Independent Invariants Research Articles

Classification of Three-Dimensional Linear Nijenhuis Operators with Functionally Independent Invariants

Classification of Three-Dimensional Linear Nijenhuis Operators with Functionally Independent Invariants

Components of curvature-squared invariants of minimal supergravity in five dimensions

We present for the first time the component structure of the supersymmetric completions for all curvature-squared invariants of five-dimensional, off-shell (gauged) minimal supergravity, including all fermions. This is achieved by using an interplay between superspace and superconformal tensor calculus techniques, and by employing results from arXiv:1410.8682 and arXiv:2302.14295. Our analysis is based on using a standard Weyl multiplet of conformal supergravity coupled to a vector and a linear multiplet compensator to engineer off-shell Poincaré supergravity. We compute all the descendants of the composite linear multiplets that describe gauged supergravity together with the three independent four-derivative invariants. These are the building blocks of the locally superconformal invariant actions. A derivation of the primary equations of motion for minimal gauged off-shell supergravity deformed by an arbitrary combination of these three locally superconformal invariants, is then provided. Finally, all the covariant descendants in the multiplets of equations of motion are obtained by applying a series of Q-supersymmetry transformations, equivalent to successively applying superspace spinor derivatives to the primary equations of motion.

Cosmological solutions in the Brans–Dicke theory via invariants of symmetry groups

We proceed to obtain an exact analytical solution of the Brans–Dicke (BD) equations for the spatially flat ([Formula: see text]) Friedmann–Lamaître–Robertson–Walker (FLRW) cosmological model in both cases of the absence and presence of the cosmological constant. The solution method that we use to solve the field equations of the BD equations is called the “invariants of symmetry groups method” (ISG method). This method is based on the extended Prelle–Singer (PS) method and it employs the Lie point symmetries, [Formula: see text]-symmetries, and Darboux polynomials (DPs). Indeed, the ISG method tries to provide two independent first-order invariants associated to the one-parameter Lie groups of transformations keeping the ordinary differential equations (ODEs) invariant, as solutions. It should be noted for integrable ODEs that the ISG method guarantees the extraction of these two invariants. In this work, for the BD equations in FLRW cosmological model, we find the Lie point symmetries, [Formula: see text]-symmetries, and DPs, and obtain the basic quantities of the extended PS method (which are the null forms and the integrating factors). By making use of the extended PS method we find two independent first-order invariants in such a way that appropriate cosmological solutions from solving these invariants as a system of algebraic equations are simultaneously obtained. These solutions are wealthy in that they include many known special solutions, such as the O’Hanlon–Tupper vacuum solutions, Nariai’s solutions, Brans–Dicke dust solutions, inflationary solutions, etc.

Effective rationality for local unitary invariants of mixed states of two qubits

We calculate the field of rational local unitary invariants for mixed states of two qubits, by employing methods from algebraic geometry. We prove that this field is rational (i.e. purely transcendental), and that it is generated by nine algebraically independent polynomial invariants. We do so by constructing a relative section, in the sense of invariant theory, whose Weyl group is a finite abelian group. From this construction, we are able to give explicit expressions for the generating invariants in terms of the Bloch matrix representation of mixed states of two qubits. We also prove similar rationality results for the local unitary invariants of symmetrically mixed states of two qubits. We also provide a sketch of how to generalize our results to the case of an arbitrary number of qubits. Our results apply to both complex-valued and real-valued invariants.

Linear coupling of transverse betatron oscillations: Dynamic stability and invariants of motion

Based on the technique of the discrete one-turn transfer maps, the problem of linear coupling between horizontal and vertical betatron oscillations in an accelerator has been treated exactly and entirely in explicit form. The stability region in the fractional part of the horizontal and the vertical betatron tune space as a function of the linear coupling strength has been obtained, and the increment/decrement of the horizontal and the vertical betatron oscillations in the case of the linear sum resonance has been shown to be approximately equal to the half of the coupling strength. The normal form parameterization of the one-turn linear map with horizontal-to-vertical coupling has been developed in detail in the spirit of Edwards and Teng formalism. The motion in the normal mode in the new normal form coordinates is decoupled by implying that two independent Courant-Snyder invariants exist, which have been found explicitly. Published by the American Physical Society 2024

On the Properties of λ-Prolongations and λ-Symmetries

In this paper, (1) We show that if there are not enough symmetries and λ-symmetries, some first integrals can still be obtained. And we give two examples to illustrate this theorem. (2) We prove that when X is a λ-symmetry of differential equation field Γ, by multiplying Γ a function μ defineded on Jn−1M, the vector fields μΓ can pass to quotient manifold Q by a group action of λ-symmetry X. (3) If there are some λ-symmetries of equation considered, we show that the vector fields from their linear combination are symmetries of the equation under some conditions. And if we have vector field X defined on Jn−1M with first-order λ-prolongation Y and first-order standard prolongations Z of X defined on JnM, we prove that gY cannot be first-order λ-prolonged vector field of vector field gX if g is not a constant function. (4) We provide a complete set of functionally independent (n−1) order invariants for V(n−1) which are n−1th prolongation of λ-symmetry of V and get an explicit n−1 order reduced equation of explicit n order ordinary equation considered. (5) Assume there is a set of vector fields Xi,i=1,...,n that are in involution, We claim that under some conditions, their λ-prolongations also in involution.

