Symmetric Field Research Articles

Unique Continuation Results for Certain Generalized Ray Transforms of Symmetric Tensor Fields

Let \(I_{m}\) denote the Euclidean ray transform acting on compactly supported symmetric m-tensor field distributions f, and \(I_{m}^{*}\) be its formal \(L^2\) adjoint. We study a unique continuation result for the operator \(N_{m}=I_{m}^{*}I_{m}\). More precisely, we show that if \(N_{m}f\) vanishes to infinite order at a point \(x_0\) and if the Saint-Venant operator W acting on f vanishes on an open set containing \(x_0\), then f is a potential tensor field. This generalizes two recent works of Ilmavirta and Mönkkönen who proved such unique continuation results for the ray transform of functions and vector fields/1-forms. One of the main contributions of this work is identifying the Saint-Venant operator acting on higher-order tensor fields as the right generalization of the exterior derivative operator acting on 1-forms, which makes unique continuation results for ray transforms of higher-order tensor fields possible. In the second half of the paper, we prove analogous unique continuation results for momentum ray and transverse ray transforms.

Spectral analysis of scattering resonances with application on high-contrast nanospheres

In this paper, we provide further spectral analysis of the general asymptotic scattering resonances formula of small 3D dielectrics of arbitrary shape with high contrast, already derived in other works to a first-order approximation. To investigate the components of a full expansion of such resonances, a breakdown is presented for the case of high-contrast nanospheres, in terms of its radius h in the interval [0, 1]. We also derive, for radially symmetric fields, an exact resonance formula for a spherical scatterer in terms of its radius, not necessarily small, and dielectric susceptibility coefficient, not necessarily high. This formula is further developed and simplified in the case of high contrast nanospheres. Such formulas are useful in imaging applications to identify objects’ properties from frequency measurements. Another application is the study of negative refractive index materials, such as metamaterials, and the anomalous localized resonance phenomenon (ALR) that is associated with cloaking and superlensing.

NORMALISED TANGENT BUNDLE, VARIETIES WITH SMALL CODEGREE AND PSEUDOEFFECTIVE THRESHOLD

AbstractWe propose a conjectural list of Fano manifolds of Picard number $1$ with pseudoeffective normalised tangent bundles, which we prove in various situations by relating it to the complete divisibility conjecture of Francesco Russo and Fyodor L. Zak on varieties with small codegree. Furthermore, the pseudoeffective thresholds and, hence, the pseudoeffective cones of the projectivised tangent bundles of rational homogeneous spaces of Picard number $1$ are explicitly determined by studying the total dual variety of minimal rational tangents (VMRTs) and the geometry of stratified Mukai flops. As a by-product, we obtain sharp vanishing theorems on the global twisted symmetric holomorphic vector fields on rational homogeneous spaces of Picard number $1$ .

Thermalization dynamics of a gauge theory on a quantum simulator.

Gauge theories form the foundation of modern physics, with applications ranging from elementary particle physics and early-universe cosmology to condensed matter systems. We perform quantum simulations of the unitary dynamics of a U(1) symmetric gauge field theory and demonstrate emergent irreversible behavior. The highly constrained gauge theory dynamics are encoded in a one-dimensional Bose-Hubbard simulator, which couples fermionic matter fields through dynamical gauge fields. We investigated global quantum quenches and the equilibration to a steady state well approximated by a thermal ensemble. Our work may enable the investigation of elusive phenomena, such as Schwinger pair production and string breaking, and paves the way for simulating more complex, higher-dimensional gauge theories on quantum synthetic matter devices.

