Finite Expansion Research Articles

Eccentric compression performance of coral concrete-filled aluminum alloy tube (CCFAT) circular short columns

Coral concrete, utilizing coral reef bodies in place of traditional aggregates, shows promising prospects use in island projects. The use of aluminum alloy-reinforced coral concrete columns effectively enhances the load-bearing capacity of these structures. This study aims to investigate the performance of aluminum alloy circular tube-reinforced coral concrete short columns under eccentric compression. The study involved designing fifteen short columns for eccentric compression experiments, focusing on key parameters such as aluminum content (α), eccentricity (e0), and coral concrete strength (fc). The study obtained failure modes, bearing capacities, and load-deformation relationships, and analyzed the impacts of aluminum content, eccentricity, coral concrete strength, and hoop coefficient (ξa) on the columns’ strength, stiffness, and ductility. Subsequently, a finite element parameter expansion analysis using Abaqus was performed. Ultimately, a correction coefficient model was developed, compared with existing international specifications, and used to establish a suitable calculation model for the bearing capacity of CCFAT eccentric short columns. The results indicated that CCFAT eccentric short columns predominantly experienced bending failure. The stresses in the aluminum alloy round tube can reach the conditional yield strength, effectively collaborating with the concrete. The load-deformation relationship did not exhibit an intensification stage, and the lateral deflection curves tended to follow sinusoidal half-wave patterns. During the initial loading stage, adherence to the plane-section assumption was observed, with the hoop effect primarily manifesting in the compression zone. The strength index showed an approximately quadratic relationship with ξa, the stiffness index showed a positive correlation with ξa, and the ductility index demonstrated a strong positive association with ξa. The correction coefficient model, along with AIJ (2001), can be employed for calculating the bearing capacity of CCFAT eccentric short columns. This study introduces the novel concept of aluminum alloy tube-reinforced coral concrete columns as a composite structure for island civil engineering construction. Beyond the benefits of traditional steel tube concrete, these composite columns are expected to reduce engineering costs, shorten construction timelines, and address durability concerns through the incorporation of marine coral stones and fragments as concrete aggregates.

Probing the Local Rapidity Distribution of a One-Dimensional Bose Gas.

One-dimensional Bose gases with contact repulsive interactions are characterized by the presence of infinite-lifetime quasiparticles whose momenta are called the "rapidities." Here, we develop a probe of the local rapidity distribution, based on the fact that rapidities are the asymptotic momenta of the particles after a long one-dimensional expansion. This is done by performing an expansion of a selected slice of the gas. We first apply this idea to a cloud in the quasicondensate regime at equilibrium in a trap. We obtain an experimental picture of the position-dependent rapidity distribution which is in fair agreement with the theory prediction. The asymptotic regime is barely reached, but we show that finite expansion time can be taken into account using the generalized hydrodynamics theory. We then apply this local probe to an out-of-equilibrium situation where the local rapidity distribution is expected to be doubly peaked-a hallmark of a nonthermal state-even though the global rapidity distribution would possess no such distinctive feature. We observe the doubly peaked local rapidity distribution.

Learning S -matrix phases with neural operators

We use Fourier neural operators (FNOs) to study the relation between the modulus and phase of amplitudes in 2→2 elastic scattering at fixed energies. Unlike previous approaches, we do not employ the integral relation imposed by unitarity, but instead train FNOs to discover it from many samples of amplitudes with finite partial wave expansions. When trained only on true samples, the FNO correctly predicts (unique or ambiguous) phases of amplitudes with infinite partial wave expansions. When also trained on false samples, it can rate the quality of its prediction by producing a true/false classifying index. We observe that the value of this index is strongly correlated with the violation of the unitarity constraint for the predicted phase and present examples where it delineates the boundary between allowed and disallowed profiles of the modulus. Our application of FNOs is unconventional: it involves a simultaneous regression-classification task and emphasizes the role of statistics in ensembles of neural operators. We comment on the merits and limitations of the approach and its potential as a new methodology in theoretical physics. Published by the American Physical Society 2024

