Vector Field Research Articles

Shaking a container full of perfect liquid; a tractable case, a torus shell, exhibits a virtual wall

Manipulation (‘shaking’) of a rigid container filled with incompressible liquid starting from stationary generally results in some displacement, or mixing, of the liquid within it. If the liquid also has zero viscosity, a ‘perfect’ or ‘Euler’ liquid, Kelvin’s theorems dramatically simplify the flow analysis. Response is instantaneous; stop the container and all liquid motion stops. In fact an arbitrary manipulation can be considered as alternating infinitesimal translations and rotations of the container. Relative to the container, the liquid is stationary during every translation. Infinitesimal rotations (an infinitesimal vector along the rotation axis) resolve into three orthogonal components in the container frame. Each generates its own infinitesimal liquid displacement vector field. Their combined consequences are usually obscure. Rather than a volume flow, a surface flow in 3D is considerably easier, the liquid slipping freely in a shell, sandwiched between two nested closed surfaces with constant infinitesimal gap. The closedness avoids extra boundaries. The two dimensionality admits a scalar streamfunction determined by the container angular velocity vector. Manipulation of the container angular velocity at will, usually leads to an infinitely rich variety of area preserving re-configurations of the liquid. In particular, any chosen point of the liquid can be moved, in the shell container frame, to any other chosen point. However, for a torus (shell) with a small enough hole (diameter < 0.195 torus diameter), there exists a virtual wall, a hypothetical axial cylinder intersecting the torus. No matter how the torus is manipulated, liquid inside the cylinder stays inside; outside stays outside. The analysis, solving for the streamfunction, is based on the relative vorticity, and the conformal mapping of a torus to a (periodic) rectangle, which lead to a fairly simple convolution integral formula for the flow.

Resolvent of vector fields and Lefschetz numbers

Resolvent of vector fields and Lefschetz numbers

Engineering the Band Topology in a Rhombohedral Trilayer Graphene Moiré Superlattice.

Moiré superlattices have become a fertile playground for topological Chern insulators, where the displacement field can tune the quantum geometry and Chern number of the topological band. However, in experiments, displacement field engineering of spontaneous symmetry-breaking Chern bands has not been demonstrated. Here in a rhombohedral trilayer graphene moiré superlattice, we use a thermodynamic probe and transport measurement to monitor the Chern number evolution as a function of the displacement field. At a quarter filling of the moiré band, a novel Chern number of three is unveiled to compete with the well-established number of two upon turning on the electric field and survives when the displacement field is sufficiently strong. The transition can be reconciled by a nematic instability on the Fermi surface due to the pseudomagnetic vector field potentials associated with moiré strain patterns. Our work opens more opportunities to active control of Chern numbers in van der Waals moiré systems.

On the tangent bundles over F-Kahlerian manifolds

The main purpose of the present paper is to study all forms of Riemannien curvatures and the harmonic Killing vector fields of a tangent bundle over an F−Kahlerian manifold endowed with a Berger type deformed Sasaki metric gBS .

Cosmological implications of the Weyl geometric gravity theory

We consider the cosmological implications of the Weyl geometric gravity theory. The basic action of the model is obtained from the simplest conformally invariant gravitational action, constructed, in Weyl geometry, from the square of the Weyl scalar, the strength of the Weyl vector, and a matter term, respectively. The total action is linearized in the Weyl scalar by introducing an auxiliary scalar field. To maintain the conformal invariance of the action the trace condition is imposed on the matter energy–momentum tensor, thus making the matter sector of the action conformally invariant. The field equations are derived by varying the action with respect to the metric tensor, the Weyl vector field, and the scalar field, respectively. We investigate the cosmological implications of the theory, and we obtain first the cosmological evolution equations for a flat, homogeneous and isotropic geometry, described by Friedmann–Lemaitre–Robertson–Walker metric, which generalize the Friedmann equations of standard general relativity. In this context we consider two cosmological models, corresponding to the vacuum state, and to the presence of matter described by a linear barotropic equation of state. In both cases we perform a detailed comparison of the predictions of the theory with the cosmological observational data, and with the standard Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} CDM model. By assuming that the presence of the Weyl geometric effects induce small perturbations in the homogeneous and isotropic cosmological background, and that the anisotropy parameter is small, the equations of the cosmological perturbations due to the presence of the Weyl geometric effects are derived. The time evolution of the metric and matter perturbations are explicitly obtained. Therefore, if Weyl geometric effects are present, the Universe would acquire some anisotropic characteristics, and its geometry will deviate from the standard FLRW one.

