reverse-time H-MB system as examples, showing that a two-soliton can also be regarded as a superposition of two single-soliton in the nonlocal cases as .

A vectorial Darboux transformation for the Fokas–Lenells system

We first reveal that there is a Riccati-type Miura transformation from the Fokas–Lenells (FL) system to the first member of the negative part of the AKNS hierarchy. We derive a vectorial Darboux transformation for the FL system by using the bidifferential graded algebra approach. Two types of reductions lead to the FL equation and nonlocal FL (nFL) equation, respectively. From vanishing seed solutions, N-soliton solutions for the FL equation and nFL equation are provided, with spectral matrices of the form of diagonal. Finally, as an application, we perform an asymptotic analysis for the bright–bright solitons of the nonlocal case.

Symmetry and quantitative stability for the parallel surface fractional torsion problem

We study symmetry and quantitative approximate symmetry for an overdetermined problem involving the fractional torsion problem in a bounded open set Ω ⊂ R n \Omega \subset \mathbb R^n . More precisely, we prove that if the fractional torsion function has a C 1 C^1 level surface which is parallel to the boundary ∂ Ω \partial \Omega then Ω \Omega is a ball. If instead we assume that the solution is close to a constant on a parallel surface to the boundary, then we quantitatively prove that Ω \Omega is close to a ball. Our results use techniques which are peculiar of the nonlocal case as, for instance, quantitative versions of fractional Hopf boundary point lemma and boundary Harnack estimates for antisymmetric functions. We also provide an application to the study of rural-urban fringes in population settlements.

The separation property for 2D Cahn-Hilliard equations: Local, nonlocal and fractional energy cases

We study the separation property for Cahn-Hilliard type equations with constant mobility and (physically relevant) singular potentials in two dimensions. That is, any solution with initial finite energy stays uniformly away from the pure phases $ \pm 1 $ from a certain time on. Beyond its physical interest, this property plays a crucial role to achieve high order Sobolev and analytic regularity of the solutions and to analyze their longtime behavior. In the local case, we streamline known arguments by exploiting the Sobolev inequality to obtain direct entropy estimates. In the nonlocal case, we provide a new proof based on De Giorgi estimates rather than the Alikakos-Moser type argument. Finally, in the spectral-fractional case, we prove nonlinear estimates and the separation property for any fractional index $ s\in (0,1) $ filling the gap between first-order (local) and zero-order (nonlocal) energy cases. In all of the aforementioned cases, our new proofs neither make use of the Trudinger-Moser inequality nor of any assumptions involving the third derivative of the entropy, as in the previous contributions. In particular, they apply for a more general class of singular potentials than the Flory-Huggins (Boltzmann-Gibbs) logarithmic density. Besides, the new methods present a series of technical advantages, which can be useful to the analysis of important physical systems that couple Cahn-Hilliard equations with other equations (e.g., reaction-diffusion equations and/or Navier-Stokes type systems) as well as their stochastic counterparts.

Discrete approximation of stationary Mean Field Games

&lt;p style='text-indent:20px;'&gt;In this paper, we focus on stationary (ergodic) mean-field games (MFGs). These games arise in the study of the long-time behavior of finite-horizon MFGs. Motivated by a prior scheme for Hamilton–Jacobi equations introduced in Aubry–Mather's theory, we introduce a discrete approximation to stationary MFGs. Relying on Kakutani's fixed-point theorem, we prove the existence and uniqueness (up to additive constant) of solutions to the discrete problem. Moreover, we show that the solutions to the discrete problem converge, uniformly in the nonlocal case and weakly in the local case, to the classical solutions of the stationary problem.&lt;/p&gt;