Nonlocal Transformation Research Articles

Analytical properties and solutions for the synchronous pulsating bubble clusters distributed on a spherical surface

The present work concerns with the nonlinear dynamics for the synchronous pulsating bubble clusters uniformly distributed on a spherical surface. First, the governing equation for such clusters with 4/6/8/12/20 coupled bubbles are established. Second, the maximum and minimum radii for the gas-filled bubble clusters are analyzed according to the first integral. Third, by introducing suitable nonlocal transformations, two novel equivalent parametric analytical solutions in the form of Weierstrass elliptic function are constructed for the gas-filled bubble clusters for a specific polytropic exponent κ=3/2 without considering the surface tension, and based on which we immediately derive the parametric analytical solution for the corresponding vapor bubble clusters. Further, to consider the case of arbitrary polytropic exponent and surface tension, we develop a direct approach to construct the parametric analytical solution using Jacobi elliptic function for gas-filled bubble clusters. It is shown that, the behaviors and results for the bubble clusters will degenerate to the corresponding ones for single bubbles as the radius of the bubble cluster approaches infinity. In addition, on the basis of the analytical results, dynamic properties and motion laws of the bubble clusters are also discussed.

A Multidimensional Tensor Low Rank Method for Magnetic Resonance Image Denoising.

In this paper, we present the Magnetic Resonance Image (MRI) denoising method via nonlocal multidimensional low rank tensor transformation constraint (NLRT). We first design a nonlocal MRI denoising method by non-local low rank tensor recovery framework. Furthermore, a multidimensional low rank tensor constraint is used to obtain low-rank prior information combined with 3-dimensional structure feature of MRI image cubes. Our NLRT can achieve denoising by retaining more image detail information. The optimization and updating process of the model is solved via the alternating direction method of multipliers (ADMM) algorithm. Several state-of-the-art denoising methods are selected for comparative experiments. In order to reflect the performance of the denoising method, Rician noise with different levels is added to the experiment to analyze the results. The experimental results prove that our NLTR has more outstanding denoising ability and can obtain better MRI images.

Nicolai maps with four-fermion interactions

Nicolai maps offer an alternative description of supersymmetric theories via nonlinear and nonlocal transformations characterized by the so-called ‘free-action’ and ‘determinant-matching’ conditions. The latter expresses the equality of the Jacobian determinant of the transformation with the one obtained by integrating out the fermions, which so far have been considered only to quadratic terms. We argue that such a restriction is not substantial, as Nicolai maps can be constructed for arbitrary nonlinear sigma models, which feature four-fermion interactions. The fermionic effective one-loop action then gets generalized to higher loops and the perturbative tree expansion of such Nicolai maps receives quantum corrections in the form of fermion loop decorations. The ‘free-action condition’ continues to hold for the classical map, but the ‘determinant-matching condition’ is extended to an infinite hierarchy in fermion loop order. After general considerations for sigma models in four dimensions, we specialize to the case of ℂPN symmetric spaces and construct the associated Nicolai map. These sigma models admit a formulation with only quadratic fermions via an auxiliary vector field, which does not simplify our analysis.

The Nicolai-map approach to supersymmetry

In 1980 Hermann Nicolai proposed a characterization of supersymmetric theories that became known as the Nicolai map. This is a particular nonlocal and nonlinear field transformation, whose perturbative expansion is given by fermion-line trees with bosonic leaves. Quantum correlation functions can by evaluated using the inversely transformed fields in the free theory. After initial promise and excitement (fuelling the author’s PhD work!), the subject all but fell dormant for 35 years. Recently however, technical progress in the construction as well as a deeper insight into the nature of the map have been achieved, from quantum mechanics to super Yang–Mills in various dimensions. I will present the Nicolai map from this modern perspective and touch on some of the current developments.

Integrable nonlocal PT-symmetric generalized -mKdV equations

Based on a soliton hierarchy associated with , we construct two integrable nonlocal PT-symmetric generalized mKdV equations. The key step is to formulate two nonlocal reverse-spacetime similarity transformations for the involved spectral matrix, and therefore, integrable nonlocal complex and real reverse-spacetime generalized -mKdV equations of fifth-order are presented. The resulting reduced nonlocal integrable equations inherit infinitely many commuting symmetries and conservation laws.