Second-order teleparallel gauge theory

In this work, we analyze second-order extensions of teleparallel theories of gravity as gauge theories for the translation group. We use Utiyama’s approach to gauge theories and show that it is possible to include second-order derivative terms in the Lagrangian of the gauge potential and preserve gauge invariance. Besides the usual field strength, a new object has to be introduced in order to preserve both gauge and diffeomorphism invariances. From this new object, we obtain a set of fourteen independent invariants which leads to equations that are linear in the fourth derivative of the tetrad field. We analyze a particular example with one of these invariants and evaluate the weak field limit, showing that the effective gravitational potential is a combination of Newton and Yukawa potentials.

J/ψ polarization in large- PT semi-inclusive deep-inelastic scattering at the EIC

We present a detailed phenomenological study of $J/\psi$ polarization in semi-inclusive deep inelastic scattering processes, focusing on the kinematics accessible at the future Electron-Ion Collider. We show theoretical estimates for the standard polarization parameters for different frames usually adopted in the literature, in the large $P_T$ region, namely $P_T\gg \Lambda_\text{QCD}$, where collinear factorization is expected to hold. We adopt both the Color Singlet Model and the Nonrelativistic QCD approach, paying special attention to the role of different sets of Long Distance Matrix Elements. Finally we present a preliminary analysis of some frame independent polarization invariants.

Curvature-squared invariants of minimal five-dimensional supergravity from superspace

We elaborate on the off-shell superspace construction of curvature-squared invariants in minimal five-dimensional supergravity. This is described by the standard Weyl multiplet of conformal supergravity coupled to two compensators being a vector multiplet and a linear multiplet. In this setup, we review the definition of the off-shell two-derivative gauged supergravity together with the three independent four-derivative superspace invariants defined in Butter et al. [J. High Energy Phys. 02 (2015) 111]. We provide the explicit expression for the linear multiplet based on a prepotential given by the logarithm of the vector multiplet primary superfield. We then present for the first time the primary equations of motion for minimal gauged off-shell supergravity deformed by an arbitrary combination of these three four-derivative locally superconformal invariants. We also identify a four-derivative invariant based on the linear multiplet compensator and the kinetic superfield of a vector multiplet, which can be used to engineer an alternative supersymmetric completion of the scalar curvature squared.

Axial and triaxial degrees of freedom in 72Zn

The unstable N=42 nucleus 72Zn has been studied using multiple safe Coulomb excitation in inverse kinematics. The experiment was performed at the REX-ISOLDE facility at CERN making first use of the silicon detector array C-REX in combination with the γ-ray spectrometer Miniball. The high angular coverage of C-REX allowed to determine the reduced transition strengths for the decay of the yrast 01+, 21+ and 41+ as well as of the 02+ and 22+ states in 72Zn. The quadrupole moments of the 21+, 41+ and 22+ states were extracted. Using model independent quadrupole invariants, the ground state of 72Zn was found to have an average deformation in the γ degree of freedom close to maximum triaxiality. In comparison to experimental data in zinc isotopes with N<40, the collectivity of the 41+ state in neutron-rich 72Zn is significantly larger, indicating a collective yrast band based on the ground state of 72Zn. In contrast, a low experimental B(E2;02+→21+) strength was determined, indicating a different structure for the 02+ state. Shell-model calculations propose a 02+ state featuring a larger fraction of the (spherical) N=40 closed-shell configuration in its wave function than for the 01+ ground state.The results were also compared with beyond mean field calculations which corroborate the large deformation in the γ degree of freedom, while pointing to a more deformed 02+ state. These experimental and theoretical findings establish the importance of the γ degree of freedom in the ground state of 72Zn, located between the 68,70Ni nuclei that have spherical ground states, and 76Ge, which has a rigid triaxial shape.

