Expansion Of Representation Research Articles

On the electron self-energy to three loops in QED

We compute the electron self-energy in Quantum Electrodynamics to three loops in terms of iterated integrals over kernels of elliptic type. We make use of the differential equations method, augmented by an ϵ-factorized basis, which allows us to gain full control over the differential forms appearing in the iterated integrals to all orders in the dimensional regulator. We obtain compact analytic expressions, for which we provide generalized series expansion representations that allow us to evaluate the result numerically for all values of the electron momentum squared. As a by-product, we also obtain ϵ-resummed results for the self-energy in the on-shell limit p2 = m2, which we use to recompute the known three-loop renormalization constants in the on-shell scheme.

Uncertain vibration response of vehicles passing through barricades based on approximate models

In vibration analysis, a vehicle system encounters dimensionality issues due to its high-dimensional uncertain parameters. An approximate model offers a viable solution for analyzing such uncertain responses. This study introduces an efficient approximate model, called PCE-HDMR, which is founded on the Legendre Polynomial Chaos Expansion (PCE) and High-Dimensional Model Representation (HDMR). Specifically, the Legendre PCE in interval space is employed to delineate the lower and upper bounds of uncertain responses. At the same time, the HDMR is harnessed to develop a high-dimensional uncertainty model that approximates the dynamic response. To demonstrate the application of PCE-HDMR, a model for a vehicle with interval parameters was constructed using a 9-DOF dynamics model for testing. In this framework, all stiffness and damping parameters are treated as interval uncertainty parameters. The numerical results validate the effectiveness of the proposed method for high-dimensional uncertain parameters, demonstrating that PCE-HDMR outperforms Monte Carlo simulation (MCS) in terms of efficiency. This study advances an effective interval uncertainty analysis approach for assessing vehicle performance, particularly when dealing with high-dimensional interval uncertainty parameters. The proposed method serves as a viable alternative for interval analysis and subsequent optimization design for complex vehicle systems characterized by high-dimensional uncertain parameters.

Hierarchical sensitivity study on the aeroelastic stability of the IEA 15 MW reference wind turbine

Increased rotor size and blade flexibility are leading to new stability characteristics for current and future wind turbines. Numerical aeroelastic models are an essential tool to understand these new mechanisms and to design next-generation wind turbines suitably. A comprehensive understanding of how model input parameters influence the stability analysis will enhance the confidence in these simulations. This article presents a study on the sensitivity of uncertain parameters on aeroelastic stability predictions of the IEA 15 MW turbine model in HAWCStab2. It uses a hierarchical approach to handle the large number of investigated model parameters. Relevant uncertainties are identified through one-at-a-time and elementary effects analyses first. The remaining parameters are fed into a variance-based uncertainty quantification (UQ), that employs polynomial chaos expansion representations as surrogates for a full nonlinear description of the sensitivities. A robust post-processing of the stability analysis results is the main challenge of the presented methodology, but it is shown how the UQ process can still be used to explore the parameter space effectively. The presented study shows that the structural blade properties have the highest sensitivity and that a set of relatively small parameter variations can lead to unstable behavior of the reference model.

Refining Heisenberg’s principle: A greedy approximation of step functions with triangular waveform dictionaries

In this paper, we consider a step function characterized by a real-valued sequence and its linear expansion representation constructed via the matching pursuit (MP) algorithm. We utilize a waveform dictionary based on the triangular function as part of this algorithm and representation. The waveform dictionary is comprised of waveforms localized in the time–frequency domain. In view of this, we prove that the triangular waveforms are more efficient than the rectangular waveforms used in a prior study by achieving a product of variances in the time–frequency domain closer to the lower bound of the Heisenberg Uncertainty Principle. We provide a MP algorithm solvable in polynomial time, contrasting the common exponential time when using Gaussian windows. We apply this algorithm on simulated data and real GDP data from 1947–2024 to demonstrate its application and efficiency.

Rotational state-changes in C5N− by collisions with He and H2

ABSTRACT The anion C5N− is one of the largest linear (C,N)-bearing chains detected in the interstellar medium. Here we present and discuss the general features of new ab initio potential energy surfaces describing the interaction of this linear anion with He and H2. We employ a Legendre Polynomials expansion representation for the former and an artificial neural network fit for the latter. We then carry out quantum scattering calculations to yield rotationally inelastic cross-sections for collisions with He and H2, using relative translational energy values in the range of 0.1–300 cm−1. We then obtained the corresponding inelastic rate coefficients as a function of temperature covering the range from 1 to 100 K. The results for these two systems are compared with each other, as well as with the earlier results on the C3N− colliding with the same partners. We found that the final inelastic rate coefficients for this anion are all fairly large, those from collisions with H2 being the largest. The consequences of such findings on their non-equilibrium rotational populations in interstellar environments are discussed in our conclusions.

