Classical Integration Research Articles

Modeling and stability analysis of a sliding bead from a problem-based learning perspective

In this work, we adopt a problem-based learning approach to develop an integrative bachelor project that relies on competences acquired from basic courses in mathematics, mechanics and computation. In this sense, the proposed project tries to pave the way from “apparently disconnected” concepts gained through previous studies toward the field of computational mechanics. On this basis, we study the motion of a particle along an arbitrary curve in space subject only to the gravitational field, the so-called sliding bead. Although this is a classic problem in mechanics, it has a substantial richness from a theoretical and practical point of view where the student reinforcing abstract skills and exploring different aspects of its numerical solution are allowed. Along this path, we start with the modeling process. This is followed by a stability analysis of the governing differential equations. Later on, we present a moderate introduction to classical numerical time integration by considering several numerical schemes and the well-known Matlab add-on Simulink. Finally, we present a brief discussion about how the proposed project allows articulating concepts of mathematics, mechanics, and computation in engineering programs of studies at horizontal and vertical levels.

All Jordanian deformations of the $AdS_5 \times S^5$ superstring

We explicitly construct and classify all Jordanian solutions of the classical Yang-Baxter equation on \mathfrak{psu}(2,2|4)𝔭𝔰𝔲(2,2|4), corresponding to Jordanian Yang-Baxter deformations of the AdS_5\times S^5AdS5 × S5 superstring. Such deformations preserve the classical integrability of the underlying sigma-model and thus are a subclass of all possible integrable deformations. The deformations that we consider are divided into two families, unimodular and non-unimodular ones. The former ensure that the deformed backgrounds are still solutions of the type IIB supergravity equations. For the simplest unimodular solutions, we find that the corresponding backgrounds preserve a number N < 32 of supercharges that can be N=12,8,6,4,0.

Development and Experimental Implementation of Optimized PI-ANFIS Controller for Speed Control of a Brushless DC Motor in Fuel Cell Electric Vehicles

This paper compares the performance of different control techniques applied to a high-performance brushless DC (BLDC) motor. The first controller is a classical proportional integral (PI) controller. In contrast, the second one is based on adaptive neuro-fuzzy inference systems (proportional integral-adaptive neuro-fuzzy inference system (PI-ANFIS) and particle swarm optimization-proportional integral-adaptive neuro-fuzzy inference system (PSO-PI-ANFIS)). The control objective is to regulate the rotor speed to its desired reference value in the presence of load torque disturbance and parameter variations. The proposed controller uses a dSPACE platform (MicroLabBox controller board). The experimental prototype comprises a PEMFC system (the Nexa Ballard FC power generator: 1.2 kW, 52 A) and a brushless DC motor BLDC of 1 kW 1000 rpm. The PSO-PI-ANFIS controller presents better performance than the PI-ANFIS and classical PI controllers due to its ability to optimize the PI-ANFIS controller’s parameters using the particle swarm optimization (PSO) algorithm. This optimization results in improved tracking accuracy and reduced overshoot and settling time.

Crafting the dynamical structure of synchronization by harnessing bosonic multilevel cavity QED

Many-body cavity QED experiments are established platforms to tailor and control the collective responses of ensembles of atoms, interacting through one or more common photonic modes. The rich diversity of dynamical phases they can host, calls for a unified framework. Here we commence this program by showing that a cavity QED simulator assembled from $N$-levels bosonic atoms, can reproduce and extend the possible dynamical responses of collective observables occurring after a quench. Specifically, by initializing the atoms in classical or quantum states, or by leveraging intra-levels quantum correlations, we craft on demand the entire synchronization/desynchronization dynamical crossover of an exchange model for $SU(N)$ spins. We quantitatively predict the onset of different dynamical responses by combining the Liouville-Arnold theorem on classical integrability with an ansatz for reducing the collective evolution to an effective few-body dynamics. Among them, we discover a synchronized chaotic phase induced by quantum correlations and associated to a first order non-equilibrium transition in the Lyapunov exponent of collective atomic dynamics. Our outreach includes extensions to other spin-exchange quantum simulators and a universal conjecture for the dynamical reduction of non-integrable all-to-all interacting systems.

From Hamiltonian Dreams to Maastricht Reality: “Whatever It Takes” 2.0?

During the COVID-19 pandemic, the institutional architecture of the euro area has been tested again. While some authors argue that European policy makers have learnt from their economic policy mistakes during the sovereign debt crisis, or even talk about the “European Hamiltonian moment”, the opportunity to fundamentally strengthen the institutional foundations of the currency union has been missed again. While public attention has been focused mostly on the creation of the so-called “Next Generation EU” (NGEU) fund, it was the ECB that quietly performed the key role of crisis manager, despite criticism of a weak initial crisis response and botched communication. Based on the synthetic framework of the classical integration theories, the principal-agent model and new intergovernmentalism, the ECB’s pandemic crisis response could be interpreted as its second “whatever it takes” moment.

