Higher-order Extension Research Articles

A Thread-Safe Lattice Boltzmann Model for Multicomponent Turbulent Jet Simulations

In this work an optimized multicomponent lattice Boltzmann (LB) model is deployed to simulate axisymmetric turbulent jets of a fluid evolving in a quiescent, immiscible environment over a wide range of dynamic regimes. The implementation of the multicomponent LB code achieves peak performance on graphic processing units (GPUs) with a significant reduction of the memory footprint, retains the algorithmic simplicity inherent to standard LB computing, and, being based on a high-order extension of the thread-safe LB algorithm, allows us to perform stable simulations at vanishingly low viscosities. The proposed approach opens attractive prospects for high-performance computing simulations of realistic turbulent flows with interfaces on GPU-based architectures.

High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries

Abstract In this article, a fourth-order finite-difference ghost-point method for the Poisson equation on regular Cartesian mesh is presented. The method can be considered the high-order extension of the second-order ghost method introduced earlier by the authors. Three different discretizations are considered, which differ in the stencil that discretizes the Laplacian and the source term. It is shown that only two of them provide a stable method. The accuracy of such stable methods is numerically verified on several test problems.

Implementing arbitrary multimode continuous-variable quantum gates with fixed non-Gaussian states and adaptive linear optics

Non-Gaussian quantum gates are essential components for optical quantum information processing. However, the efficient implementation of practically important multimode higher-order non-Gaussian gates has not been comprehensively studied. We propose a measurement-based method to directly implement general, multimode, and higher-order non-Gaussian gates using only fixed non-Gaussian ancillary states and adaptive linear optics. Compared to existing methods, our method allows for a more resource-efficient and experimentally feasible implementation of multimode gates that are important for various applications in optical quantum technology, such as the two-mode cubic quantum nondemolition gate or the three-mode continuous-variable Toffoli gate, and their higher-order extensions. Our results will expedite the progress toward fault-tolerant universal quantum computing with light. Published by the American Physical Society 2024

Carrollian supersymmetry and SYK-like models

This work challenges the conventional notion that in a spacetime dimension higher than one, a supersymmetric Lagrangian invariably consists of purely bosonic terms, purely fermionic terms, as well as boson-fermion mixing terms. By recasting a relativistic Lagrangian in terms of its nonrelativistic and ultrarelativistic sectors, we reveal that an ultrarelativistic (Carrollian) supersymmetric Lagrangian exhibits an exotic feature; that is, it can exist without a purely bosonic contribution. Based on this result, we demonstrate a link between higher-dimensional Carrollian and (0+1)-dimensional quantum mechanical models, yielding higher-order extensions of supersymmetric Sachdev-Ye-Kitaev (SYK) models in which purely bosonic higher-order terms are absent. Given that supersymmetry plays an essential role in improving the quantum behavior and solubility, our findings may lead to interesting applications in non-AdS holography. Published by the American Physical Society 2024

A generic type system for higher-order Ψ-calculi

The Higher-Order Ψ-calculus framework (HOΨ) by Parrow et al. is a generalisation of many first- and higher-order extensions of the π-calculus. In this paper we present a generic type system for HOΨ-calculi. It satisfies a subject reduction property and can be instantiated to yield both existing and new type systems for calculi, that can be expressed as HOΨ-calculi. In this paper, we consider the type system for termination in HOπ by Demangeon et al. Moreover, we derive a new type system for the ρ-calculus of Meredith and Radestock and present a type system for non-interference for mobile code.

High-order dilated nested arrays with increased degrees of freedom and reduced mutual coupling

In a companion paper [32], a sparse array called the dilated nested array (DNA) was introduced, which owns the same number of uniform degrees of freedom (DOFs) as the nested array but has the first and third constituent sub-arrays dense, since their sensors are uniformly spaced out by the critical inter-sensor spacing (2×λ/2). Therefore, in this paper we introduce two further dilations to address this shortcoming. In the first dilation - referred to as the one-sided dilated nested array (OS-DNA), Qf possible high-order extensions starting from the 2nd-order extension (Q=2) to the (Qf+1)th-order extension are applied to the third parent sub-array, and in the second dilation - referred to as the two-sided dilated nested array (TS-DNA), an additional extension is applied to the first parent sub-array. While the high-order extensions of the OS-DNA as well as the TS-DNA strictly preserve the same number of uniform DOFs of the parent DNA, the first extension of the OS-DNA eliminates all the sensor pairs with separation 2 in the third parent sub-array, whereas the Qth-order extensions (for 2<Q≤Qf+1) increase the number of nonuniform DOFs as Q increases. The TS-DNA then eliminates all the sensor pairs with spacing 2 in the first sub-array of the (Qf+1)th-order extension and concurrently maintains the higher number of nonuniform DOFs of this parent order. As such, the TS-DNA, named the super dilated nested array (SDNA) as well, still retains the number of uniform and nonuniform DOFs of the highest-order extension of the OS-DNA and at the same time enjoys the ideal critical weights of the co-prime array. Many theoretical properties are proved and extensive simulations are included to demonstrate the superior performance of these arrays.

