Arbitrary Vector Field Research Articles

On improving sound source localization in the wind tunnel

When traveling in an open-jet wind tunnel, the path of an acoustic wave is affected by the flow causing a shift of source positions in acoustical maps of phased arrays outside the flow. In this paper, we start by comparing several well-known approaches to correct travel times between microphones and assumed sources, used, for example, by beamforming algorithms in such an environment. The methods under consideration include the original 1D-Amiet/Bahr formulation, a standard 2.5D-planar approach that assumes the constancy of angular frequency and wave-number-vector components along a plane, and finally, a 3D-ray-tracing method. Common to the former two of these methods is the assumption that the boundary layer of the flow, the so-called shear layer, is infinitely thin and refracts the acoustical ray, whereas, in principle, the latter algorithm allows for an arbitrary (but sufficiently smooth) vector field modeling the flow. Going through these methods, we discuss travel-time results relative to each other and in terms of differences in source localization using beamforming maps. In particular, we model the flow of an existing wind tunnel and discuss results of real-world measurements comparing the 2.5D-planar approach with the 3D-ray-tracing method.

A note on field equations in generalized theories of gravity

In the work [Phys. Rev. D 84, 124 041 (2011)], to obtain a simple and economic formulation of field equations for generalised theories of gravity described by the Lagrangian −gLgαβ,Rμνρσ , the key equality ∂L/∂gμνRαβκω=2PμλρσRνλρσ was derived. In this note, it is demonstrated that such an equality can be directly derived from an off-shell Noether current associated with an arbitrary vector field. As byproducts, a generalized Bianchi identity related to the divergence for the expression of field equations, together with the Noether potential, is obtained. On the basis of the above, we further propose a systematic procedure to derive the equations of motion from the Noether current, and then this procedure is extended to more general higher-order gravities endowed with the Lagrangian encompassing additional terms of the covariant derivatives of the Riemann tensor. To our knowledge, both the detailed expressions for field equations and the Noether potential associated with such theories are first given at a general level. All the results reveal that using the Noether current to determine field equations establishes a straightforward connection between the symmetry of the Lagrangian and the equations of motion and such a remedy even can avoid calculating the derivative of the Lagrangian density with respect to the metric.

Metafiber transforming arbitrarily structured light

Structured light has proven useful for numerous photonic applications. However, the current use of structured light in optical fiber science and technology is severely limited by mode mixing or by the lack of optical elements that can be integrated onto fiber end-faces for wavefront engineering, and hence generation of structured light is still handled outside the fiber via bulky optics in free space. We report a metafiber platform capable of creating arbitrarily structured light on the hybrid-order Poincaré sphere. Polymeric metasurfaces, with unleashed height degree of freedom and a greatly expanded 3D meta-atom library, were 3D laser nanoprinted and interfaced with polarization-maintaining single-mode fibers. Multiple metasurfaces were interfaced on the fiber end-faces, transforming the fiber output into different structured-light fields, including cylindrical vector beams, circularly polarized vortex beams, and arbitrary vector field. Our work provides a paradigm for advancing optical fiber science and technology towards fiber-integrated light shaping, which may find important applications in fiber communications, fiber lasers and sensors, endoscopic imaging, fiber lithography, and lab-on-fiber technology.

Generalized Minkowski Type Integral Formulas for Compact Hypersurfaces in Pseudo-Riemannian Manifolds

We obtain some generalized Minkowski type integral formulas for compact Riemannian (resp., spacelike) hypersurfaces in Riemannian (resp., Lorentzian) manifolds in the presence of an arbitrary vector field that we assume to be timelike in the case where the ambient space is Lorentzian. Some of these formulas generalize existing formulas in the case of conformal and Killing vector fields. We apply these integral formulas to obtain interesting results concerning the characterization of such hypersurfaces in some particular cases such as when the ambient space is Einstein admitting an arbitrary (in particular, conformal or Killing) vector field, and when the hypersurface has a constant mean curvature.

Boundary and bulk notions of transport in the AdS3/CFT2 correspondence

We construct operators in holographic two-dimensional conformal field theory, which act locally in the code subspace as arbitrary bulk spacelike vector fields. Key to the construction is an interplay between parallel transport in the bulk spacetime and in kinematic space. We outline challenges, which arise when the same construction is extended to timelike vector fields. We also sketch several applications, including boundary formulations of the bulk Riemann tensor, dreibein, and spin connection, as well as an application to holographic complexity.