Simulating thermal density operators with cluster expansions and tensor networks

We provide an efficient approximation for the exponential of a local operator in quantum spin systems using tensor-network representations of a cluster expansion. We benchmark this cluster tensor network operator (cluster TNO) for one-dimensional systems, and show that the approximation works well for large real- or imaginary-time steps. We use this formalism for representing the thermal density operator of a two-dimensional quantum spin system at a certain temperature as a single cluster TNO, which we can then contract by standard contraction methods for two-dimensional tensor networks. We apply this approach to the thermal phase transition of the transverse-field Ising model on the square lattice, and we find through a scaling analysis that the cluster-TNO approximation gives rise to a continuous phase transition in the correct universality class; by increasing the order of the cluster expansion we find good values of the critical point up to surprisingly low temperatures.

On the Design Flow of the Fractional-Order Analog Filters Between FPAA Implementation and Circuit Realization

This work explicitly states the design flows of the fractional-order analog filters that researchers throughout the literature have used. Two main flows are studied: the FPAA implementation and the circuit realization. Partial-fraction expansion representation is used to prepare the approximated fractional-order response for implementation on FPAA. The generalization of the second-order active RC analog filters based on opamp from the integer-order domain to the fractional-order domain is presented. The generalization is studied from both mathematical and circuit realization points of view. It is found that the great benefit of the fractional-order domain is that it adds more degrees of freedom to the filter design process. Simulation and experimental results match the expected theoretical analysis.

THE GLOBAL PETROLEUMSCAPE AND ITS IMPACT ON DESIGN PRACTICE

Over the last century the petroleum industry’s rapid growth has been accompanied by a steady flow of aggressively promoted petroleum-based products. The petroleumscape’s spatial expansion and visual representation achieved widespread citizen buy-in. Following World War II the use of plastic materials in the building industry significantly increased through efforts from architects and industry leaders. The House of the Future, built by MIT architects, the Monsanto Chemical Company, and Disneyland exemplified a modern lifestyle: clean, functional, and fun. The architectural and technocratic dream of a mass-produced, fully plastic house that seemed possible in the post-war years did not survive the subsequent commercialisation of the plastics industry in the 1960s and 70s.

On a Wick-type stochastic parabolic equations with random potentials

Stochastic parabolic equations with random potentials, where the Wick product is used to give sense to the product of generalized stochastic processes, are considered. The existence and uniqueness of solutions are proved via the chaos expansion method from white noise analysis and explicit estimates on the coefficients in the chaos expansion representation of the solutions are provided.

Deciphering the maximal transcendentality principle via bootstrap

We prove the principle of maximal transcendentality for a class of form factors, including the general two-loop minimal form factors, the two-loop three-point form factor of tr(F2), and the two-loop four-point form factor of tr(F3). Our proof is based on a recently developed bootstrap method using the representation of master integral expansions, together with some unitarity cuts that are universal in general gauge theories. The maximally transcendental parts of the two-loop four-gluon form factor of tr(F3) are obtained for the first time in both planar mathcal{N} = 4 SYM and pure YM theories. This form factor can be understood as the Higgs-plus-four-gluon amplitudes involving a dimension-seven operator in the Higgs effective theory. In this case, we find that the maximally transcendental part of the mathcal{N} = 4 SYM result is different from that of pure YM, and the discrepancy is due to the gluino-loop contributions in mathcal{N} = 4 SYM. In contrast, the scalar-loop contributions have no maximally transcendental parts. Thus, the maximal transcendentality principle still holds for the form factor results in mathcal{N} = 4 SYM and QCD, after a proper identification of the fundamental quarks and adjoint gluinos as nf→ 4Nc. This seems to be the first example of the maximally transcendental principle that involves fermion-loop contributions. As another intriguing observation, we find that the four-point form factor of the half-BPS tr(ϕ3) operator is precisely a building block in the form factor of tr(F3).

Fast Stabiliser Simulation with Quadratic Form Expansions

This paper builds on the idea of simulating stabiliser circuits through transformations of {quadratic form expansions}. This is a representation of a quantum state which specifies a formula for the expansion in the standard basis, describing real and imaginary relative phases using a degree-2 polynomial over the integers. We show how, with deft management of the quadratic form expansion representation, we may simulate individual stabiliser operations in O(n2) time matching the overall complexity of other simulation techniques \cite{Aaronson2004,Anders2006,Bravyi2016}. Our techniques provide economies of scale in the time to simulate simultaneous measurements of all (or nearly all) qubits in the standard basis. Our techniques also allow single-qubit measurements with deterministic outcomes to be simulated in constant time. We also describe throughout how these bounds may be tightened when the expansion of the state in the standard basis has relatively few terms (has low `rank'), or can be specified by sparse matrices. Specifically, this allows us to simulate a `local' stabiliser syndrome measurement in time O(n), for a stabiliser code subject to Pauli noise --- matching what is possible using techniques developed by Gidney \cite{gidney2021stim} without the need to store which operations have thus far been simulated.