Optical vectorial-mode parity Hall effect: a case study with cylindrical vector beams

The vectorial optical field (VOF) assumes a pivotal role in light-matter interactions. Beyond its inherent polarization topology, the VOF also encompasses an intrinsic degree of freedom associated with parity (even or odd), corresponding to a pair of degenerate orthogonal modes. However, previous research has not delved into the simultaneous manipulation of both even and odd parities. In this study, we introduce and validate the previously unexplored parity Hall effect for vectorial modes using a metasurface design. Our focus lies on a cylindrical vector beam (CVB) as a representative case. Through the tailored metasurface, we effectively separate two degenerate CVBs with distinct parities in divergent directions, akin to the observed spin states split in the spin Hall effect. Additionally, we provide experimental evidence showcasing the capabilities of this effect in multi-order CVB demultiplexing and parity-demultiplexed CVB-encoded holography. This effect unveils promising opportunities for various applications, including optical communication and imaging.

On the integration of $$L^0$$-Banach $$L^0$$-modules and its applications to vector calculus on $$\textsf{RCD}$$ spaces

AbstractA finite-dimensional $$\textsf{RCD}$$ RCD space can be foliated into sufficiently regular leaves, where a differential calculus can be performed. Two important examples are given by the measure-theoretic boundary of the superlevel set of a function of bounded variation and the needle decomposition associated to a Lipschitz function. The aim of this paper is to connect the vector calculus on the lower dimensional leaves with the one on the base space. In order to achieve this goal, we develop a general theory of integration of $$L^0$$ L 0 -Banach $$L^0$$ L 0 -modules of independent interest. Roughly speaking, we study how to ‘patch together’ vector fields defined on the leaves that are measurable with respect to the foliation parameter.

Optical skyrmions in the Bessel profile

Optical skyrmions formed in terms of polarization are topological quasi-particles, and they have garnered much interest in the optical community owing to their unique inhomogeneous polarization structure and simplicity in their experimental realization. These structures belong to the Poincaré beams satisfying the stable topology. We theoretically investigated the non-diffracting and self-healing Poincaré beams based on the superposition of two orthogonal Bessel modes by the longitudinal mode matching technique. These Poincaré beams are topologically protected, and we suggest them as optical skyrmions in the corresponding Stokes vector fields. These optical skyrmions are quasi-skyrmions, and their range of propagation depends on the range of superposed Bessel modes. We have shown longitudinal mode matching of superposed Bessel beams is a necessary condition for the generation of propagation-invariant and non-diffracting skyrmions. The proposed longitudinal mode matching technique facilitates the generation of skyrmions with tunable position and range without any on-axis intensity modulations along the propagation axis. A suitable experimental configuration is suggested to realize variable order skyrmions in Bessel modes. The suggested experimental configuration can produce optical skyrmions even in ultra-short laser pulses with high mode conversion efficacy. This work can provide a new direction for the generation of skyrmions with completely new textures and features with reference to existing skyrmions originating from Laguerre-Gaussian modes.

Left-handed metamaterial based on circular split ring (CSRR) resonator for microwave sensing Application

This paper presents a new metamaterial structure specifically designed to operate in the L, S, and C frequency bands. It is characterized by a small unit cell size of 15mm × 15 mm. The metamaterial's electromagnetic properties are thoroughly examined, utilizing FR-4 substrate material and copper circular split ring resonators (CSRRs). The Computer Simulation Technology (CST) software uses the finite integration technique (FIT) to precisely model the effective parameters over a frequency range of 0–6 GHz. In simulations and experiments, the structure demonstrates negative refractive index, effective permittivity, and effective permeability. The electric and magnetic field distributions within the unit cell are accurately characterized using vector fields. In addition, the equivalent circuit is simulated and depicted using ADS software, with comparisons drawn between the measured data and the simulated data. The experimental validation, conducted using six varying thicknesses of FR4 materials, verifies the resemblance between the simulated and the measured transmission coefficients. Moreover, the effective medium ratio (EMR) of the metamaterial is calculated to be 12.74, indicating its promising capabilities for microwave sensing applications. The examination of the sensor model uncovers such metrics as quality factors, frequency detection resolution (FDR) values, and sensitivities, all of which enhance our comprehension of the material's performance. This discovery has the potential to advance the development of sensing metamaterials, which could significantly impact various applications and enhance research in the field.