Discovering explicit Reynolds-averaged turbulence closures for turbulent separated flows through deep learning-based symbolic regression with non-linear corrections

This work introduces a novel data-driven framework to formulate explicit algebraic Reynolds-averaged Navier–Stokes (RANS) turbulence closures. Recent years have witnessed a blossom in applying machine learning (ML) methods to revolutionize the paradigm of turbulence modeling. However, due to the black-box essence of most ML methods, it is currently hard to extract interpretable information and knowledge from data-driven models. To address this critical limitation, this work leverages deep learning with symbolic regression methods to discover hidden governing equations of Reynolds stress models. Specifically, the Reynolds stress tensor is decomposed into linear and non-linear parts. While the linear part is taken as the regular linear eddy viscosity model, a long short-term memory neural network is employed to generate symbolic terms on which tractable mathematical expressions for the non-linear counterpart are built. A novel reinforcement learning algorithm is employed to train the neural network to produce best-fitted symbolic expressions. Within the proposed framework, the Reynolds stress closure is explicitly expressed in algebraic forms, thus allowing for direct functional inference. On the other hand, the Galilean and rotational invariance are craftily respected by constructing the training feature space with independent invariants and tensor basis functions. The performance of the present methodology is validated through numerical simulations of three different canonical flows that deviate in geometrical configurations. The results demonstrate promising accuracy improvements over traditional RANS models, showing the generalization ability of the proposed method. Moreover, with the given explicit model equations, it can be easier to interpret the influence of input features on generated models.

Invariant-Parameterized Exact Evolution Operator for SU(2) Systems with Time-Dependent Hamiltonian.

We report the step-by-step construction of the exact, closed and explicit expression for the evolution operator U(t) of a localized and isolated qubit in an arbitrary time-dependent field, which for concreteness we assume to be a magnetic field. Our approach is based on the existence of two independent dynamical invariants that enter the expression of SU(2) by means of two strictly related time-dependent, real or complex, parameters. The usefulness of our approach is demonstrated by exactly solving the quantum dynamics of a qubit subject to a controllable time-dependent field that can be realized in the laboratory. We further discuss possible applications to any SU(2) model, as well as the applicability of our method to realistic physical scenarios with different symmetry properties.

Criteria for opening an axial cavity in cross-wedge rolling

The application of empirical, phenomenological deformation and energy theories of metal fracture in relation to the process of cross-wedge rolling is considered. A distinctive feature of cross-wedge rolling in comparison with other metal pressure treatment processes is the complex stress-strain state in the deformation focus and the possibility of opening the axial cavity. The existing empirical criteria of destruction are given. The scheme of a new calibration test based on crosswedge rolling and the fracture criterion proposed by Zbigniew Pater are considered. It is shown that the Zbigniew Pater’s empirical criterion for the destruction of Zbigniew Pater provides a correct determination of the moment of opening the cavity during cross-wedge rolling. The improved phenomenological deformation theory is based on the theory of fracture developed by V. L. Kolmogorov. It is proposed to evaluate the plasticity of the metal depending on the stress state in the form of two independent invariants of the stress tensor: the average stress and the parameter of the third invariant of the stress tensor. The dependence of the limit value of accumulated deformation, at which destruction occurs, on the stress state is constructed in the form of a plasticity surface. The phenomenological deformation criterion of metal destruction during plastic deformations allows us to determine the moment of opening the cavity during cross-wedge rolling and the plasticity resource of the metal.

THREE-TERM REPRESENTATIONS OF POWER TENSOR SERIES IN THE THEORY OF CONSTITUTIVE RELATIONS

A class of power tensor series (constitutive relations) with coefficients (material functions), which are functions of three independent invariants, is considered in three-dimensional space. Based on the Hamilton–Cayley formula the exact expressions in the form of matrix series are found for the coefficients of three-term representations of such power series. The relationship of the coefficients of direct and inverse three-term constitutive relations is derived. The cases of tensor linearity, or quasi-linearity, as well as the independence of material functions from invariants are discussed.

Gauge invariants of linearized gravity with a general background metric

In linearized gravity with distributed matter, the background metric has no generic symmetries, and decomposition of the metric perturbation into global normal modes is generally impractical. This complicates the identification of the gauge-invariant part of the perturbation, which is a concern, for example, in the theory of dispersive gravitational waves (GWs) whose energy–momentum must be gauge-invariant. Here, we propose how to identify the gauge-invariant part of the metric perturbation and the six independent gauge invariants per se for an arbitrary background metric. For the Minkowski background, the operator that projects the metric perturbation on the invariant subspace is proportional to the well-known dispersion operator of linear GWs in vacuum. For a general background, this operator is expressed in terms of the Green’s operator of the vacuum wave equation. If the background is smooth, it can be found asymptotically using the inverse scale of the background metric as a small parameter.