The theory of symmetric tensor field: From fractons to gravitons and back

We consider the theory of a symmetric tensor field in 4D, invariant under a subclass of infinitesimal diffeomorphism transformations, where the vector diff parameter is the 4-divergence of a scalar parameter. The resulting gauge symmetry characterizes the “fracton” quasiparticles and identifies a theory which depends on a dimensionless parameter, which cannot be reabsorbed by a redefinition of the tensor field, despite the fact that the theory is free of interactions. This kind of “electromagnetic gauge symmetry” is weaker that the original diffeomorphism invariance, in the sense that the most general action contains, but is not limited to, linearized gravity, and we show how it is possible to switch continuously from linearized gravity to a mixed phase where both gravitons and fractons are present, without changing the degrees of freedom of the theory. The gauge fixing procedure is particularly rich and rather peculiar, and leads to the computation of propagators which in the massive case we ask to be tachyonic-free, thus constraining the domain of the parameter of the theory. Finally, a closer contact to fractons is made by the introduction of a parameter related to the “rate of propagation”. For a particular value of this parameter the theory does not propagate at all, and we guess that, for this reason, the resulting theory should be tightly related to the fracton excitations.

Calculation of the electron-optical scheme of a new type mirror energy analyzer of charged particles

Further studies of the electron-optical properties of electrostatic multipole-cylindrical fields, synthesized from the fields of a cylindrical mirror and circular multipoles, are continued in the work. The implementation of electron spectroscopy methods is based on the use of complex equipment, one of the main elements of which is an electron energy analyzer of low and medium energies. Application of the multipole approach to the synthesis of deflecting fields makes it possible to develop effective methods for energy analysis of charged particle beams. The electron-optical scheme of new type mirror energy analyzer of charged particle beams based on an electrostatic axially symmetric octupole-cylindrical field is proposed in the work. An axially symmetric octupole-cylindrical field is constructed as a superposition of a basic cylindrical field and a circular octupole. When the fields were added, the central circle of the octupole was combined with the zero equipotential of the logarithmic field. The motion of charged particles in the electrostatic octupole-cylindrical field. An integrodifferential equation for the motion of charged particles in an electrostatic octupolecylindrical field is derived. Calculation of trajectories in an energy analyzer with an octupole-cylindrical field was performed on the basis of the method of expansion into a fractional-power series of the particle motion equation presented in the integrodifferential form. Coefficients of the series, representing the trajectory of motion in an analytical form, accessible for further studies of the electron-optical characteristics of the octupole-cylindrical field, are obtained. Based on an octupole-cylindrical field, high luminosity energy analyzers can be built to determine the composition of charged particle beams with energies from units of eV to tens of keV in space plasma.

증기 터빈의 제어밸브에서 발생하는 유체유발소음의 수치해석 연구

A numerical study of control valves is performed to identify the effects of a plug shape on the flow field and noise characteristics. Two different configurations of control valves are used to compare their performance and noise sources. Model A consists of a quick opening plug, a seat, and a pipe line. Model B has the same components as model A, but the vicinity of the quick opening plug is cut to be flat. Asymmetric velocity and pressure fields are formed in model B, while a symmetric flow field is formed in model A. The asymmetric flow field is caused by backflow near the plug which is cut in model B. An analysis of noise is performed through fast Fourier transform of noise sources. As a result, noise sources in the outlet cavity and throttle in model B are found not significantly influenced, while noise sources are found to be dramatically increased near the wall of the seat and the right part of the plug. Furthermore, high-frequency noise, which is insifnificant in model A, is amplified.

Fractons, dipole symmetries and curved spacetime

We study complex scalar theories with dipole symmetry and uncover a no-go theorem that governs the structure of such theories and which, in particular, reveals that a Gaussian theory with linearly realised dipole symmetry must be Carrollian. The gauging of the dipole symmetry via the Noether procedure gives rise to a scalar gauge field and a spatial symmetric tensor gauge field. We construct a worldline theory of mobile objects that couple gauge invariantly to these gauge fields. We systematically develop the canonical theory of a dynamical symmetric tensor gauge field and arrive at scalar charge gauge theories in both Hamiltonian and Lagrangian formalism. We compute the dispersion relation of the modes of this gauge theory, and we point out an analogy with partially massless gravitons. It is then shown that these fractonic theories couple to Aristotelian geometry, which is a non-Lorentzian geometry characterised by the absence of boost symmetries. We generalise previous results by coupling fracton theories to curved space and time. We demonstrate that complex scalar theories with dipole symmetry can be coupled to general Aristotelian geometries as long as the symmetric tensor gauge field remains a background field. The coupling of the scalar charge gauge theory requires a Lagrange multiplier that restricts the Aristotelian geometries.