Romanovski–Jacobi spectral collocation schemes for distributed order differential problems

The article highlights the growing importance of computational mathematics in tackling intricate real-world phenomena governed by distributed order fractional dynamics. These dynamics, characterized by memory effects and non-local interactions, are pervasive in various fields such as physics, biology, finance, and engineering. Despite their prevalence, analytical solutions for distributed order fractional differential equations remain difficult to find, necessitating the development of robust numerical methods. The paper explores the effectiveness of Romanovski–Jacobi spectral collocation techniques in this context and provides a thorough analysis of how to use them to solve distributed order fractional differential equations. By leveraging these schemes, researchers gain valuable insights into efficiently analyzing and simulating fractional systems, thereby advancing our understanding of complex dynamics across diverse domains. It provides precise numerical findings by using finite expansion to evaluate residuals. The precision of the approach is illustrated by numerical simulations, especially in distributed order fractional differential equations. Additionally, we provide a few numerical tests to demonstrate the method’s ability to preserve the underlying problem’s non-smooth solution.

Finite and high-temperature series expansion via many-body perturbation theory: application to Heisenberg spin-1/2 XXZ chain

We present a new algorithm to evaluate the grand potential at high and finite temperatures using many-body perturbation theory. This algorithm enables us to calculate the contribution of any Hugenholtz or Feynman vacuum diagrams and formulate the results as a sum of divided differences. Additionally, the proposed method is applicable to any interaction in any dimension, allowing us to calculate thermodynamic quantities efficiently at any given temperature, particularly at high temperatures.Furthermore, we apply this algorithm to the Heisenberg spin-1/2 XXZ chain. We obtain all coefficients of the high-temperature expansion of the free energy and susceptibility per site of this model up to the sixth order.

Robust Recursive Least-Squares Fixed-Point Smoother and Filter using Covariance Information in Linear Continuous-Time Stochastic Systems with Uncertainties

This study develops robust recursive least-squares (RLS) fixed-point smoothing and filtering algorithms for signals in linear continuous-time stochastic systems with uncertainties. The algorithms use covariance information, such as the cross-covariance function of the signal with the observed value and the autocovariance function of the degraded signal. A finite Fourier cosine series expansion approximates these functions. Additive white Gaussian noise is present in the observation of the degraded signal. A numerical simulation compares the estimation accuracy of the proposed robust RLS filter with the robust RLS Wiener filter, showing similar mean square values (MSVs) of the filtering errors. The MSVs of the proposed robust RLS fixed-point smoother are also compared to those of the proposed robust RLS filter.

Spectral solutions for fractional Klein–Gordon models of distributed order

The Klein–Gordon equation is a fundamental theoretical physics concept, governing the behavior of relativistic quantum particles with spin-zero. Its numerical solution is crucial in fields like quantum field theory, particle physics, and cosmology. The study explores numerical methodologies for solving this equation, highlighting their significance and challenges. This study uses the collocation method to approximate fractional Klein–Gordon models of distributed order based on Shifted Jacobi orthogonal polynomials and Shifted fractional order Jacobi orthogonal functions. While, the distributed term (integral term) was treat using Legendre–Gauss–Lobatto quadrature. It assesses residuals through finite expansion and yields accurate numerical results. The method is more factual and fair when initial and boundary conditions are enforced. Numerical simulations are presented to demonstrate the method’s accuracy, particularly in fractional Klein–Gordon models of distributed order. Furthermore, we offer a few numerical test scenarios to show that the method is able to maintain the non-smooth solution of the underlying issue.

Error bounds for kernel-based approximations of the Koopman operator

We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic simulations. We derive both an exact expression for the variance of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds for the finite-data estimation error. Moreover, we derive a bound on the prediction error of observables in the RKHS using a finite Mercer series expansion. Further, assuming Koopman-invariance of the RKHS, we provide bounds on the full approximation error. Numerical experiments using the Ornstein-Uhlenbeck process illustrate our results.