Catenaries and minimal surfaces of revolution in hyperbolic space

We introduce the concept of extrinsic catenary in the hyperbolic plane. Working in the hyperboloid model, we define an extrinsic catenary as the shape of a curve hanging under its weight as seen from the ambient space. In other words, an extrinsic catenary is a critical point of the potential functional, where we calculate the potential with the extrinsic distance to a fixed reference plane in the ambient Lorentzian space. We then characterize extrinsic catenaries in terms of their curvature and as a solution to a prescribed curvature problem involving certain vector fields. In addition, we prove that the generating curve of any minimal surface of revolution in the hyperbolic space is an extrinsic catenary with respect to an appropriate reference plane. Finally, we prove that one of the families of extrinsic catenaries admits an intrinsic characterization if we replace the extrinsic distance with the intrinsic length of horocycles orthogonal to a reference geodesic.

Free curves, eigenschemes, and pencils of curves

AbstractLet . A reduced plane curve is free if its associated module of tangent derivations is a free ‐module, or equivalently if the corresponding sheaf of vector fields tangent to splits as a direct sum of line bundles on . In general, free curves are difficult to find, and in this paper, we describe a new method for constructing free curves in . The key tools in our approach are eigenschemes and pencils of curves, combined with an interpretation of Saito's criterion in this context. Previous constructions typically applied only to curves with quasihomogeneous singularities, which is not necessary in our approach. We illustrate our method by constructing large families of free curves.

Geometric interpretation of Tensor-Vector-Scalar theory in a Kaluza-Klein reference fluid

Abstract Gravitational alternatives to dark matter require additional fields or assumptions beyond general relativity while continuing to agree with tight solar system constraints. Modified Newtonian Dynamics (MOND), for example, predicts the Tully-Fisher relation for galaxies more accurately than dark matter models while limiting to Newtonian gravity in the solar system. On the other hand, MOND does a poor job predicting larger scale observations such as the Cosmic Microwave Background and Matter Power Spectra. Tensor-Vector-Scalar (TeVeS) theory is a relativistic generalization of MOND that accounts for these observations without dark matter. In this paper, I derive a generalized TeVeS from Kaluza-Klein theory in one extra dimension as a consequence of $n=0$ Kaluza-Klein modes. In the KK theory, MOND is a special case of a slicing condition in the 5D ADM formalism enforced by a reference fluid as in the Isham-Kucha\v{r} method which may arise from a broken displacement symmetry. This has two benefits: first is means that TeVeS is compatible with Kaluza-Klein dark matter theory, which is a strong candidate for Weakly Interacting Massive Particles (WIMPs), the other is that it provides an elegant mechanism for the scalar and vector fields. It constrains most of the freedom in the definition of TeVeS which does not have a field theoretic motivation. This is important because the Kaluza-Klein theory predicts that spin-2 tensor modes must propagate at the speed of light, in agreement with observation, from theoretical constraints while TeVeS has to match this observation empirically. Furthermore, it removes need for the interpolating function in MOND and the Lorentz-violating condition on the vector field to be physical since they are analogous to a gauge condition and depend on state of motion.

Statistical conformal Killing vector fields for FLRW space-time

Abstract The classification of conformal Killing vector fields for FLRW space-time from Riemannian point of view was done by Maartens-Maharaj in \cite{Maartens1986}. In this paper, we introduce conformal Killing vector fields from statistical point of view and classify them for FLRW space-time. We also, obtain the conditions for vanishing the Lie derivative of affine connections with respect to conformal Killing vector fields. Finally, we study the Lie derivative of torsion tensor with respect to conformal Killing vector fields.

Introduction to gradient h-almost η-Ricci soliton warped product

In this paper, we introduce the new concept of gradient h-almost η-Ricci soliton. We discuss here a steady or expanding gradient h-almost η-Ricci soliton warped product Bn ×f Fm, m 1. We show that the warping function f of this warped product attains minimum as well as maximum and it will definitely be a Riemannian product under certain conditions. We also describe some suitable restrictions to these constructions for the compact base of this warped product. Later, we study h-almost η-Ricci soliton and gradient h-almost η-Ricci soliton on warped product manifolds including a concurrent vector field.