The Sasa–Satsuma equation with high-order discrete spectra in space-time solitonic regions: soliton resolution via the mixed ∂¯ -Riemann–Hilbert problem

In this paper, we investigate the Cauchy problem of the Sasa–Satsuma (SS) equation with initial data belonging to the Schwartz space. The SS equation is one of the integrable higher-order extensions of the nonlinear Schrödinger equation and admits a 3 × 3 Lax representation. With the aid of the ∂¯ -nonlinear steepest descent method of the mixed ∂¯ -Riemann–Hilbert problem, we give the soliton resolution and long-time asymptotics for the Cauchy problem of the SS equation with the existence of second-order discrete spectra in the space-time solitonic regions.

Soliton unveilings in optical fiber transmission: Examining soliton structures through the Sasa–Satsuma equation

The exploration of higher-order and multicomponent extensions of the nonlinear Schrödinger equation holds significant importance across diverse applications, notably within the field of optics. Among these equations, the integrable Sasa–Satsuma equation stands out for its captivating soliton solutions. The Sasa–Satsuma equation serves as a mathematical representation for describing the propagation of femtosecond light pulses through optical fibers. The breather, multiwave, combined dark–bright, singular, dark, bright and periodic singular optical soliton solutions are derived in this paper by using the 1φ(ϑ),φ′(ϑ)φ(ϑ) and multivariate generalized exponential rational integral function approach. Furthermore, the paper emphasizes the essential fiber parameters essential for the emergence of these structures and offers visual representations of chosen solutions to elucidate their physical characteristics. The novelty of this work lies in the inaugural application of these methods to the Sasa–Satsuma equation and gives a significant advancement in the understanding of optical phenomena. This study not only opens avenues for further research but also introduces a fresh perspective on the Sasa–Satsuma model in the field of nonlinear physics with its applications in optical systems.

Higher-order properties and extensions for indirect MRAC and APPC of linear systems

Higher-order properties and extensions for indirect MRAC and APPC of linear systems

Learning higher-order features for relation prediction in knowledge hypergraph

Knowledge Hypergraph (KHG) is a higher-order extension of the Knowledge Graph (KG), and its relation prediction is based on known data to predict unknown higher-order relations, thereby providing useful knowledge services. However, the existing KHG relation algorithms still have some limitations: (i) most studies only consider the influence of the direct neighbors, and (ii) they ignore the complex interactions existing inside higher-order facts. Based on this, we propose a KHG relation prediction model HoGCNF2 based on higher-order hypergraph convolutional network and feature fusion. Dual-channel hypergraph convolutional network considers the significant and higher-order information propagation of entities. Feature fusion strategy considers different types of higher-order structures. Besides, attention mechanism adaptively assigns weights to the learned embeddings. Extensive experiments demonstrate the superiority of HoGCNF2 on different datasets. Specifically, the MRR result improves by 2.6% on the unfixed dataset FB-AUTO, and improves by 9.7% on the fixed dataset WikiPeople-4. Our implementations are publicly available at: https://doi.org/10.24433/CO.5584354.v1.

