Weak Limit Research Articles

Weak gravitational lensing and fundamental frequencies of Einstein–Euler–Heisenberg black hole

This paper is suggested to analyze the gravitational weak lensing and fundamental frequencies in the framework of the Einstein–Euler–Heisenberg (EEH) black hole. We compute the deflection angle of light by the EEH black hole in weak field limits. Which represents that the bending of light is a global and topological effect. For this purpose, we deduce the Gaussian curvature and apply the Gauss–Bonnet theorem (GBT). Furthermore, we determine the deflection angle at which light is deflected by a plasma medium. We also look into how an EEH black hole behaves graphically in vacuum and plasma medium. Moreover, we study the fundamental frequencies with three different models of EEH black hole.

On second order differential inclusion driven by quasi-variational–hemivariational inequalities

In this paper we investigate an evolution problem (SDQHVI) which constitutes of the second order differential inclusion driven by a quasi-variational–hemivariational inequality (QHVI) with perturbation operator in Banach spaces. Unlike the existing literature on differential inclusions, on the one hand, the second order differential operator is not assumed to be compact; on the other hand, the solvability of the inequality is proved under non-coercive condition. More precisely, first, it is shown that the solution set of the QHVI with perturbation operator is nonempty, bounded, closed and convex under the KKM theorem and the Minty formulation. Then, an associated multivalued map with the solution set of the QHVI with perturbation operator is introduced, and we prove that it is upper semicontinuous and measurable. Finally, by utilizing Kakutani–Fan–Glicksberg fixed point theorem, a weak topology technique and properties of strong continuous cosine operators, we prove the existence of mild solutions for SDQHVI.

Comparison Theorems for Weak Topologies (3)

Weak topology on a nonempty set X is defined as the smallest or weakest topology on X with respect to which a given (fixed) family of functions on X is continuous. In an effort to construct and analyze strictly stronger weak topologies, only four weak topologies (namely, the Arens topology, the Mackey topology, the weakened topology, and the strong topology) have been compared before now. Also, all these weak topologies were constructed with an eye on only the polars of linear spaces (i.e. with a focus on only linear functionals on linear spaces). To that extent, the study of strictly stronger weak topologies before now can be described as a study of linear topological spaces. Here in Part 3 of our Comparison Theorems for Weak Topologies: We established and clarified the place of our comparison theorems in the context of the weakened topology, the Arens topology, the Mackey topology, and the strong topology that have up to now been compared. We showed that there are many other weak topologies between the four already compared weak topologies—constructible even by the use of polars. In particular, we showed that we can we find a weak topology stronger than the one hitherto known and referred to as the strong topology. Then we showed that if two weak topologies, generated by one family of functions on a set, are strictly comparable, then there exist in a range space two strictly comparable topologies which induce the weak topologies. The rather simplistic view that” The weak topology is Hausdorff” has been held by many authors for very long time. We changed that narrative here, as we stated and proved (with examples) that:” Some weak topologies are Hausdorff while others are not.”

Local structure of convex surfaces

A point on the surface of a convex body and a supporting plane to the body at this point are under consideration. A plane parallel to this supporting plane and cutting off part of the surface is drawn. The limiting behaviour of the cut-off part of the surface as the cutting plane approaches the point in question is investigated. More precisely, the limiting behavior of the appropriately normalized surface area measure in $S^2$ generated by this part of the surface is studied. The cases when the point is regular and singular (a conical or a ridge point) are considered. The supporting plane can be positioned in different ways with respect to the tangent cone at the point: its intersection with the cone can be a vertex, a line (if a ridge point is considered), a plane angle (which can degenerate into a ray or a half-plane), or a plane (if the point is regular and, correspondingly, the cone degenerates into a half-space). In the case when the intersection is a ray, the plane can be tangent (in a one- or two-sided manner) or not tangent to the cone. It turns out that the limiting behaviour of the measure can be different. In the case when the intersection of the supporting plane and the cone is a vertex or in the case of a (one- or two-sided) tangency, the weak limit always exists and is uniquely determined by the plane and the cone. In the case when the intersection is a line or a ray with no tangency, there may be no limit at all. In this case all possible weak partial limits are characterized. Bibliography: 13 titles.