Compatibility of the method of brackets with classical integration rules

Abstract The method of brackets is a symbolic approach to the computation of integrals over R n {{\mathbb{R}}}^{n} based on a deep result by Ramanujan. Its usefulness to obtain new and difficult integrals has been demonstrated many times in the last few years. This article shows that this method is consistent with most classical rules for the computation of integrals, such as the fundamental theorem of calculus, the Laplace transform, the reduction formula for the integration of functions with spherical symmetry, the Cauchy-Schlömilch transformation, and explicit evaluations for multivariate integrals of product of Bessel functions as obtained by Exton and Srivastava. This work is part of a program dedicated to the derivation of solid theoretical grounds for the use of this attractive integration method.

Temporal integration is a robust feature of perceptual decisions.

Making informed decisions in noisy environments requires integrating sensory information over time. However, recent work has suggested that it may be difficult to determine whether an animal's decision-making strategy relies on evidence integration or not. In particular, strategies based on extrema-detection or random snapshots of the evidence stream may be difficult or even impossible to distinguish from classic evidence integration. Moreover, such non-integration strategies might be surprisingly common in experiments that aimed to study decisions based on integration. To determine whether temporal integration is central to perceptual decision making, we developed a new model-based approach for comparing temporal integration against alternative 'non-integration' strategies for tasks in which the sensory signal is composed of discrete stimulus samples. We applied these methods to behavioral data from monkeys, rats, and humans performing a variety of sensory decision-making tasks. In all species and tasks, we found converging evidence in favor of temporal integration. First, in all observers across studies, the integration model better accounted for standard behavioral statistics such as psychometric curves and psychophysical kernels. Second, we found that sensory samples with large evidence do not contribute disproportionately to subject choices, as predicted by an extrema-detection strategy. Finally, we provide a direct confirmation of temporal integration by showing that the sum of both early and late evidence contributed to observer decisions. Overall, our results provide experimental evidence suggesting that temporal integration is an ubiquitous feature in mammalian perceptual decision-making. Our study also highlights the benefits of using experimental paradigms where the temporal stream of sensory evidence is controlled explicitly by the experimenter, and known precisely by the analyst, to characterize the temporal properties of the decision process.

EESD special issue: AI and data‐driven methods in earthquake engineering – (Part 1)

EESD special issue: AI and data‐driven methods in earthquake engineering – (Part 1)

ϕ-FEM: an optimally convergent and easily implementable immersed boundary method for particulate flows and Stokes equations

We present an immersed boundary method to simulate the creeping motion of a rigid particle in a fluid described by the Stokes equations discretized thanks to a finite element strategy on unfitted meshes, called ϕ-FEM, that uses the description of the solid with a level-set function. One of the advantages of our method is the use of standard finite element spaces and classical integration tools, while maintaining the optimal convergence (theoretically in the H1 norm for the velocity and L2 for pressure; numerically also in the L2 norm for the velocity).

Higher-spin Yang–Mills, amplitudes and self-duality

The existence of interacting higher-spin theories is tightly constrained by many no-go theorems. In this paper, we construct a chiral, higher-spin generalization of Yang–Mills theory in flat space which avoids these no-go theorems and has non-trivial tree-level scattering amplitudes with some higher-spin external legs. The fields and action are complex, so the theory is non-unitary and parity-violating, yet we find surprisingly compact formulae for all-multiplicity tree-level scattering amplitudes in the maximal helicity violating (MHV) sector, where the two negative helicity particles have identical but arbitrary spin. This is possible because the theory admits a perturbative expansion around its self-dual sector. Using twistor theory, we prove the classical integrability of this self-dual sector and show that it can be described on spacetime by an infinite tower of interacting massless scalar fields. We also give a twistor construction of the full theory and use it to derive the formula for the MHV amplitude.

A Stratonovich integral for anticipating processes

A stochastic integral for anticipating integrands was introduced by Ayed and Kuo in 2008. Riemann–Stieltjes sums were considered, where the adapted part of the integrand was evaluated at the left endpoints of the subintervals, while the instantly independent part was evaluated at the right endpoints. Since then, many results have been proved, such as formulas for differentials. In this paper, the Stratonovich counterpart of the Ayed–Kuo integral is investigated. In its simplest version, it is proved that, analogously to the classical stochastic integration theory for adapted processes, the fundamental theorem of calculus holds. Consequences, extensions, and limitations are discussed in detail.