Weighted Radon transforms of vector fields, with applications to magnetoacoustoelectric tomography

Currently, theory of ray transforms of vector and tensor fields is well developed, but the Radon transforms of such fields have not been fully analyzed. We thus consider linearly weighted and unweighted longitudinal and transversal Radon transforms of vector fields. As usual, we use the standard Helmholtz decomposition of smooth and fast decreasing vector fields over the whole space. We show that such a decomposition produces potential and solenoidal components decreasing at infinity fast enough to guarantee the existence of the unweighted longitudinal and transversal Radon transforms of these components. It is known that reconstruction of an arbitrary vector field from only longitudinal or only transversal transforms is impossible. However, for the cases when both linearly weighted and unweighted transforms of either one of the types are known, we derive explicit inversion formulas for the full reconstruction of the field. Our interest in the inversion of such transforms stems from a certain inverse problem arising in magnetoacoustoelectric tomography (MAET). The connection between the weighted Radon transforms and MAET is exhibited in the paper. Finally, we demonstrate performance and noise sensitivity of the new inversion formulas in numerical simulations.

Viscosity flux recursion of flow motions in pseudo-hyperbolic space

In this study, we examine the time-dependent motion of a particle in pseudo-hyperbolic (p-hyperbolic) space with the help of some flow equations that bring many geometrical solutions to physical problems in areas fluid dynamics, magnetic fluxes, and wave propagation. First, we consider this moving particle as a magnetic flux that makes the Heisenberg ferromagnet spin chain (Hfsc) movement and obtain the p-hyperbolic frame equations representing the flux movement. Then, we obtain some motion equations, assuming that an arbitrary vector field we choose in this space represents the vortex filament flow. Here, we obtain some results by examining the flow motion we have chosen for two different situations, with and without viscosity. More clearly, results such as the time-dependent equation of curvature of an unviscous vortex filament flow and the curvature-dependent equation of the viscosity coefficient of a viscous vortex filament flow are obtained.

On generalized almost para-Hermitian spaces

Recently, a generalized almost Hermitian metric on an almost complex manifold (M, J) is determined as a generalized Riemannian metric (i.e. an arbitrary bilinear form) G which satisfies G(JX, JY) = G(X,Y), where X and Y are arbitrary vector fields on M. In the same manner we can study a generalized almost para-Hermitian metric and determine almost para-Hermitian spaces. Some properties of these spaces and special generalized almost para-Hermitian spaces including generalized para-Hermitian spaces as well as generalized nearly para-K?hler spaces are determined. Finally, a non-trivial example of generalized almost para-Hermitian space is constructed.

The variation of a functional on the Riemannian submanifolds and its applications

The aim of this paper is to derive the first variation of a functional on an oriented submanifold in the Riemannian manifold involving an arbitrary vector field. After obtaining the first variation formula, a notion of $\sigma$-mean curvature is naturally introduced. We also find an important identity involving the $\sigma$-mean curvature and the divergence of the tangent component of the vector field. As an application, we get a general formula of the $\sigma$-mean curvature for the hypersurfaces in a class of Finsler manifolds called the general $(\alpha,\beta)$-manifolds (introduced in [38]). Hence the vanishing $\sigma$-mean curvature characterizes the minimal hypersurfaces under the Busemann-Hausdorff measure ([29]) and the Holmes-Thompson measure ([23]). We also give a general formula for the hypersurface in a Randers manifold involving the navigation data without any restriction on the vector field. In terms of the identity, we prove some nonexistence theorems of the closed orientable minimal submanifold in some special non-Minkowskian Finsler manifolds.

Exact solution of generalized KG-oscillator under Lorentz-violating effects by a fixed vector field subject to interaction potential

In this work, the relativistic quantum oscillator under the Lorentz symmetry violation environment defined by an arbitrary constant vector field [Formula: see text] is analyzed. We solve the generalized Klein–Gordon oscillator with different vector field configurations and obtained the bound-states solutions. After that, we insert a static Coulomb-type scalar potential in the wave equation and solve the generalized Klein–Gordon oscillator analytically. We discuss the effects of Lorentz symmetry violation and the Coulomb-type potential on the energy spectrum of these oscillator fields.