Coherent structures in elastic turbulent planar jets

At low Reynolds numbers, the flow of a Newtonian planar jet remains laminar, thus easy to characterize. In contrast, the presence of elasticity (e.g., attained dissolving polymers in a Newtonian solvent) enables a highly-complex turbulent-like behavior termed elastic turbulence. In this work, we run data-driven modal decomposition algorithms on high-fidelity data collected from the simulation of an elastic turbulent planar jet. The large-scale motions are expressed as a finite expansion of modes that condense the dominant dynamics. The modes associated with lower frequencies weight the most on the reconstruction of the original data, thus they are further decomposed in space to investigate their implications on the sustainment of the elastic turbulent state. Our findings suggest that slower dynamics are crucial for the sustainment of elastic turbulence, which is connected to the interaction of spanwise-coherent structures, steady in space, with spanwise-periodic traveling waves, causing the breakdown of the structures close to the inlet.

A projection method based on the piecewise Chebyshev cardinal functions for nonlinear stochastic ABC fractional integro‐differential equations

In this study, the Atangana–Baleanu fractional derivative in the Caputo type (as a kind of non‐local and non‐singular derivative) is used to define a new class of stochastic fractional integro‐differential equations. A projection method (more precisely, a Galerkin approach) based on the piecewise Chebyshev cardinal functions is developed to solve these stochastic fractional equations. To construct this method, the operational matrices of fractional and stochastic integrals of these basis functions are obtained and used in the established method. By approximating the solution of the problem with a finite expansion of the expressed basis functions (in which the expansion coefficients are unknown), a system of algebraic equations is obtained. By solving this system, the expansion coefficients and subsequently the solution of the original stochastic fractional problem are obtained. The convergence analysis of the proposed method is investigated, theoretically and numerically. The accuracy of the established procedure is illustrated by solving several numerical examples.

Discrete Chebyshev polynomials for the numerical solution of stochastic fractional two-dimensional Sobolev equation

In this work, the stochastic fractional two-dimensional Sobolev equation is introduced and a collocation method is proposed to solve it. The discrete Chebyshev polynomials are used as a proper family of basis functions to establish this collocation method. Some operational matrices related to conventional and stochastic integrals, as well as fractional and ordinary derivatives of these polynomials, are extracted and successfully used in making the expressed method. More precisely, by approximating the problem solution via a finite expansion of the expressed polynomials (where the expansion coefficients are unknown) and substituting it into the stochastic fractional problem, as well as by employing the above obtained matrices, a system of linear algebraic equations is obtained. Finally, the expansion coefficients and subsequently the solution of the original stochastic fractional problem are found by solving this system. The correctness of the established method is examined by solving some numerical examples.

Shallow Moments to Capture Vertical Structure in Open Curved Shallow Flow

Shallow water models are successfully used for simulating geophysical flows like river floods, tsunamis, sediment transport, or debris flows. Depth-averaged models are in general attractive due to their low computational cost. However, information on the vertical velocity is required a priori, typically by assuming a parametrized profile using a time-independent function, e.g., a constant. A systematic generalization of depth-averaged flow models is given by the shallow moment method. This method retains transient information about the vertical flow profile by using a finite Legendre expansion for said profile with time and space-dependent coefficients. The shallow moment approach allows to include vertical information and generates a hierarchy of equations that in the limit, recover the reference system before depth-averaging. Even a low number of basis functions significantly increases the model’s capability to resolve vertical information and, whenever this is relevant for the modeling task at stake, significantly improves the predictive power compared to classical shallow water systems. In the paper, we show a two-dimensional simulation result based on the shallow moment system to predict processes unobservable in classical shallow flow, namely secondary flow and the redistribution of velocity profiles, which are of relevance in curved channelized flow as experimentally demonstrated by Steffler.