On a generalization of time-accurate and highly-stable explicit operators for stiff problems

In this work we propose a generalization of the family of Time-Accurate and highly-Stable Explicit (TASE) operators recently introduced by Calvo, Montijano and Rández (2021). In this family the TASE operator of order p depends on p free real parameters. Here we consider operators that can also be defined by complex parameters occurring in conjugate pairs. Despite this choice the calculations continue to be done only in real arithmetic, thus not burdening the computational cost of the previous version of the family. Conversely, this generalization leads to improve both the accuracy and stability properties of explicit Runge-Kutta schemes supplemented with TASE operators. Numerical experiments showing the competitiveness of the methods proposed in this paper with respect to classical integrators for stiff problems are also presented.

A predictive microstructure inspired approach for anisotropic damage, residual stretches and hysteresis in biodegradable sutures

We propose a predictive model for the mechanical behavior of biodegradable polymers of interest for biomedical applications. Starting from a detailed description of the network behavior of the copolymer material, taking care of bonds breaking and recrosslinking effects, folded–unfolded transitions and network topological constraints, we deduce a macroscopic law for the complex mechanical behavior of many biomedical materials with a particular focus on absorbable suture threads and a perspective to new material design. A crucial novelty of the model is the careful description of the observed microscopic anisotropic damage induced by the deformation, here described based on the classical microsphere integration approach. The resulting energy, characterized by few material parameters, with a physically clear interpretation, is successfully adopted to predict our cyclic experiments on poliglecaprone sutures with anisotropic damage, permanent stretch and internal hysteresis. By also studying other suture materials behavior we show that the predictivity properties of the model can be extended also to materials with a different mechanical response and thus also possibly applied for the design of new, high-performance biomedical materials.

Numerical integration rules with improved accuracy close to discontinuities

This work is devoted to the construction and analysis of a new nonlinear technique that allows obtaining accurate numerical integrations of any order using data that contains discontinuities, and when the integrand is only known at grid points. The novelty of the technique consists in the inclusion of correction terms with a closed expression that depend on the size of the jumps of the function and its derivatives at the discontinuities, that are supposed to be known. The addition of these terms allows recovering the accuracy of classical numerical integration formulas close to the discontinuities, as these correction terms account for the error that the classical integration formulas commit up to their accuracy at smooth zones. Thus, the correction terms can be added during the integration or as post-processing, which is useful if the main calculation of the integral has been already done using classical formulas. We include several numerical experiments that confirm the theoretical conclusions reached in this article.

Combination of physics‐based and data‐driven modeling for nonlinear structural seismic response prediction through deep residual learning

AbstractDespite great progress in seeking accurate numerical approximator to nonlinear structural seismic response prediction using deep learning approaches, tedious training process and large volume of structural response data under earthquakes for training and validation are often prohibitively accessible. In our methodology, the main innovation can be seen in the incorporation of deep neural networks (DNNs) into a classical numerical integration method by using a hybridized integration time‐stepper. In this way, the linear physics information of the structure and the obscure nonlinear dynamics are smoothly combined. We propose to use residual network (ResNet) to learn time‐stepping schemes specifically for the nonlinear state variables of the system. Our Physics‐DNN hybridized integration (PDHI) time‐stepping scheme provides important advantages over current pure data‐driven approaches, including (i) a flexible framework incorporating known time‐invariant physics information, (ii) requirement of structural seismic response data being circumvented by simple short bursts of trajectories collected from underlying nonlinear components, and (iii) efficiency in training and validation process. Besides, our results indicate that a simple feedforward or convolutional architecture outperforms recurrent networks to fulfill the requirement of prediction accuracy as well as long‐range memory in structural dynamic analysis. Several numerical and experimental examples are presented to demonstrate the performance of the method.

On fuzzy fractional integral operators having exponential kernels and related certain inequalities for exponential trigonometric convex fuzzy-number valued mappings

The most important operator in fractional theory that enables the classical theory of integrals to be generalized is the Riemann-Liouville fractional integrals. In this paper, we have introduced new fractional operators in the fuzzy environment known as fuzzy Riemann-Liouville fractional integrals having exponential kernels. All classical fractional integrals that depend upon exponential kernels are exceptional cases of this new one. Moreover, we have defined a new class of convex mappings which is known as exponential trigonometric convex fuzzy-number valued mappings. With the help of this class and the newly proposed fuzzy fractional integral operator, the well-known Hermite-Hadamard type and related inequalities are taken into account in this work. Moreover, some new versions of midpoint Hermite-Hadamard-type inequalities are also established. By applying these definitions, we have amassed some novel and classical exceptional cases that serve as implementations of the key findings. For the purpose of proving the viability of the fuzzy order relations put forth in this research, some nontrivial examples of fuzzy numbered valued convexity are also provided.