Comparison Theorems for Weak Topologies (1)

Weak topology on a nonempty set X is defined as the smallest or weakest topology on X with respect to which a given (fixed) family of functions on X is continuous. Let τw be a weak topology generated on a nonempty set X by a family {fα,:α ∈ ∆} of functions, together with a corresponding family {(Xα, τα):α ∈ ∆} of topological spaces. If for some α0 ∈ ∆, τα0 on Xα0 is not the indiscrete topology and fα0 meets certain requirements, then there exists another topology τw1 on X such that τw1 is strictly weaker than τw and fα is τw1-continuous, for all α ∈ ∆. Here in Part 1 of our Comparison Theorems for Weak Topologies it is observed that: (a) the new topology τw1 on X deserves to be called a weak topology (with respect to the fixed family of functions) in its own right. Hence, we call τw1 a strictly weaker weak topology on X, than τw; (b) the usual weak, weak star, and product topologies have chains of pairwise strictly comparable (respectively) weak, weak star, and product topologies around them. All the necessary and sufficient conditions for the existence of τw1 in relation to τw are established. Ample examples are given to illustrate (at appropriate places) the various issues discussed.

Hyperplane-Open Weak Topologies in Rn

A general procedure of constructing hyperplane-open weak topology on Rn is exposed. The process also leads to the formulation of matrix-open weak topology on Rn.This is a very interesting discovery since we never expected that a matrix of points in the Cartesian plane could come out as an open set.

Existence of large-data global weak solutions to kinetic models of nonhomogeneous dilute polymeric fluids

We prove the existence of large-data global-in-time weak solutions to a general class of coupled bead-spring chain models with finitely extensible nonlinear elastic (FENE) type spring potentials for nonhomogeneous incompressible dilute polymeric fluids in a bounded domain in $ \mathbb{R}^d $, $ d = 2 $ or $ 3 $. The class of models under consideration involves the Navier–Stokes system with variable density, where the viscosity coefficient depends on both the density and the polymer number density, coupled to a Fokker–Planck equation with a density-dependent drag coefficient. The proof is based on combining a truncation of the probability density function with a two-stage Galerkin approximation and weak compactness and compensated compactness techniques to pass to the limits in the sequence of Galerkin approximations and in the truncation level.

On the lack of compactness in the axisymmetric neo-Hookean model

Abstract We provide a fine description of the weak limit of sequences of regular axisymmetric maps with equibounded neo-Hookean energy, under the assumption that they have finite surface energy. We prove that these weak limits have a dipole structure, showing that the singular map described by Conti and De Lellis is generic in some sense. On this map, we provide the explicit relaxation of the neo-Hookean energy. We also make a link with Cartesian currents showing that the candidate for the relaxation we obtained presents strong similarities with the relaxed energy in the context of $\mathbb {S}^2$ -valued harmonic maps.

Gravitational waves in New General Relativity

The models of New General Relativity have recently got attention of research community, and there are some works studying their dynamical properties. The formal aspects of this investigation have been mostly restricted to the primary constraints in the Hamiltonian analysis. However, it is by far not enough for counting their degrees of freedom or judging whether they are any good and viable. In this paper we study linearised equations in vacuum around the trivial Minkowski tetrad. By taking the approach of cosmological perturbation theory we show that the numbers of primary constraints are very easily seen without any need of genuine Hamiltonian techniques, and give the full count of linearised degrees of freedom in the weak field limit of each and every version of New General Relativity without matter.

Ordered patterns in electroactive polymer ionic liquid blends: effect of long range interactions.

Designing multifunctional soft materials via pattern formation has been a major challenge for scientists and engineers. Soft materials based on polymers are the perfect candidates for designing such materials as they are not only easy to handle, but also offer diverse combinations of mechanical and chemical properties. Here, we present a polymer-based ternary system and reveal, using modelling and simulations, the mechanisms for creating patterned surfaces. Specifically, we consider polymer ionic liquid (PIL) blends and demonstrate that exposure to a uniform electric field results in the formation of ordered patterns through phase separation. Our approach is based on reaction-diffusion phenomena and utilizes Poisson-Boltzmann-Nernst-Planck equations to capture the long-range interactions of ionic liquids in both weak and strong segregation limits. Furthermore, we elucidate that the ordered patterns in our PIL blend can be tuned by changing the direction of the electric field. From the structural characterization point of view, we reveal that the presence of the electric field significantly enhances the domain growth rate and their respective ordering in a remarkable fashion. We believe this non-invasive technique is a significant step towards the development of ordered structures at microscopic length scales and can be utilized for micro-scale fabrication from soft materials.