Gradient properties of perturbative multiscalar RG flows to six loops

The gradient property of the renormalisation group (RG) flow of multiscalar theories is examined perturbatively in d=4 and d=4−ε dimensions. Such theories undergo RG flows in the space of quartic couplings λI. Starting at five loops, the relevant vector field that determines the physical RG flow is not the beta function traditionally computed in a minimal subtraction scheme in dimensional regularisation, but a suitable modification of it, the B function. It is found that up to five loops the B vector field is gradient, i.e. BI=GIJ∂A/∂λJ with A a scalar and GIJ a rank-two symmetric tensor of the couplings. Up to five loops the beta function is also gradient, but it fails to be so at six loops. The conditions under which the B function (and hence the RG flow) is gradient at six loops are specified, but their verification rests on a separate six-loop computation that remains to be performed.

A geometric framework for interstellar discourse on fundamental physical structures

This paper considers the possibility that abstract thinking and advanced synthesis skills might encourage extraterrestrial civilizations to accept communication with mankind on Earth. For this purpose, a notation not relying upon the use of alphabet and numbers is proposed, in order to denote just some basic geometric structures of current physical theories: vector fields, [Formula: see text]-form fields, and tensor fields of arbitrary order. An advanced civilization might appreciate the way here proposed to achieve a concise description of electromagnetism and general relativity, and hence it might accept the challenge of responding to our signals. The abstract symbols introduced in this paper to describe the basic structures of physical theories are encoded into black and white bitmap images that can be easily converted into short bit sequences and modulated on a carrier wave for radio transmission.

Almost Yamabe solitons satisfying certain conditions on the associated vector field

We study an almost Yamabe soliton (Mn , g, V, ρ) and obtain various conditions on V in terms of infinitesimal harmonic transformation, 2-Killing, and an inequality, to show that V is Killing. In addition, we have provided characterizations of a 2-Killing vector field V on any Riemannian manifold in terms of the strain tensor and divergence of V.

In situ build surface topography determination in electron beam powder bed fusion

AbstractElectron optical imaging is the most promising process monitoring method in electron beam powder bed fusion. State of the art in modern machines is the installation of a single detector in the top center of the build chamber. Exemplary applications are the reconstruction of digital twins of manufactured parts to compare their dimensional accuracy or analysing the top surface of each layer to identify surface features like pores or material transport. Multi-detector systems are currently under research and have shown great potential in reconstructing the surface topography in situ. A recently developed ray tracing model, describing the image formation process, allows to formulate design guide lines for multi-detector systems and provides a method for the computation of the normal vector field of the build surface. This work utilizes the recent progress and presents a newly developed four-detector system and an updated computation chain, which enable build surface topography reconstruction in situ in every layer of a build process. The computation chain contains a normal integration algorithm, which employs Tikhonov regularization to cope with measurement irregularities. The integration method is validated with ex situ measured as-built surfaces. Additionally, first applications are demonstrated and connections to process parameter changes illustrated.

On submanifolds of a Sasakian manifold

In this paper, we study submanifolds of a Sasakian manifolds provide with a torqued vector field and also accepting a Ricci-Yamabe soliton of both Sasakian manifold and Sasakian spaceform. Further, we obtain some important results which categorize the submanifolds admitting a Ricci-Yamabe soliton of Sasakian spaceform.

Lignin-Derived Lightweight Carbon Aerogels for Tunable Epsilon-Negative Response.

Electromagnetic (EM) metamaterials have garnered considerable attention due to their capacity to achieve negative parameters, significantly influencing the integration of natural materials with artificially structural media. The emergence of carbon aerogels (CAs) offers an opportunity to create lightweight EM metamaterials, notable for their promising EM shielding or absorption effects. This paper introduces an efficient, low-cost method for fabricating CAs without requiring stringent drying conditions. By finely tuning the ZnCl2/lignin ratio, the porosity is controlled in CAs. This control leads to an epsilon-negative response in the radio-frequency region, driven by the intrinsic plasmonic state of the 3D carbon network, as opposed to traditional periodic building blocks. This approach yields a tunable and weakly epsilon-negative response, reaching an order of magnitude of -103 under MHz frequencies. Equivalent circuit analysis highlights the inductive characteristics of CAs, correlating their significant dielectric loss at low frequencies. Additionally, EM simulations are performed to evaluate the distribution of the electric field vector in epsilon-negative CAs, showcasing their potential for effective EM shielding. The lignin-derived, lightweight CAs with their tunable epsilon-negative response hold promise for pioneering new directions in EM metamaterials and broadening their application in diverse extreme conditions.