Higher-Order Corrections to the Effective Field Theory of Low-Energy Axions

Dark matter (DM) can be composed of a collection of axions, or axion-like particles (ALPs), whose existence is due to the spontaneous breaking of the Peccei–Quinn U(1) symmetry, which is the most compelling solution of the strong CP-problem of quantum chromodynamics (QCD). Axions must be spin-0 particles with very small masses and extremely weak interactions with themselves as well as with the particles that constitute the Standard Model. In general, the physics of axions is detailed by a quantum field theory of a real scalar field, ϕ. Nevertheless, it is more convenient to implement a nonrelativistic effective field theory with a complex scalar field, ψ, to characterize the mentioned axions in the low-energy regime. A possible application of this equivalent description is for studying the collapse of cold dark matter into more complex structures. There have been a few derivations of effective Lagrangians for the complex field ψ, which were all equivalent after a nonlocal space transformation between ϕ and ψ was found, and some other corrections were introduced. Our contribution herein is to further provide higher-order corrections; in particular, we compute the effective field theory Lagrangian up to order (ψ*ψ)5, also incorporating the fast-oscillating field fluctuations into the dominant slowly varying nonrelativistic field.

Dynamics of the modified emden and pseudo-modified emden equations: position-dependent mass, invariance and exact solvability

We consider the modified Emden equation (MEE) and introduce its most general solution, using the most general solution for the simple harmonic oscillator’s linear dynamical equation (i.e. the initial conditions shall be identified by the PDM-MEE problem at hand). We use a general nonlocal point transformation and show that modified Emden dynamical equation is transformed to describe position-dependent mass (PDM) classical particles. Two PDM-MEE-type classical particles are used as illustrative examples, and their exact solutions are reported. Under specific parametric considerations, the phase-space trajectories are reported for the MEE-type and for PDM-MEE-type classical particles.

On an integrable family of oscillators with linear and quadratic damping

In this work we study integrability of a family of nonlinear oscillators with linear and quadratic damping. Equations from this family often appear in various applications in physics, mechanics and biology. We demonstrate that certain nonlocal transformations preserve integrating factors for the considered family of equations and provide an explicit expression that connects integrating factors of two nonlocally related oscillators. We apply these results and construct an integrable family of oscillators with linear and quadratic damping, which is connected to an equation from the Painlevé–Gambier classification. We show that members of this family possess an integrating factor, a first integral in terms of the hypergeometric function and a pair of invariant curves. In order to explicitly illustrate our results, we construct new integrable examples of two biologically and physically relevant dynamical systems.

Matrices characterizing non-local OAM transformations of light beams.

Matrices characterizing the orbital angular momentum (OAM) transformations of deterministic and random light beams by commonly used OAM-modulating optical systems are revealed. Such matrices are the counterparts of the Jones and the Mueller matrices in polarization optics defined for the input and output OAM indices and the radial variables. In particular, matrices of systems leading to OAM mode shift, dispersion, and coupling are discussed.

Duality, criticality, anomaly, and topology in quantum spin-1 chains

In quantum spin-1 chains, there is a nonlocal unitary transformation known as the Kennedy-Tasaki transformation ${U}_{\mathrm{KT}}$, which defines a duality between the Haldane phase and the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry-breaking phase. In this paper, we find that ${U}_{\mathrm{KT}}$ also defines a duality between a topological Ising critical phase and a trivial Ising critical phase, which provides a ``hidden symmetry breaking'' interpretation of the topological criticality. Moreover, since the duality relates different phases of matter, we argue that a model with self-duality (i.e., invariant under ${U}_{\mathrm{KT}}$) is natural to be at a critical or multicritical point. We study concrete examples to demonstrate this argument. In particular, when $H$ is the Hamiltonian of the spin-1 antiferromagnetic Heisenberg chain, we prove that the self-dual model $H+{U}_{\mathrm{KT}}H{U}_{\mathrm{KT}}$ is exactly equivalent to a gapless spin-$1/2$ XY chain, which also implies an emergent quantum anomaly. On the other hand, we show that the topological and trivial Ising critical phases that are dual to each other meet at a multicritical point which is indeed self-dual.