Fourier Series Expansion and Integral Representation of Apostol-Type Frobenius–Euler Polynomials of Complex Parameters and Order α

In this paper, the Fourier series expansions of Apostol-type Frobenius–Euler polynomials of complex parameters and order α are derived, and consequently integral representations of these polynomials are established. This paper provides some techniques in computing the symmetries of the defining equation of Apostol-type Frobenius–Euler polynomials resulting in their expansions and integral representations.

Before and After the Fall of Communism: The Molotov–Ribbentrop Pact and Soviet Foreign Policy in 1939–1940 in Soviet and Post-Soviet Russian Textbooks

ABSTRACT The Molotov–Ribbentrop Pact is a controversial event in the history of the Second World War. Some scholars regard the Pact as essential to the security of the Soviet Union; others consider it a reason to equate the Soviet Union with Nazi Germany as the main initiator of the Second World War. Comparison of the historical representations of the Soviet territorial expansion in 1939–1940 in Soviet and post-Soviet Russian textbooks can uncover the conceptual transformations in historical consciousness since the fall of Communism, and reveal: does the current evaluation of the Pact in Russia and Eastern European states vary?

Elucidating appealing features of differentiable auto-correlation functions: A study on the modified exponential kernel

Research on stochastic processes in recent decades has pointed out that, in the context of modeling spatial or temporal uncertainties, auto-correlation functions that are differentiable at the origin have advantages over functions that are not differentiable. For instance, the non-differentiability of e.g., single exponential auto-correlation functions yields non-smooth sample paths. Such sample paths might not be physically possible or may yield numerical difficulties when used as random parameters in partial differential equations (such as encountered in e.g., mechanical equilibrium problems). Further, it is known that due to the non-differentiability of certain auto-correlation functions, more terms are required in the series expansion representations of the associated stochastic processes. This makes these representations less efficient from a computational standpoint.This paper elucidates some additional appealing features of auto-correlation functions which are differentiable at the origin. Further, it focuses on enhancing the arguments in favor of these functions already available in literature. Specifically, attention is placed on single exponential, modified exponential and squared exponential auto-correlation functions, which can be shown to be all part of the Whittle–Matérn family of functions. To start, it is shown that the power spectrum of differentiable kernels converges faster to zero with increasing frequency as compared to non-differentiable ones. This property allows capturing the same percentage of the total energy of the spectrum with a smaller cut-off frequency, and hence, less stochastic terms in the harmonic representation of stochastic processes. Further, this point is examined with regards to the Karhunen–Loève series expansion and first and second order Markov processes, generated by auto-regressive representations. The need for finite differentiability is stressed and illustrated.

A surrogate model for estimating uncertainty in marine riser fatigue damage resulting from vortex-induced vibration

Of interest in this work is the quantification of uncertainty in fatigue damage of a top-tensioned riser (TTR) resulting from vortex-induced vibration (VIV). Using a distributed wake oscillator model in time-domain simulations of a riser in uniform current, followed by rainflow cycle-counting, fatigue damage is computed. A polynomial chaos expansion (PCE) representation of damage is derived that treats three wake oscillator model variables as direct stochastic input: (i) motion amplitude; (ii) the ratio of vortex-shedding frequency to riser natural frequency; and (iii) current velocity. Damage at different locations along the selected riser is assessed. Damage predictions, obtained using the efficient PCE framework, are compared against the more computationally expensive alternative that involves Monte Carlo Simulation (MCS). This study suggests that PCE offers a tractable, versatile and accurate option for VIV-related fatigue damage prediction.

Evaluating the benefits of kinetic Monte Carlo and microkinetic modeling for catalyst design studies in the presence of lateral interactions