THE Q-ANALOGUES OF NONSINGULAR FRACTIONAL OPERATORS WITH MITTAG-LEFFLER AND EXPONENTIAL KERNELS

This paper is devoted to introducing a new [Formula: see text]-fractional calculus in the framework of Atangana–Baleanu ([Formula: see text]) and Caputo–Fabrizio ([Formula: see text]) operators. First, an appropriate [Formula: see text]-Mittag-Leffler function is defined, and then [Formula: see text]-analogues of fractional derivatives of Atangana–Baleanu–Riemann ([Formula: see text]) and Atangana–Baleanu–Caputo ([Formula: see text]) are derived. Next, the [Formula: see text]-analogues of proper fractional integrals in the [Formula: see text] sense are proved. Several important properties of these definitions are investigated by using the [Formula: see text]-Laplace transform. Additionally, a suitable [Formula: see text]-exponential function is defined, and the [Formula: see text]-analogues of [Formula: see text] fractional derivatives with their inverse operators are introduced. The higher-order extension of the [Formula: see text]-analogues of [Formula: see text] and [Formula: see text] fractional operators is discussed. Finally, a demonstrative example is enhanced to check the effectiveness of [Formula: see text]-[Formula: see text] calculus. We believe that these outcomes will be the care of many researchers in the field of fractional calculus.

Approximate deconvolution discretisation

A new strategy is presented for the construction of high-order spatial discretisations extracted from a lower-order basic discretisation. The key consideration is that any spatial discretisation of a derivative of a solution can be expressed as the exact differentiation of a corresponding ‘filtered’ solution. Hence, each numerical discretisation method may be directly linked to a unique spatial filter, expressing the truncation error of the basic method. By approximately deconvolving the implied filter of the basic numerical discretisation an augmented high-order method can be obtained. In fact, adopting a deconvolution of the implied filter of suitable higher order enables the formulation of a new spatial discretisation method of correspondingly higher order. This construction is illustrated for finite difference (FD) discretisation schemes, solving partial differential equations in fluid mechanics. Knowing the implied filter of the basic discretisation, one can derive a corresponding higher order method by approximately eliminating the implied spatial filter to a certain desired order. We use deconvolution to compensate for the implied filter. This corresponds to a ‘sharpening’ of numerical solution features before the application of the basic FD method. The combination will be referred to as Approximate Deconvolution Discretisation (ADD). The accuracy of the deconvolved FD scheme depends on the order of approximation of the deconvolution filter. We present the ‘sharpening’ of several well-known FD operators for first- and second-order derivatives and quantify the achieved accuracy in terms of the modified wavenumber spectrum. Examples include high-order extensions up to new schemes with spectral accuracy. The practicality of the deconvolved FD schemes is illustrated in various ways: (i) by investigation of exactly solvable advection and diffusion problems, (ii) by tracking the evolution of the numerical solution to the Taylor-Green vortex problem and (iii) by showing that ADD yields spectral accuracy for the Burgers equation and for double-jet flow of an incompressible fluid.

Higher-Order Property-Directed Reachability

The property-directed reachability (PDR) has been used as a successful method for automated verification of first-order transition systems. We propose a higher-order extension of PDR, called HoPDR, where higher-order recursive functions may be used to describe transition systems. We formalize HoPDR for the validity checking problem for conjunctive nu-HFL(Z), a higher-order fixpoint logic with integers and greatest fixpoint operators. The validity checking problem can also be viewed as a higher-order extension of the satisfiability problem for Constrained Horn Clauses (CHC), and safety property verification of higher-order programs can naturally be reduced to the validity checking problem. We have implemented a prototype verification tool based on HoPDR and confirmed its effectiveness. We also compare our HoPDR procedure with the PDR procedure for first-order systems and previous methods for fully automated higher-order program verification.

Cornering extended Starobinsky inflation with CMB and SKA

Starobinsky inflation is an attractive, fundamental model to explain the Planck measurements, and its higher-order extension may allow us to probe quantum gravity effects. We show that future CMB data combined with the 21cm intensity map from SKA will meaningfully probe such an extended Starobinsky model. A combined analysis will provide a precise measurement and intriguing insight into inflationary dynamics, even accounting for correlations with astrophysical parameters.

A higher-order extension of Atangana–Baleanu fractional operators with respect to another function and a Gronwall-type inequality

This paper aims to extend the Caputo–Atangana–Baleanu (ABC) and Riemann–Atangana–Baleanu (ABR) fractional derivatives with respect to another function, from fractional order mu in (0,1] to an arbitrary order mu in (n,n+1], n=0,1,2,dots . Also, their corresponding Atangana–Baleanu (AB) fractional integral is extended. Additionally, several properties of such definitions are proved. Moreover, the generalization of Gronwall’s inequality in the framework of the AB fractional integral with respect to another function is introduced. Furthermore, Picard’s iterative method is employed to discuss the existence and uniqueness of the solution for a higher-order initial fractional differential equation involving an ABC operator with respect to another function. Finally, examples are given to illustrate the effectiveness of the main findings. The idea of this work may attract many researchers in the future to study some inequalities and fractional differential equations that are related to AB fractional calculus with respect to another function.