Dualization of higher derivative heterotic supergravities in 6D and 10D

There exist two four-derivative extensions of N = (1, 0) supergravity in six dimensions. A particular combination of them is known to dualize to the analog of the Bergshoeff-de Roo (BdR) action in 10D. Here we first show that the two extensions are not related to each other by any field redefinitions. Next, we dualize them separately thereby obtaining a two parameter dual theory. This is done directly at the level of the action, thus avoiding the laborious method of integrating equations of motion of the dualized theory into an action. To explore whether a similar phenomenon exists in 10D, we study the dualization of the BdR action in 10D in detail. We find an obstacle in the separation of the result into a sum of two independent invariants because of the presence of terms which do not lift from 6D to 10D. We also compare the dual of the BdR action with an existing result obtained in superspace. We find that the bosonic actions agree modulo field redefinitions.

Projective invariants of images

The method of equivariant moving frames is employed to construct and completely classify the differential invariants for the action of the projective group on functions defined on the two-dimensional projective plane. While there are four independent differential invariants of order $\leq 3$ , it is proved that the algebra of differential invariants is generated by just two of them through invariant differentiation. The projective differential invariants are, in particular, of importance in image processing applications.

Digital rock analysis to estimate stress-sensitive rock permeabilities

Reservoir deformations due to stress changes may have a substantial impact on the effective flow properties of reservoir rocks, which may potentially impact productivity and injectivity performance. Absolute permeability has been a primary factor in measuring hydraulic conductivity of porous materials. In order to investigate the impact of the granular structure and the extent of intergranular cementation on stress-dependent absolute permeability, we present a workflow that utilizes limited knowledge about the rock sample in the form of micro-Computed Tomography images to simulate stress-induced deformations. This analysis is then followed by Lattice Boltzmann Methods to simulate fluid flow in the deformed medium. While previous works using digital rocks ignored the impact of intergranular cements, we should consider the impact of these cements on the stress sensitivity of absolute permeability of the rock. It is notable that changes in the rock properties like the nature of stress would be a continuous change, so measurements at few stress points may not be enough to describe the overall response in any arbitrary loadings. Despite empirical formulae that are limited to few measurements, we proposed a method based on the equation of states to estimate rock permeability through three independent strain invariants. Digital rock physics, finite element analysis, and Lattice Boltzmann Methods were integrated to obtain a quick estimation of the rock permeability using micro-CT images without the need for repetitive laboratory experiments. Our results show that absolute permeability decreases significantly with increasing first and third strain invariants while it slightly increases with the second strain invariant. The stress sensitivity of absolute permeability was found to be negatively correlated with Young’s modulus of the intergranular cement. We observed that incorporating more intergranular cement will make the absolute permeability more sensitive to stress changes.

Ghost instability in the teleparallel gravity model with parity violations

In this paper we consider the parity violating gravity model within the framework of teleparallel gravity. The parity violations are caused by the couplings of a scalar field to the scalar invariants which are parity-odd and quadratic in the torsion tensor. Totally there are two such type independent invariants, and one of them is the Nieh-Yan density. Through investigations on the cosmological perturbations of this model, we find that in general it suffers from the difficulties of ghost instability in the scalar and vector perturbations. But in the special case only the coupling to the Nieh-Yan density exists, this model is ghost free and reduces to the Nieh-Yan modified Teleparallel Gravity model. We also analyze the severity of the ghost instability by studying the perturbations around the Minkowski background.

Invariants in copolar interferometry: An Abelian gauge theory

An $N$-element interferometer measures correlations among pairs of array elements. Closure invariants associated with closed loops among array elements are immune to multiplicative, element-based ("local") corruptions that occur in these measurements. Till recently, it has been unclear how a complete set of independent invariants can be analytically determined. We view the local, element-based corruptions in co-polar correlations as gauge tranformations belonging to the gauge group $\textrm{GL}(1,\mathbb{C})$. Closure quantities are then naturally gauge invariant. We use this to provide a simple and effective formalism, and identify the complete set of independent closure invariants from co-polar interferometric correlations using only quantities defined on $(N-1)(N-2)/2$ elementary and independent triangular loops. The $(N-1)(N-2)/2$ closure phases and $N(N-3)/2$ closure amplitudes (totaling $N^2-3N+1$ real invariants), familiar in astronomical interferometry, naturally emerge from this formalism, which unifies what has required separate treatments until now. We do not require auto-correlations, but can easily include them if reliably measured. This unified view clarifies issues relating to noise and inference of object model parameters. It also allows us to extend the rule of parallel transport associated with Pancharatnam phase in optics to apply to amplitudes as well. The framework presented here extends to $\textrm{GL}(2,\mathbb{C}$) for full polarimetric interferometry as presented in a companion paper, which generalizes and clarifies earlier work. Our findings are relevant to state of the art co-polar and full polarimetric very long baseline interferometry measurements to determine features very near the event horizons of blackholes at the centers of M87, Centaurus~A, and the Milky Way.