Linearizabiliy and Lax representations for cubic autonomous and non-autonomous nonlinear oscillators

In this work we consider a family of cubic, with respect to the first derivative, second-order ordinary differential equations. We study linearizability conditions for this family of equations via generalized nonlocal transformations. We construct linearizability criteria in the explicit form for some particular cases of these transformations. We show that each linearizable equation admits a quadratic rational first integral. Moreover, we demonstrate that generalized nonlocal transformations under certain conditions preserve Lax integrability for equations from the considered family. Consequently, we find that any linearizable equation from the considered family possesses a Lax representation. This provides a connection between two different approaches for studying integrability and demonstrates that the Lax technique can be effectively applied to finding and classifying dissipative integrable autonomous and non-autonomous nonlinear oscillators. We illustrate our results by several examples of linearizable equations, including a non-autonomous generalization of the Rayleigh–Duffing oscillator, a family of chemical oscillators and a generalized Duffing–Van der Pol oscillator.

Deterministic improvements of quantum measurements with grouping of compatible operators, non-local transformations, and covariance estimates

Obtaining the expectation value of an observable on a quantum computer is a crucial step in the variational quantum algorithms. For complicated observables such as molecular electronic Hamiltonians, one of the strategies is to present the observable as a linear combination of measurable fragments. The main problem of this approach is a large number of measurements required for accurate estimation of the observable’s expectation value. We consider three previously studied directions that minimize the number of measurements: (1) grouping commuting operators using the greedy approach, (2) involving non-local unitary transformations for measuring, and (3) taking advantage of compatibility of some Pauli products with several measurable groups. The last direction gives rise to a general framework that not only provides improvements over previous methods but also connects measurement grouping approaches with recent advances in techniques of shadow tomography. Following this direction, we develop two measurement schemes that achieve a severalfold reduction in the number of measurements for a set of model molecules compared to previous state-of-the-art methods.

A study of periodic solutions and periodic background solutions for the reverse-space–time modified nonlinear Schrödinger equation

The modified nonlinear Schrödinger (MNLS) equation is used to describe the effect of femtosecond pulses and nonlinear dispersion in long single-mode fibers and the propagation of modulated Alfvén waves along a magnetic field in a cold plasma. Under investigation in this paper is its nonlocal reverse-space–time type, extending it to the more general case, which has much physical significance in the field of nonlocal nonlinear dynamical systems. Through constructing nonlocal type Darboux transformations (DTs), we will derive a series of new solutions as follows: (1)Bright and dark solitons on plane wave backgrounds, single and double periodic solutions; (2)Breather, rogue wave solutions and their coexistence mechanism on single periodic backgrounds via odd-fold DTs; (3) Bright and dark breather-like solutions on single periodic backgrounds, breather, rogue wave solutions and their degenerate and coexistence forms on double periodic backgrounds via even-fold DTs. In addition, we will discuss dynamic behaviors of those solutions through graphic simulations.

Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics

We present a new Integrated Finite Element Neural Network framework (I-FENN), with the objective to accelerate the numerical solution of nonlinear computational mechanics problems. We leverage the swift predictive capability of neural networks (NNs) and we embed them inside the finite element stiffness function, to compute element-level state variables and their derivatives within a nonlinear, iterative numerical solution. This process is conducted jointly with conventional finite element methods that involve shape functions: the NN receives input data that resembles the material point deformation and its output is used to construct element-level field variables such as the element Jacobian matrix and residual vector. Here we introduce I-FENN to the continuum damage analysis of quasi-brittle materials, and we establish a new non-local gradient-based damage framework which operates at the cost of a local damage approach. First, we develop a physics informed neural network (PINN) to resemble the non-local gradient model and then we train the neural network offline. The network learns to predict the non-local equivalent strain at each material point, as well as its derivative with respect to the local strain. Then, the PINN is integrated in the element stiffness definition and conducts the local to non-local strain transformation, whereas the two PINN outputs are used to construct the element Jacobian matrix and residual vector. This process is carried out within the nonlinear solver, until numerical convergence is achieved. The resulting method bears the computational cost of the conventional local damage approach, but ensures mesh-independent results and a diffused non-local strain and damage profile. As a result, the proposed method tackles the vital drawbacks of both the local and non-local gradient method, respectively being the mesh-dependence and additional computational cost. We showcase through a series of numerical examples the computational efficiency and generalization capability of I-FENN in the context of non-local continuum damage, and we discuss the future outlook. The PINN training code and associated training data files have been made publicly available online, to enable reproducibility of our results.