Effects of Lorentz symmetry violation by a constant vector field subject to Cornell-type potential on relativistic quantum oscillator

In this paper, we analyze a relativistic quantum oscillator model under a Lorentz symmetry violation (LSV) environment defined by an arbitrary constant vector field [Formula: see text] subject to Cornell-type scalar potential. We first introduce the Cornell-type scalar potential by modifying the mass term via [Formula: see text], and then momentum vector [Formula: see text] in the wave equation. We solve the Klein–Gordon oscillator analytically for different vector field configurations and obtained the bound-states energy profiles and the wave function. We discuss the influences of a vector field on the energy eigenvalues and the wave function of these oscillators.

Relativistic quantum oscillator model under the effects of the violation of Lorentz symmetry by an arbitrary fixed vector field

We study a spin-zero relativistic quantum oscillator model under the violation of Lorentz symmetry defined by an arbitrary constant vector field . We solve the Klein-Gordon oscillator using the Nikiforov-Uvarov method and obtain the bound-states solutions of the quantum system. We see that vector field configurations modify the energy profile of the oscillator fields.

Явное представление сокращенных в размерности уравнений Эйлера и Навье – Стокса несжимаемой жидкости в интегральной форме

In this paper, we have obtained “stationary” systems of integraldifferential equations, which are consequences of the non-stationary Euler and Navier - Stokes equations of an incompressible fluid, and for which there are no time derivatives. If we set a correct problem for them, then we can determine the entire non-stationary flow in the volume without solving the nonstationary problem. It is enough to specify time-varying data only on some surface of this stream. The method of reduction of overdetermined systems of differential equations, proposed earlier by the authors, is used. In this method, in the case of a successful choice of the additional constraint equation, the overdetermined systems of differential equations are reduced to PDE systems of dimension less than that of the original PDE systems. The Euler and Navier-Stokes equations themselves act as the constraint equations, and the dimension is reduced for the integral equations, which are obtained using the Helmholtz theorem on the expansion of an arbitrary vector field into vortex and potential components. The peculiarity of this work lies in the fact that all the equations reduced in dimension are obtained in an explicit form, in contrast to the previous works of the authors, where up to 200–300 equations with reduced dimensions were proposed. In reduced systems of integral-differential equations, there is integration over space, therefore, reduction over spatial variables by given method is impossible. If this method is generalized a little, then we can obtain a reduced system of integral equations, where the unknowns have no derivatives with respect to space, but integration over space remains. Non-stationary new integral equations are also obtained, which determine the evolution of the flow. In Appendices A and B, sufficient conditions for the correctness of the reduced system of integral-differential equations are derived. Conditions are found under which the Euler equations follow directly from these reduced equations. It is shown how the time-varying data on some surface of this stream should be related. The change of variables, which is used in the reduction of the Navier - Stokes equations, is also investigated he resulting integral equations can be used to study complex vortices in the atmosphere. For example, if devices measuring data on its surface are installed on a flying object, then, knowing the vorticity profile ω0(r) at a certain moment in time, it is possible, by solving these equations, to track vortex activity at a distance from this object in real time.

Cosmological evolution and dark energy in osculating Barthel\u2013Randers geometry

We consider the cosmological evolution in an osculating point Barthel–Randers type geometry, in which to each point of the space-time manifold an arbitrary point vector field is associated. This Finsler type geometry is assumed to describe the physical properties of the gravitational field, as well as the cosmological dynamics. For the Barthel–Randers geometry the connection is given by the Levi-Civita connection of the associated Riemann metric. The generalized Friedmann equations in the Barthel–Randers geometry are obtained by considering that the background Riemannian metric in the Randers line element is of Friedmann–Lemaitre–Robertson–Walker type. The matter energy balance equation is derived, and it is interpreted from the point of view of the thermodynamics of irreversible processes in the presence of particle creation. The cosmological properties of the model are investigated in detail, and it is shown that the model admits a de Sitter type solution, and that an effective cosmological constant can also be generated. Several exact cosmological solutions are also obtained. A comparison of three specific models with the observational data and with the standard Lambda CDM model is also performed by fitting the observed values of the Hubble parameter, with the models giving a satisfactory description of the observations.

On Metric Contact Pairs with Certain Semi-Symmetry Conditions

Blair et al. [7] introduced the notion of bicontact manifold in the context of Hermitian geometry. Bande and Hadjar [1] studied on this notion under the name of contact pairs. These type of structures have important properties and their geometry is some different from classical contact structures. In this paper, we study on some semi-symmetry properties of the normal contact pair manifolds. We prove that a Ricci semi-symmetric (or concircularly Ricci semi-symmetric) normal metric contact pair manifold is a generalized quasi-Einstein manifold. Also, we classify normal metric contact pair manifolds as a generalized quasi-Einstein manifold with certain semi-symmetry conditions and for the concircular curvature tensor , the Riemannian curvature tensor , and an arbitrary vector field .