A note on the adapted weak topology in discrete time

A note on the adapted weak topology in discrete time

Studies on invariant measures of fractional stochastic delay Ginzburg-Landau equations on Rn.

This paper is concerned with invariant measures of fractional stochastic delay Ginzburg-Landau equations on the entire space $ \mathbb{R}^n $. We first derive the uniform estimates and the mean-square uniform smallness of the tails of solutions in corresponding space. Then we deduce the weak compactness of a set of probability distributions of the solutions applying the Ascoli-Arzel$ \grave{a} $. We finally prove the existence of invariant measures by applying Krylov-Bogolyubov's method.

Weak limit of homeomorphisms in W1,n−1: Invertibility and lower semicontinuity of energy

Let Ω, Ω′ ⊂ ℝn be bounded domains and let fm: Ω → Ω′ be a sequence of homeomorphisms with positive Jacobians Jfm > 0 a.e. and prescribed Dirichlet boundary data. Let all fm satisfy the Lusin (N) condition and supm ∫Ω( |D fm|n - 1 + A( |cof D fm|) + φ(Jf)) < ∞, where A and φ are positive convex functions. Let f be a weak limit of fm in W1,n−1. Provided certain growth behaviour of A and φ, we show that f satisfies the (INV) condition of Conti and De Lellis, the Lusin (N) condition, and polyconvex energies are lower semicontinuous.

Closing lemmas for interval translation maps

A interval translation mapping (or a circle translation mapping) is studied. Such maps can be regarded as interval exchange maps with overlaps. It is known that for any mapping of that type admits a Borel probability invariant non-atomic measure. This measure can be constructed as a weak limit of invariant measures of maps with periodic parameters. Those measures, are just normalized Lebesgue ones on a family of sub-sectors. For such limit measures, in the case of a shift of the arcs of the circle, it is shown that any point of their supports can be made periodic by arbitrarily small change of the parameters of the system without changing the number of segments. For any invariant measure, it is deduced from Poincar´es Recurrence Theorem shows every point can be made periodic by a small change in the parameters of the system, with the number of intervals increasing by two at most.

Controlled Extended Branciari Quasi-b-Metric Spaces, Results, and Applications to Riesz-Caputo Fractional Differential Equations and Nonlinear Matrix Equations

We introduce the concept of controlled extended Branciari quasi-b-metric spaces, as well as a Gq-implicit type mapping. Under this new space setting, we derive some new fixed points, periodic points, right and left Ulam–Hyers stability, right and left weak well-posed properties, and right and left weak limit shadowing results. Additionally, we use these findings to solve the fractional differential equations of a Riesz–Caputo type with integral anti-periodic boundary values, as well of nonlinear matrix equations. All ideas, results, and applications are properly illustrated with examples.

Confinement and nonlinear electrodynamics: Asymptotic Schwarzschild charged black hole

While most nonlinear electrodynamics (NED) models tend to Maxwell’s linear theory in the weak field regime, very recently a model of NED has been proposed in the context of heavy quark–antiquark confinement that gives the same limit in the strong field regime. This form of correction to Maxwell’s linear theory is shown to be applicable in such a confined system. In this model, the radial electric field of a point charge is described by E=qr2+ζqr which consists of a Coulomb term plus a term proportional to the NED parameter ζ. Here in this paper, we extend the previous study on the model and accordingly, first, present more on the properties of the model and then introduce the magnetic black hole solution obtained in the gravity coupled minimally to this NED. The solution due to the unique property of the NED is asymptotically Schwarzschild black hole. In addition to the thermodynamics of the black hole, we investigate the circular speed–radius curve of the hypothetical star orbiting the mentioned black hole. It is shown that again due to the correction of Maxwell’s theory in the weak field limit these curves are equivalent to the corresponding curves in the Schwarzschild black hole.