Popular computational catalyst design strategies rely on the identification of reactivity descriptors, which can be used along with Brønsted−Evans−Polanyi (BEP) and scaling relations as input to a microkinetic model (MKM) to make predictions for activity or selectivity trends. The main benefit of this approach is related to the inherent dimensionality reduction of the large material space to just a few catalyst descriptors. Conversely, it is well documented that a small set of descriptors is insufficient to capture the intricacies and complexities of a real catalytic system. The inclusion of coverage effects through lateral adsorbate-adsorbate interactions can narrow the gap between simplified descriptor predictions and real systems, but mean-field MKMs cannot properly account for local coverage effects. This shortcoming of the mean-field approximation can be rectified by switching to a lattice-based kinetic Monte Carlo (kMC) method using cluster expansion representation of adsorbate−adsorbate lateral interactions.Using the prototypical CO oxidation reaction as an example, we critically evaluate the benefits of kMC over MKM in terms of trend predictions and computational cost when using only a small set of input parameters. After confirming that in the absence of lateral interactions the kMC and MKM approaches yield identical trends and mechanistic information, we observed substantial differences between the two kinetic models when lateral interactions were introduced. The mean-field implementation applies coverage corrections directly to the descriptors, causing an artificial overprediction of the activity of strongly binding metals. In contrast, the cluster expansion in kMC implementation can differentiate among the highly active metals but it is very sensitive to the set of included interaction parameters. Considering that computational screening relies on a minimal set of descriptors, for which MKM makes reasonable trend predictions at a ca. three orders of magnitude lower computational cost than kMC, the MKM approach does provide a better entry point for computational catalyst design.

Time Domain Sampling of the Radial Functions in Spherical Harmonics Expansions

Spherical harmonics representations are widely adopted in applications such as analysis, manipulation, and synthesis of wave fields that are either captured or simulated. Recent studies have shown that, for broadband fields, a time domain representation of spherical harmonics expansions can benefit from computational efficiency and favorable transient properties. For practical usage, an accurate discrete-time modeling of spherical harmonics expansions is indispensable. The main challenge is to model the so called radial functions which describe the radial and frequency dependencies. In homogeneous cases, they are commonly realized as finite impulse response filters where the coefficients are obtained by sampling the continuous-time representations. This article investigates the temporal and spectral properties of the resulting discrete-time radial functions. The spectral distortions caused by aliasing are evaluated both analytically and numerically, revealing the influence of the distance from the expansion center, sampling frequency, and fractional sample delay. It is also demonstrated how the aliasing can be reduced by employing a recently introduced band limitation method.

Effect of Family firms’ religious values on their Business Ethics and other outcomes like Bribery, Expansion, Discrimination, Philanthropy, and Product representation: The mediatory Role of Rule-based and Practice-based approaches.

Effect of Family firms’ religious values on their Business Ethics and other outcomes like Bribery, Expansion, Discrimination, Philanthropy, and Product representation: The mediatory Role of Rule-based and Practice-based approaches.

On the recurrent computation of fractional operator with Mittag-Leffler kernel

The paper considers the problem of computing the values of the Atangana-Baleanu derivative in Caputo sense which arises while solving fractional partial differential equations. In such case the values of the derivative are needed to be calculated in recurrent manner with increasing value of time or space variable. We consider series expansion and integral representations of the Mittag-Leffler function that is an integral kernel in the Atangana-Baleanu derivative. Expanding into series the argument of the Mittag-Leffler function and applying a variable separation technique we obtain an efficient recurrent computational algorithm when the value of argument increases. The proposed algorithm has close-to-constant computational complexity comparing to linear complexity of the algorithms that perform direct numerical integration while computing the values of the Atangana-Baleanu derivative. For the proposed algorithm we present accuracy and performance estimates checking them computing integrals of the Mittag-Leffler function and solving time-fractional convection-diffusion equation using some finite-difference scheme.

Thermoelastic modeling and comparative analysis of biomass sensors under rippling deformation and magnetic field

This study is intended for investigating the linear and nonlinear responses of carbon nanotube-based mass sensors in thermal and magnetic environment. For accurate modeling, the carbon nanotube rippling deformation effect is considered and various representations for the thermal expansion are examined and compared to each other. By utilizing Euler-Bernoulli beam theory assumptions and Eringen's nonlocal elasticity theory, the nonlinear reduced-order model is developed on the basis of the extended Hamilton's principle. The results show that the natural and resonant frequencies and frequency shifts of the system are strongly dependent on the magnetic field and thermal expansion representation. The method of multiple scale is used to determine the modulation equation including the von Karman and rippling nonlinearities. The results show a very good agreement between the perturbation solution and the numerical integration results for specific conditions of the forcing, temperature difference, and quality factor. A comparative study between the linear and nonlinear mass sensing approaches is performed to show their limits of applicability. It is demonstrated that the linear approach may result in erroneous detection of the deposited mass. The obtained results indicate that the longitudinal magnetic field enhances the dynamic stability of the carbon nanotube mechanical resonator in the pre-buckling oscillation regime, while the dynamic stability of the nanoscale resonator is decreased in the presence of the magnetic field for the post-buckling configuration. Also, the ripple-based nonlinearity is accompanied by an increase in the mass responsivity of the resonator. On the contrary, mass sensitivity of the carbon nanotube resonator is diminished by considering the von Kármán geometric nonlinearity. This study shows the importance of considering the nonlinear effects on the system's sensitivity from frequency and amplitude sensing techniques.