Pseudoprocesses related to higher-order equations of vibrations of rods

In this paper we study Fresnel pseudoprocesses whose signed measure density is a solution to a higher-order extension of the equation of vibrations of rods. We also investigate space-fractional extensions of the pseudoprocesses related to the Riesz operator. The measure density is represented in terms of generalized Airy functions which include the classical Airy function as a particular case. We prove that the Fresnel pseudoprocess time-changed with an independent stable subordinator produces genuine stochastic processes. In particular, if the exponent of the subordinator is chosen in a suitable way, the time-changed pseudoprocess is identical in distribution to a mixture of stable processes. The case of a mixture of Cauchy distributions is discussed and we show that the symmetric mixture can be either unimodal or bimodal, while the probability density function of an asymmetric mixture can possibly have an inflection point.

Orbital stability of periodic peakons for a new higher-order μ-Camassa–Holm equation

The consideration here is a higher-order μ-Camassa–Holm equation, which is a higher-order extension of the μ-Camassa–Holm equation and retains some properties of the μ-Camassa–Holm equation and the modified μ-Camassa–Holm equation. By utilizing the inequalities with the maximum and minimum of solutions related to the first three conservation laws, we establish that the periodic peakons of this equation are orbitally stable under small perturbations in the energy space.

Minimal Massive Supergravity.

Minimal massive gravity in three dimensions propagates a single massive spin-2 mode around an anti-de Sitter vacuum. It is distinguished by allowing for vacua with positive central charges of the asymptotic conformal algebra and a bulk graviton of positive energy. We present a new action for the model (and its higher order extensions) in terms of a dreibein and an independent spin connection. From this, we construct its supersymmetric extension. Surprisingly, all vacua complying with bulk and boundary unitarity appear to break supersymmetry spontaneously. In contrast, all supersymmetric vacua have a negative central charge whenever the bulk graviton has positive energy.

A Generic Type System for Higher-Order Ψ-calculi

The Higher-Order $\Psi$-calculus framework (HO$\Psi$) is a generalisation of many first- and higher-order extensions of the $\pi$-calculus. It was proposed by Parrow et al. who showed that higher-order calculi such as HO$\pi$ and CHOCS can be expressed as HO$\Psi$-calculi. In this paper we present a generic type system for HO$\Psi$-calculi which extends previous work by H\"uttel on a generic type system for first-order $\Psi$-calculi. Our generic type system satisfies the usual property of subject reduction and can be instantiated to yield type systems for variants of HO{\pi}, including the type system for termination due to Demangeon et al. Moreover, we derive a type system for the $\rho$-calculus, a reflective higher-order calculus proposed by Meredith and Radestock. This establishes that our generic type system is richer than its predecessor, as the $\rho$-calculus cannot be encoded in the $\pi$-calculus in a way that satisfies standard criteria of encodability.

A class of high-order weighted compact central schemes for solving hyperbolic conservation laws

We propose a class of weighted compact central schemes for solving hyperbolic conservation laws. The linear version can be considered as a high-order extension of the central Lax–Friedrichs scheme and the central conservation element and solution element scheme. On every cell, the solution is approximated by a Pth-order polynomial of which all the DOFs are stored and updated separately. The cell average is updated by a classical finite volume scheme which is constructed based on space-time staggered meshes such that the fluxes are continuous across the interfaces of the adjacent control volumes and, therefore, the local Riemann problem is bypassed. The kth-order spatial derivatives are updated by a central difference of the (k−1)th-order spatial derivatives at cell vertices. All the space-time information is calculated by the Cauchy–Kovalewski procedure. By doing so, the schemes are able to achieve arbitrarily uniform space-time high-order on a compact stencil consisting of only neighboring cells with only one explicit time step. In order to capture discontinuities without spurious oscillations, a weighted essentially non-oscillatory type limiter is tailor-made for the schemes. The limiter preserves the compactness and high-order accuracy of the schemes. The schemes' accuracy, robustness, and efficiency are verified by several numerical examples of scalar conservation laws and the compressible Euler equations.