Searching for Planets Orbiting Fomalhaut with JWST/NIRCam

We report observations with the JWST/NIRCam coronagraph of the Fomalhaut (α PsA) system. This nearby A star hosts a complex debris disk system discovered by the IRAS satellite. Observations in F444W and F356W filters using the round 430R mask achieve a contrast ratio of ∼4 × 10−7 at 1″ and ∼4 × 10−8 outside of 3″. These observations reach a sensitivity limit of <1 M Jup across most of the disk region. Consistent with the hypothesis that Fomalhaut b is not a massive planet but is a dust cloud from a planetesimal collision, we do not detect it in either F356W or F444W (the latter band where a Jovian-sized planet should be bright). We have reliably detected 10 sources in and around Fomalhaut and its debris disk, all but one of which are coincident with Keck or Hubble Space Telescope sources seen in earlier coronagraphic imaging; we show them to be background objects, including the “Great Dust Cloud” identified in Mid-Infrared Instrument (MIRI) data. However, one of the objects, located at the edge of the inner dust disk seen in the MIRI images, has no obvious counterpart in imaging at earlier epochs and has a relatively red [F356W]–[F444W] > 0.7 mag (Vega) color. Whether this object is a background galaxy, brown dwarf, or a Jovian-mass planet in the Fomalhaut system will be determined by an approved Cycle 2 follow-up program. Finally, we set upper limits to any scattered light from the outer ring, placing a weak limit on the dust albedo at F356W and F444W.

Limit Distributions of Products of Independent and Identically Distributed Random 2 × 2 Stochastic Matrices: A Treatment with the Reciprocal of the Golden Ratio

Consider a sequence (Xn)n≥1 of i.i.d. 2×2 stochastic matrices with each Xn distributed as μ. This μ is described as follows. Let (Cn,Dn)T denote the first column of Xn and for a given real r with 0&lt;r&lt;1, let r−1Cn and r−1Dn each be Bernoulli distributions with parameters p1 and p2, respectively, and 0&lt;p1,p2&lt;1. Clearly, the weak limit of the sequence μn, namely λ, is known to exist, whose support is contained in the set of all 2×2 rank one stochastic matrices. In a previous paper, we considered 0&lt;r≤12 and obtained λ explicitly. We showed that λ is supported countably on many points, each with positive λ-mass. Of course, the case 0&lt;r≤12 is tractable, but the case r&gt;12 is very challenging. Considering the extreme nontriviality of this case, we stick to a very special such r, namely, r=5−12 (the reciprocal of the golden ratio), briefly mention the challenges in this nontrivial case, and completely identify λ for a very special situation.

The open string pair production, its enhancement and the physics behind

Why adding a collinear magnetic field to the electric one on a D3 brane in a system of two parallel separated D3 branes can enhance the open string pair production? How is the open string pair production rate related to the QED ones in the weak field limit? We answer these questions and report the somehow expected but still remarkable relation between the weak field non-perturbative stringy rate and the corresponding QED ones in this letter.

Geometric Stochastic Resonance in an Asymmetric T-Shaped Chamber

The investigation of a Brownian particle subjected to an AC force that diffuses within a T-shaped chamber was conducted. This T-shaped chamber is composed of a strip cavity and a trapezoidal cavity positioned below it. The interplay between the AC force and asymmetric geometry creates a spatially bistable potential perpendicular to the AC force. With the assistance of noise, the particles can transition between two stable states and oscillate along the AC force at corresponding amplitudes at every spatially stable state. The asymmetric geometry facilitates the trapezoid cavity’s ability to more easily trap the Brownian particle than the upper strip cavity in the weak noise limit. Our observations reveal that proper noise can ensure the particle’s efficient trapping within the upper strip cavity and synchronization with the AC force, indicating the occurrence of geometric stochastic resonance. The T-shaped chamber serves as a simplified model, aiding in the further understanding of geometric stochastic resonance induced by irregular geometries and enabling the manipulation of microscopic particles in various small-scale systems.