2D photonic crystal hexagonal ring resonator-based all-optical logic gates

This paper proposes a novel and compact design of two-dimensional photonic crystal (PhC) based all-optical logic gates with a unique combination of line defects and ring resonator. The proposed design comprises air holes in a Si background arranged in hexagonal lattice geometry. The finite difference time domain (FDTD) and plane wave expansion (PWE) methods are used to analyze and calculate the required performance metrics. The peculiar property of the proposed structure is that the same structure is used for the realization of all All-optical logic gates and is symmetric about the origin. The principle operation used is the linear interference method in which the phase of the input beams is kept at φ=00 while the appropriate phase is introduced at the additional port known as reference port R so that they may interfere constructively or destructively. The All-Optical logic gates implemented using a proposed single structure are AND, OR, XOR, NOT, NAND, XNOR, and NOR, showing an ON to OFF contrast ratio of 12 dB, 20.63 dB, 4.44 dB, 2.07 dB, 4.44 dB, 12 dB, and 6.05 dB respectively. Moreover, AND, OR, and XNOR gate offers a response period of 0.353 ps and a bit rate of 2.83 Tb/s.

Inferring the probability distribution over strain tensors in polycrystals from diffraction based measurements

Polycrystals illuminated by high-energy X-rays or neutrons produce diffraction patterns in which the measured diffraction peaks encode the individual single crystal strain states. While state of the art X-ray and neutron diffraction approaches can be used to routinely recover per grain mean strain tensors, less work has been produced on the recovery of higher order statistics of the strain distributions across the individual grains. In the setting of small deformations, we consider the problem of estimating the crystal elastic strain tensor probability distribution from diffraction data. For the special case of multivariate Gaussian strain tensor probability distributions, we show that while the mean of the distribution is well defined from measurements, the covariance of strain has a null-space. We show that there exist exactly 6 orthogonal perturbations to this covariance matrix under which the measured strain signal is invariant. In particular, we provide analytical parametrisations of these perturbations together with the set of possible maximum-likelihood estimates for a multivariate Gaussian fit to data. The parametric description of the null-space provides insights into the strain PDF modes that cannot be accurately estimated from the diffraction data. Understanding these modes prevents erroneous conclusions from being drawn based on the data. Beyond Gaussian strain tensor probability densities, we derive an iterative radial basis regression scheme in which the strain tensor probability density is estimated by a sparse finite basis expansion. This is made possible by showing that the operator mapping the strain tensor probability density onto the measured histograms of directional strain is linear, without approximation. The utility of the proposed algorithm is demonstrated by numerical simulations in the setting of single crystal monochromatic X-ray scattering. The proposed regression methods were found to robustly reject outliers and accurately predict the strain tensor probability distributions in the presence of Gaussian measurement noise.

Romanovski–Jacobi polynomials for the numerical solution of multi-dimensional multi-order time fractional telegraph equations

In this paper, the one- and two-dimensional multi-order time fractional telegraph equations are introduced. Two collocation methods based on the one- and two-dimensional Romanovski–Jacobi polynomials are proposed to solve them. In this way, some operational matrices related to the classical and fractional derivatives of these polynomials are achieved at first. Then, the solution of the problem under consideration is approximated by a finite expansion of these polynomials with unknown coefficients. Next, an algebraic system of equations is created in terms of the expansion coefficients by applying the derived operational matrices and the collocation technique. Eventually, by solving this system and determining the expressed coefficients, a solution for the multi-order time fractional telegraph equation under consideration is obtained. The correctness of the developed schemes are checked by solving some examples.

Arithmetics in Generalised Cantor Base Systems

For alternate Cantor real base numeration systems we generalise the result of Frougny and Solomyak on arithmetics on the set of numbers with finite expansions. We generalise the notion of finiteness property, denoted ( F ). We also provide several necessary conditions, and comment on a sufficient condition of this property. The sufficient condition allows us to find a set of generalised Cantor bases with Property ( F ). The used construction also provides a method for performing addition of finite expansions in Cantor real bases.