Supersymmetric large-order perturbation with the Nicolai map

In rigidly supersymmetric quantum theories, the Nicolai map allows one to turn on a coupling constant (from zero to a finite value) by keeping the (free) functional integration measure but subjecting the fields to a particular nonlocal and nonlinear transformation. A recursive perturbative construction of the Nicolai-transformed field configuration expresses it as a power series in the coupling, with its coefficient function at order n being a sum of particular tree diagrams. For a quantum-mechanical example, the size of these tree diagrams (under a certain functional norm) is estimated by the (n+1)st power of the field size, and their number grows like n−3/2×4.967n. Such an asymptotic behaviour translates to a finite convergence radius for the formal perturbative expansion of the Nicolai map, which establishes its non-perturbative existence. The known factorial growth of the number of Feynman diagrams for quantum correlators is reproduced by the combinatorics of free-field Wick contractions as usual. We expect our results to extend to higher dimensions, including super Yang–Mills theory.

Two loop ghost free quantisation of Wilson loops in [formula omitted] supersymmetric Yang-Mills

We report a perturbative calculation of the expectation value of the infinite straight line Maldacena-Wilson loop in N=4 supersymmetric Yang-Mills theory to order g6. Thus, we extend the previous perturbative result by one order. The vacuum expectation value is reformulated in terms of a non-linear and non-local transformation, the Nicolai map, mapping the full functional measure of the interacting theory to that of a free bosonic theory. The results are twofold. The perturbative cancellations of the different contributions to the Maldacena-Wilson loop are by no means trivial and seem to resemble those of the circular Maldacena-Wilson loop at order g4. Furthermore, we argue that our approach to computing quantum correlation functions is competitive with more standard diagrammatic techniques.

Nonlocal Transformations with Additional Variables. Forced Symmetries

Nonlocal Transformations with Additional Variables. Forced Symmetries

All‐Dielectric Metasurface Empowered Optical‐Electronic Hybrid Neural Networks

AbstractOptical computing has a series of advantages over its electronic counterpart, e.g., low energy consumption, high speed, and intrinsic parallelism. Diffraction deep neural networks (D2NNs) are a prominent example capable of processing images directly without addressing the spatial locations of each element. Despite the great successes, the D2NNs typically utilize the multilayer framework and face the severe challenge of misalignment in the optical region. Herein, a single metasurface‐based optical‐electronic hybrid neural network (OENN) is proposed and experimentally demonstrated. The OENN is composed of a titanium dioxide (TiO2) metasurface and a fully‐connected electronic layer. The combination of nonlocal neural layer and nonlinear transformation has significantly expanded the neural network capacity. Consequently, the classification accuracy on handwritten digits recognition can still be as high as 98.05% without employing the architecture of cascaded metasurfaces. The OENN shall shed light on the practical applications of optical computing in the visible spectrum.

Effect of nonlocal transformations on the linearizability and exact solvability of the nonlinear generalized modified Emden-type equations

The nonlinear generalized modified Emden-type equations (GMEE) are known to be linearizable into simple harmonic oscillator (HO) or damped harmonic oscillators (DHO) via some nonlocal transformations. Hereby, we show that the structure of the nonlocal transformation and the linearizability into HO or DHO determines the nature/structure of the dynamical forces involved (hence, determine the structure of the dynamical equation). Yet, a reverse engineering strategy is used so that the exact solutions of the emerging GMEE are nonlocally transformed to find the exact solutions of the HO and DHO dynamical equations. Consequently, whilst the exact solution for the HO remains a textbook one, the exact solution for the DHO (never reported elsewhere, to the best of our knowledge) turns out to be manifestly the most explicit and general solution that offers consistency and comprehensive coverage for the associated under-damping, critical-damping, and over-damping cases (i.e. no complex settings for the coordinates and/or the velocities are eminent/feasible). Moreover, for all emerging dynamical system, we report illustrative figures for each solution as well as the corresponding phase-space trajectories as they evolve in time.

The nonlocal potential transformation method for solitary wave packets of a shock-breaking dynamics system

In this work, we employ the potential similarity transformation method to derive some solitary wave packet solutions for the Calogero-Bogoyavlenskii-Schiff (CBS) equation. Exploiting nonsingular local multipliers, a set of local conservation laws is presented for the equation. The nonlocally related partial differential equation (PDE) systems were observed. Considering the nonlocal-related PDE systems, we present 46 analytic solutions of the CBS equation that involves several arbitrary functions, enabling the spread of the equation’s solution to understand Riemann waves dynamics extensively. These are helpful in severalapplications such as electromagnetic waves in an optical tsunami in fibers, magneto-sonic and ion in plasma, the acoustic waves in compressible fluids, surface, and interval waves in oceans, tidal and tsunami in rivers.