Интеграционные процессы в мировой экономике: нооподход и развитие евразийского пространства

Within the current conditions and realities – namely, aggravation of inter-country conflicts, continuous chains of financial and economic crises, unpredictable epidemics and pandemic, increasing threats and risks to the existence of humanity – issues of integration and solidarity are becoming particularly relevant. At the same time, the existing theories of integration and approaches to its interpretation, identification of stages and evaluation of results are very specific due to the concentration mainly on economic and political-economic processes, as well as often because of their “sharpening” for the European Union. The purpose of this paper is to develop an alternative (to the “classical” one) approach to the determination and analysis of integration based on the theory of noonomy worked out by S. D. Bodrunov. The article provides evidence of an overestimating the results of European integration, based mainly on the creation in theory and practice of its idealized image of this continuously progressive and exemplary model that stands out from the rest of the world. The basic provisions and conclusions of noonomy, the “parallels” of this theory and the results of its “conjugation” with the theory of changing technological modes and the concept of changing world economic modes, worked out by S. Y. Glazyev, are presented. We enter the concept of “noointegration”, highlight its main stages and analyze the differences and correlation of its processes with classical integration. It is explained why noo-integration, without denying economic integration at the pre-noonomic stage of public life, turns out to be of a higher priority and more useful for the harmonious joint development of the Eurasian space and its participating countries than the classical integration model.

A generalization of first direct method of pursuit for differential inclu-sions

To solve the problem of pursuit in linear differential games, L.S. Pontryagin suggested two direct methods. Direct methods are of great importance in the development of the theory of differential games and in control theory under the conditions of uncertainty. It turned out to be useful also in solving the problem of control synthesis. Pontryagin direct methods have proved themselves as an effective means for solving problems of pursuit- evasion and control. These use integrals, having a number of significant differences from the classical integral. One of the differences consists in the use of multivalued mapping. Pontryaginʼs second direct method, based on concept of the alternating integral, which has no analogs in integration of real function. In definition of alternating integral participate of integration of setvalued mappings and geometric difference (Minkovski difference) of sets. These operations make difficulties for computation of alternating integral. From this point of view, the integral used by the first direct method has a simpler construction. Therefore, the question naturally arises of generalization the first direct method of pursuit. In this paper it will be studied a generalization of the first direct method for pursuit games, being described by differential inclusions where F is a continuous multivalued mapping. This method will be called the modified first direct method of pursuit for differential inclusions. In particular, the class of stroboscopic strategies, the trajectory of the system are determined. For these classes games, it is proved that if the starting point belongs to the modified first integral (the integral from the multivalued mapping, which is present in the definition of the modified fist direct metod), then this is necessary and sufficient condition for completing the game in a fixed time instant in the class of stroboscobic strategies. The problem of computation this integral is important. In the present article it has also been proved that the union operations in the definition of the modified first integral can be narrowed down to the class of compact-valued mappings

Integrable Systems: In the Footprints of the Greats

In his 1842 lectures on dynamics C.G. Jacobi summarized difficulties with differential equations by saying that the main problem in the integration of differential equations appears in the choice of right variables. Since there is no general rule for finding the right choice, it is better to introduce special variables first, and then investigate the problems that naturally lend themselves to these variables. This paper follows Jacobi’s prophetic observations by introducing certain “meta” variational problems on semi-simple reductive groups G having a compact subgroup K. We then use the Maximum Principle of optimal control to generate the Hamiltonians whose solutions project onto the extremal curves of these problems. We show that there is a particular sub-class of these Hamiltonians that admit a spectral representation on the Lie algebra of G. As a consequence, the spectral invariants associated with the spectral curve produce a large number of integrals of motion, all in involution with each other, that often meet the Liouville complete integrability criteria. We then show that the classical integrals of motion associated, with the Kowalewski top, the two-body problem of Kepler, and Jacobi’s geodesic problem on the ellipsoid can be all derived from the aforementioned Hamiltonian systems. We also introduce a rolling geodesic problem that admits a spectral representation on symmetric Riemannian spaces and we then show the relevance of the corresponding integrals on the nature of the curves whose elastic energy is minimal.

Introduction: Assimilation, integration or transnationalism? An overview of theories of migrant incorporation

Introduction: Assimilation, integration or transnationalism? An overview of theories of migrant incorporation