FeynGrav 2.0

We present a new version of FeynGrav. The present version supports Feynman rules for matter with non-vanishing mass and SU(N) Yang-Mills model. We revisit the gauge fixing procedure for gravity and derive interaction rules valid for an arbitrary gauge fixing parameter. We provide a few simple examples of calculations to illustrate package usage. Program summaryProgram Title: FeynGravCPC Library link to program files:https://doi.org/10.17632/rwnrbbdkkv.1Developer's repository link:github.com/BorisNLatosh/FeynGravLicensing provisions: GPLv3Programming language: Wolfram Mathematica 10 and higherExternal routines/libraries: FeynCalc [1,2]Nature of problem: FeynGrav is a software package that simplifies calculations involving the Feynman rules for gravity. These rules are complex and consist of numerous terms, but FeynGrav makes it possible to work with a specific set of rules within the FeynCalc environment. The package includes interaction rules for regular matter with different spin values, as well as for the SU(N) Yang-Mills model in curved spacetime and general relativity. FeynGrav provides scripts to generate the relevant interaction rules and libraries of pre-generated expressions, making it a powerful tool for researchers in the field of theoretical physics.Solution method: I propose a framework that provides a straightforward way to obtain expressions for the corresponding Feynman rules by efficiently utilizing perturbative expansions. In particular, within the perturbative approach to quantum gravity, certain quantities admit finite perturbative expansions, while others have infinite perturbative expansions. I derive formulas that allow the generation of terms with infinite expansions for any given order of perturbation theory. In all actions describing gravitational interactions, all derivatives can be placed on terms with finite perturbative expansions via integration by parts. In this way, the proposed framework gives a simple algorithm for deriving Feynman rules.Additional comments including restrictions and unusual features: Scripts responsible for generating interaction rules are placed in the “Rules” folder. They can be used separately and only require FeynCalc to run. The “Libs” folder contains the “FeynGravLibrariesGenerator.wl” script and the libraries it generates. “FeynGrav.wl” is the main package file that imports data from the libraries and uses it to work with the interaction rules. The file “FeynGrav_Examples.nb” contains examples of calculations.

Exponential integrator for stochastic strongly damped wave equation based on the Wong–Zakai approximation

We consider strong convergence of numerical approximations for a stochastic strongly damped wave equation driven by a class of additive space–time noises characterized by a parameter β∈(−1,2]. With the help of the Wong–Zakai (WZ) approximation to the noise, i.e., replacing the driven noise with its finite spectral expansion truncation, we obtain an approximate equation. Based on the consistency and high regularity of the approximate equation, we develop two exponential integrators for time-stepping discretization and use the spectral Galerkin method in space to develop full-discrete schemes. Performing error estimates in the strong sense, we show that the optimal strong order in time of the proposed WZ-approximation-based exponential Euler scheme is min{1,1+β/2−ϵ}, which is |β|/2 order higher than the order of min{1,1+β} in the existing works when β∈(−1,0). Moreover, we prove that the proposed WZ-approximation-based exponential trapezoidal scheme is of min{3/2,1+β/2−ϵ}-order strong convergence in time when β∈[−1/4,2], so it can break the first order barrier and obtain a higher accurate numerical solution. Numerical examples are performed to verify and develop the theoretical findings.

A new boson expansion theory utilizing a norm operator

Abstract We propose a new boson expansion method using a norm operator. The small parameter expansion, in which the boson approximation becomes the zeroth-order approximation, requires the double commutation relations between phonon operators that are not closed between the phonon excitation modes adopted as boson excitations. This results in an infinite expansion regardless of whether the type of the boson expansion is Hermitian or non-Hermitian. The small parameter expansion does not hold when the commutation relations are closed. The norm operator is expressed as a function of the number operator in the physical subspace, which enables us to obtain substantially a finite boson expansion regardless of the Hermitian or non-Hermitian type. We also point out the problems of the conventional boson expansion methods. The normal-ordered linked-cluster expansion theory has failed to refute Marshalek’s claim that KT-1 and KT-2 are of chimerical boson expansion. The Dyson boson expansion theory does not have exceptional superiority over other types. Previous studies using the boson expansion methods should be re-examined.

On the trace of the wave group and regularity of potentials

For the wave equation $\partial_t^2-\Delta+V$ on $\mathbb{R}^d$ with compactly supported, real valued potential $V$, we establish a sharp relation between Sobolev regularity of $V$ and the existence of finite order expansions as $t\rightarrow 0$ for the relative trace of the wave group.