Arbitrary Vector Beam Encoding Using Single Modulation for Information Security Applications

Features of vector beam such as its spatial degree of freedom and intensity-based measurements are useful in information encoding but often require a complex optical-setup. In this letter, the concept of modulating a single-polarization component for encoding in arbitrary vector field through a non-interferometric approach has been introduced. Utilizing this concept, an efficient information security scheme has been demonstrated. The experimental results verify the feasibility of a practical system.

Off-Shell Noether Currents and Potentials for First-Order General Relativity

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

A general framework for validated continuation of periodic orbits in systems of polynomial ODEs

<p style='text-indent:20px;'>In this paper a parametrized Newton-Kantorovich approach is applied to continuation of periodic orbits in arbitrary polynomial vector fields. This allows us to rigorously validate numerically computed branches of periodic solutions. We derive the estimates in full generality and present sample continuation proofs obtained using an implementation in Matlab. The presented approach is applicable to any polynomial vector field of any order and dimension. A variety of examples is presented to illustrate the efficacy of the method.

On magnetostrophic dynamos in annular cores

ABSTRACTMagnetostrophic dynamos are studied in an annular core, adapting the seminal work of Taylor [The magnetohydrodynamics of a rotating fluid and the Earth's dynamo problem. Proc. R. Soc. London A. 1963, 274, 274] for a fluid-filled core. The model consists of an inviscid fluid core and a concentric solid inner core. The fluid is supposed to obey the Boussinesq equations of motion and is driven into motion by a flow-forcing function consisting of the buoyancy force of an adverse radial temperature gradient, opposed by the Lorentz force of the self-sustained magnetic field. Coriolis forces act but inertial and viscous forces are ignored. Taylor (1963) showed how such a “magnetostrophic dynamo” can be found when there is no solid inner core, but his ideas have to be non-trivially generalised when an inner core is present. That is undertaken in this paper. In the 1993, CUP book, “Theory of Solar and Planetary Dynamos”, Hollerbach and Proctor gave examples in which the zonal flow created by a specified flow-forcing function may be singular on the “tangent cylinder”, an imaginary cylinder tangential to the inner core and parallel to the polar axis. It is shown here how this singularity is related to the flow-forcing function, and how discontinuities of other components of the fluid velocity on the tangent cylinder are determined by that function. In appendix A, an identity is established between the leading terms in the Fourier expansion of two of the cylindrical components of an arbitrary vector field. In appendix B, eight examples are given relevant to annular dynamos. In appendix C, equatorial symmetry is considered.

Продолжения аффинной связности и горизон­тальных векторов

The linear frame bundle over a smooth manifold is considered. The mapping dе defined by the differentials of the first-order frame e is a lift to the normal N, i. e., a space complementing the first-order tangent space to the second-order tangent space to this bundle. In particular, the map­ping defined by the differentials of the vertical vector of this frame is a vertical lift into normal N. The lift dе allows us to construct a prolongation both of the tangent space and its vertical subspace into the second-order tangent space, more precisely into the normal N. The normal lift dе defines the normal prolon­gation of the tangent space (i. e. the normal N) and its vertical subspace. The vertical lift defines the vertical prolongation of the tangent space and its vertical subspace. The differential of an arbitrary vector field on the linear frame bundle is a complete lift from the first-order tangent space to the second-order tangent space to this bundle. It is known that the action of vector fields as differential operators on functions coincides with action of the differentials of these functions as 1-forms on these vector fields. Horizontal vectors played a dual role in the fibre bundle. On the one hand, the basic horizontal vectors serve as opera­tors for the covariant differentiation of geometric objects in the bundle. On the other hand, the differentials of these geometric objects can be con­sidered as forms (including tangential-valued ones) and their values on basic horizontal vectors give covariant derivatives of these geometric ob­jects. For objects which covariant derivatives require the second-order con­nection, the covariant derivatives are equal to the values of the differen­tials of these objects on horizontal vectors in prolonged affine connectivi­ty. Prolongations of the basic horizontal vectors, i. e., the second-order horizontal vectors for prolonged connection, were constructed. The sec­ond-order tangent space is represented as a straight sum of the first-order tangent space, vertical prolongations of the vertical and horizontal sub­spaces, and horizontal prolongation of the horizontal subspace.