AN ITERATIVE METHOD TO SOLVING THE PROBLEM OF RECONSTRUCTING A 2-TENSOR FIELD FROM LIMITED DATA

In this paper, we consider problem of recovery of a two-dimensional symmetric 2-tensor field. Data are values of ray transform in domain limited by angles. We apply the iterative Gerchberg-Papoulis algorithm to this problem. Steps of the algorithm are performed in two spaces: the main space of tensor field components and the Fourier space. The actions performed are based on back-projection, direct and inverse Fourier transforms and the use of a priori information. Numerical experiments on recovery of test 2-tensor fields showed good results.

Phase Transitions in the Blume–Capel Model with Trimodal and Gaussian Random Fields

We study the effect of different symmetric random field distributions: trimodal and Gaussian on the phase diagram of the infinite range Blume–Capel model. For the trimodal random field, the model has a very rich phase diagram. We find three new ordered phases, multicritical points like tricritical point (TCP), bicritical end point (BEP), critical end point (CEP) along with some multi-phase coexistence points. We also find re-entrance at low temperatures for some values of the parameters. On the other hand for the Gaussian distribution the phase diagram consists of a continuous line of transition followed by a first order transition line, meeting at a TCP. The TCP vanishes for higher strength of the random field. In contrast to the trimodal case, in Gaussian case no new phase emerges.

Beltrami’s completeness for 𝕃p symmetric matrix fields

Geymonat extended Gurtin’s result on Beltrami’s completeness to [Formula: see text]-case. The first objective of this paper is to generalize Geymonat’s result to [Formula: see text]-case with tangential and normal boundary conditions. Maggiani et al. proposed an extension of Beltrami’s-type decomposition for symmetric matrix fields in [Formula: see text] when [Formula: see text] is simply connected domain and of class [Formula: see text]. The second objective of this paper is to present more extensions of the above decomposition when the domain [Formula: see text] is only of class [Formula: see text] and not necessarily simply connected. The third objective of this paper is to use the previous characterizations of matrix fields to solve some variants of the bi-Laplacian problem.

Effects of unsymmetrical magnetic field on discharge characteristics of Hall thruster with large height-radius ratio

The large height-radius ratio design of the Hall thruster increases the difference of radial distributions of magnetic field and plasma parameters in the channel, so that the optimal discharge effect of the thruster cannot be guaranteed under the symmetrical magnetic field configuration relative to the channel centerline. In this work, the effects of the unsymmetrical magnetic field on discharge characteristics of Hall thruster with large height-radius ratio were studied through experiments and PIC simulations. The results show that the thruster can realize stable discharge in a wide power range under unsymmetrical magnetic fields. Also, the unsymmetrical magnetic field plays a positive role in improving the discharge performance and prolonging the service life of the thruster. The inward-inclined magnetic field can effectively improve the discharge performance of the thruster, and the discharge efficiency can be increased by 1.5% under the rated working condition compared with the symmetrical magnetic field. The advantage in performance is mainly attributed to the improvement of ionization efficiency of propellant. The outward-inclined magnetic field is more conducive to prolong the service life of the thruster, and the peak value of ion power deposition on the inner wall can be reduced by 40% under the rated working condition due to the outward inclination of the ionization zone and the reduction of the radial electric field. However, under unsymmetrical magnetic field, the ionization zone deviates from the channel center, and the high ionization rate zone is more dispersed in distribution, thus leading to the increase in plume divergence angle. This study verifies that the unsymmetrical magnetic field has certain advantages under the structure of large height-radius ratio, which could be used as a guidance for the optimal design of Hall thruster with large height-radius ratio.

Dual description of gauge theories from an iterative Noether approach

An iterative Noether scheme, advocated by Deser, is used to introduce gauge invariant couplings to nonrelativistic matter with global symmetries related to usual charge conservation and dipole conservation recently discussed in fractonic theories. No reference to any gauge principle, fractonic or otherwise, is required. A dual description is found where the theory is defined either in terms of the usual vector gauge field or, alternatively, in terms of higher derivatives of a symmetric tensor field given in fractonic theories. A connection between these two descriptions is obtained by providing an explicit map between the vector and tensor fields. This method yields a novel ‘minimal’ prescription involving tensor fields which is identified with the usual minimal prescription that involves vector fields, by using the mapping. It also spells out the structure of the pure gauge field action in both formulations. Extension of the abelian U(1) invariance to the nonabelian SU(N) invariance is done for the standard formulation.

Recurrent ancient geomagnetic field anomalies shed light on future evolution of the South Atlantic Anomaly

The strength of the geomagnetic field has decreased rapidly over the past two centuries, coinciding with an increasing field asymmetry due to the growth of the South Atlantic Anomaly. The underlying processes causing the decrease are debated, which has led to speculation that the field is about to reverse. Here, we present a geomagnetic field model based on indirect observations over the past 9,000 y and identify potential ancient analogs. The model is constructed using a probabilistic approach that addresses problems with age uncertainties and smoothing of sedimentary data that have hampered previous attempts. We find evidence for recurrent hemispherical field asymmetries, related to quasiperiodic millennial-scale variations in the dipole moment. Our reconstruction indicates that minima in the dipole moment tend to coincide with geomagnetic field anomalies, similar to the South Atlantic Anomaly. We propose that the period around 600 BCE, characterized by a strongly asymmetric field, could provide an analog to the present-day field. The analogy implies that the South Atlantic Anomaly will likely disappear in next few hundred years, accompanied by a return to a more symmetric field configuration and possibly, a strengthening of the axial dipole field.

Proton acceleration in vacuum using frequency-chirped laser pulses

Recent investigations demonstrate the achievement of protons with energies of several hundred MeV via vacuum laser acceleration mechanisms. However, the symmetric electric field from an unchirped laser pulse is not efficient in generating a high energy proton beam. It is known that a proton can gain energy from a chirped laser pulse through the optimal phase synchronization between the proton and the laser field. In this study, we investigate the proton acceleration by different frequency-chirped laser pulses in vacuum both analytically and numerically. We consider four different situations constituting the unchirped, linear, quadratic, and sinusoidal chirped laser pulses. For the unchirped laser case, the pulse symmetry results in no net energy gain for the proton from the laser field. Conversely, using frequency-chirped laser pulses renders high energy to the proton compared with the unchirped laser pulse. We compared three different chirped cases, and found that a sinusoidal chirped laser pulse renders maximum acceleration to the protons compared to the linear and quadratic chirped situations. There exists an optimal chirp parameter for each type of chirped laser pulse.

A Simple proto nebula constituted of plasma under magneto hydrodynamics conditions

Using simplifying assumptions, a region or time in the universe is conceived, which allows for a state constituted solely by a gaseous plasma subject to the current laws of gravity and electromagnetism to exist. Under these conditions, it is proposed that the presence of a portion of that medium, organized in a highly symmetric magnetic field and plasma structure, possesses the symmetry of a torus. We resort to the fact that magneto hydrodynamics (MHD) is the common state of this medium and that it has the property of attaching to it the mass. (I.e. in the language of MHD the mass of the torus is assumed to be magnetized.) The solution to this MHD equilibrium of the matter of the torus, which prevents it from coalescing through the gravitational pool, is presented. Further, analysis shows that a gravitational field generated by the torus may be capable of attracting the other, non-magnetized matter, under the influence of the torus’ gravitational pool. In this way it is shown that the torus shaped MHD structure, under the discussed conditions, satisfies essential properties, like having large regions where the Keplerian mass motion cannot be present. A scenario consistent with a few key properties of proto-nebulae in equilibrium is presented and considered.

Relativistic dynamical inversion in manifestly covariant form

The relativistic dynamical inversion technique, a novel tool for finding analytical solutions to the Dirac equation, is written in explicitly covariant form. It is then shown how the technique can be used to make a change from Cartesian to spherical coordinates of a given Dirac spinor. Moreover, the most remarkable feature of the method presented here, which is the ease of performing a nontrivial change of reference frames, is demonstrated. Such a feature constitutes a potentially powerful tool for finding novel solutions to the Dirac equation. Furthermore, a whole family of normalizable analytic solutions to the Dirac equation is constructed. More specifically, we find exact solutions for the case of a Dirac electron in the presence of a magnetic field as well as a solution comprising a combination of a spherically symmetric electric field and magnetic fields. These solutions shed light on the possibility of separating the positive- and negative-energy parts of localized Dirac spinors in the presence as well as in the absence of magnetic fields. The presented solutions provide an illustration of the connection between the geometrical properties of the spinor and spin-orbit coupling for normalizable spinorial wave functions.

Semiclassical gravitational collapse of a radially symmetric massless scalar quantum field

We present a method to study the semiclassical gravitational collapse of a radially symmetric scalar quantum field in a coherent initial state. The formalism utilizes a Fock space basis in the initial metric, is unitary and time reversal invariant up to numerical precision. It maintains exact compatibility of the metric with the expectation values of the energy momentum tensor in the scalar field coherent state throughout the entire time evolution. We find a simple criterion for the smallness of discretization effects, which is violated when a horizon forms. As a first example, we study the collapse of a specific state in the angular momentum $l=0$ approximation. Outside the simulated volume it produces a Schwarzschild metric with $r_s \sim 3.5 \ell_p$. We see behaviour that is compatible with the onset of horizon formation both in the semiclassical and corresponding classical cases in a regime where we see no evidence for large discretization artefacts. In our example setting, we see that quantum effects accelerate the possible horizon formation and move it radially outward. We find that this effect is robust against variations of the radial resolution, the time step, the volume, the initial position and shape of the inmoving state, the vacuum subtraction, the discretization of the time evolution operator and the integration scheme of the metric. We briefly discuss potential improvements of the method and the possibility of applying it to black hole evaporation. We also briefly touch on the extension of our formalism to higher angular momenta, but leave the details and numerics for a forthcoming publication.

A Note on Gauged Baby Skyrmions

In this paper we study a gauged version of the two dimensional Skyrme model of nuclear physics; the field configurations are couples (u, A), where u:{mathbb {R}}^2rightarrow S^2 is a map constant at infinity, which can be classified by its topological degree, and A:{mathbb {R}}^2rightarrow {mathbb {R}}^2 is the gauge field. We prove the existence of rotationally symmetric field configurations which minimize the gauged Skyrme energy on every topological sector (gauged baby skyrmions). Moreover we study the behavior of these skyrmions in the case of weak or strong coupling with the gauge field. In particular, we show that, as observed by many authors by means of numerical simulations, in the strong coupling regime the magnetic flux associated with the gauge field becomes quantized.

Floating particles mixing characteristics in an eccentric stirred tank coupled with dislocated fractal impellers

Abstract The mixing characteristics of floating particle dispersion process in an eccentric stirred tank with dislocated fractal impellers were investigated using computational fluid dynamics (CFD) and experimental analyses. Solid concentration distribution, axial solid concentration profile, cloud height, solid integrated velocity, power consumption and just drawdown speed were investigated. Results showed that dislocated fractal impeller can enhance solid integrated velocity and fluid turbulent fluctuation intensity compared with dislocated pitched blade impeller, and eccentric agitation coupled with dislocated fractal impeller could destroy the typical circulation loops and symmetric flow field and improve the axial circulation efficiency of floating particles on the basis of dislocated fractal impeller. Eccentric agitation coupled with dislocated fractal impeller could further enhance the floating particle dispersion homogeneity (MI) and decrease the just drawdown speed (N jd) on the basis of dislocated fractal impeller and dislocated pitched blade impeller at the specific power consumption. Meanwhile, eccentric ratio of 0.3 or 0.4 was optimal for the floating particle mixing process in this work.