Approximation Theory Research Articles

Transitions in stochastic non-equilibrium systems: Efficient reduction and analysis

A central challenge in physics is to describe non-equilibrium systems driven by randomness, such as a randomly growing interface, or fluids subject to random fluctuations that account e.g. for local stresses and heat fluxes in the fluid which are not related to the velocity and temperature gradients. For deterministic systems with infinitely many degrees of freedom, normal form and center manifold theory have shown a prodigious efficiency to often completely characterize how the onset of linear instability translates into the emergence of nonlinear patterns, associated with genuine physical regimes. However, in presence of random fluctuations, the underlying reduction principle to the center manifold is seriously challenged due to large excursions caused by the noise, and the approach needs to be revisited.In this study, we present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds, on one hand, and energy estimates measuring the defect of parameterization of the high-modes, on the other. To operate for fluid problems subject to stochastic stirring forces, these error estimates are derived under assumptions regarding dissipation effects brought by the high-modes in order to suitably counterbalance the loss of regularity due to the nonlinear terms. As a result, the approach enables us to predict, from reduced equations of the stochastic fluid problem, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation, as long as the noise's intensity and the eigenvalue's magnitude of the mildly unstable mode scale accordingly.In the case of SPDEs forced by a multiplicative noise in the orthogonal subspace of e.g. its mildly unstable mode, our parameterization formulas show that the noise gets transmitted to this mode via non-Markovian coefficients, and that the reduced equation is only stochastically driven by the latter. These coefficients depend explicitly on the noise path's history, and their memory content is self-consistently determined by the intensity of the random force and its interaction through the SPDE's nonlinear terms. Applications to a stochastic Rayleigh-Bénard problem are detailed, for which conditions for a stochastic pitchfork bifurcation (in large probability) to occur, are clarified.

Learning high-dimensional parametric maps via reduced basis adaptive residual networks

We propose a scalable framework for the learning of high-dimensional parametric maps via adaptively constructed residual network (ResNet) maps between reduced bases of the inputs and outputs. When just few training data are available, it is beneficial to have a compact parametrization in order to ameliorate the ill-posedness of the neural network training problem. By linearly restricting high-dimensional maps to informed reduced bases of the inputs, one can compress high-dimensional maps in a constructive way that can be used to detect appropriate basis ranks, equipped with rigorous error estimates. A scalable neural network learning framework is thus to learn the nonlinear compressed reduced basis mapping. Unlike the reduced basis construction, however, neural network constructions are not guaranteed to reduce errors by adding representation power, making it difficult to achieve good practical performance. Inspired by recent approximation theory that connects ResNets to sequential minimizing flows, we present an adaptive ResNet construction algorithm. This algorithm allows for depth-wise enrichment of the neural network approximation, in a manner that can achieve good practical performance by first training a shallow network and then adapting. We prove universal approximation of the associated neural network class for Lν2 functions on compact sets. Our overall framework allows for constructive means to detect appropriate breadth and depth, and related compact parametrizations of neural networks, significantly reducing the need for architectural hyperparameter tuning. Numerical experiments for parametric PDE problems and a 3D CFD wing design optimization parametric map demonstrate that the proposed methodology can achieve remarkably high accuracy for limited training data, and outperformed other neural network strategies we compared against.

A note on dyadic approximation in Cantor’s set

We consider the convergence theory for dyadic approximation in the middle-third Cantor set, K, for approximation functions of the form ψτ(n)=n−τ (τ⩾0). In particular, we show that for values of τ beyond a certain threshold we have that almost no point in K is dyadically ψτ-well approximable with respect to the natural probability measure on K. This refines a previous result in this direction obtained by the first, third, and fourth named authors.

Optimal regularity for variational solutions of free transmission problems

In this article we study functionals of the type considered in [36], i.e.J(v):=∫B1(A(x,u)|∇u|2+f(x,u)u+Q(x)λ(u))dx here A(x,u)=A+(x)χ{u>0}+A−(x)χ{u<0}, f(x,u)=f+(x)χ{u>0}+f−(x)χ{u<0} and λ(x,u)=λ+(x)χ{u>0}+λ−(x)χ{u≤0}. We prove the optimal C0,1− regularity of minimizers of the functional indicated above (with precise estimates) when the coefficients A± are continuous functions and μ≤A±≤1μ for some 0<μ<1, with f∈LN(B1) and Q bounded. We do this by presenting a new compactness argument and approximation theory similar to the one developed by L. Caffarelli in [9] to treat the regularity theory for solutions to fully nonlinear PDEs. Moreover, we introduce the Ta,b operator that allows one to transfer minimizers from the transmission problems to the Alt-Caffarelli-Friedman type functionals, in small scales, allowing this way the study of the regularity theory of minimizers of Bernoulli type free transmission problems.

Comparative study of galaxy clustering using halo approximation and mean-field theory in the light of modified theories of gravity

In this manuscript, we have explored the large-scale structure (LSS) of the Universe using the halo approximation of galaxies and the mean-field (mean-field approximation) theory of galaxy clusters. Employing the modified gravitational potential, which relies on Boltzmann’s statistics, we have analyzed the effect of galaxy clusters on the large-scale structure of the Universe. With both of these approximations, we have observed the significance of the modified potential. Also, the gravitational partition function obtained from the applied approximations has been used to determine the thermodynamic properties of these galaxy clusters. Besides this, the impact of these approximations on the derived quantities, such as Helmholtz free energy, entropy, internal energy, pressure, and chemical potential, was observed and is also evident from their respective plots along with their distribution functions. Importantly, the divergence of configurational integrals has been removed through the said approach.

The continuity in q of the Lupaş q-analogues of the Bernstein operators

The Lupaş q-analogue Rn,q of the Bernstein operator is the first known q-version of the Bernstein polynomials. It had been proposed by A. Lupaş in 1987, but gained the popularity only 20 years later, when q-analogues of classical operators pertinent to the approximation theory became an area of intensive research. In this work, the continuity of operators Rn,q with respect to parameter q in the strong operator topology and in the uniform operator topology has been investigated. The cases when n is fixed and n→∞ have been considered.

Rényi Entropies of Multidimensional Oscillator and Hydrogenic Systems with Applications to Highly Excited Rydberg States.

The various facets of the internal disorder of quantum systems can be described by means of the Rényi entropies of their single-particle probability density according to modern density functional theory and quantum information techniques. In this work, we first show the lower and upper bounds for the Rényi entropies of general and central-potential quantum systems, as well as the associated entropic uncertainty relations. Then, the Rényi entropies of multidimensional oscillator and hydrogenic-like systems are reviewed and explicitly determined for all bound stationary position and momentum states from first principles (i.e., in terms of the potential strength, the space dimensionality and the states's hyperquantum numbers). This is possible because the associated wavefunctions can be expressed by means of hypergeometric orthogonal polynomials. Emphasis is placed on the most extreme, non-trivial cases corresponding to the highly excited Rydberg states, where the Rényi entropies can be amazingly obtained in a simple, compact, and transparent form. Powerful asymptotic approaches of approximation theory have been used when the polynomial's degree or the weight-function parameter(s) of the Hermite, Laguerre, and Gegenbauer polynomials have large values. At present, these special states are being shown of increasing potential interest in quantum information and the associated quantum technologies, such as e.g., quantum key distribution, quantum computation, and quantum metrology.

One-Dimensional Dyadic Wavelets

One-Dimensional Dyadic Wavelets

Rheology of peristaltic flow in couple stress fluid in an inclined tube: Heat and mass transfer analysis

The recent investigations ensure that the effect of an endoscope on the peristaltic flow is very important for medical diagnosis and it has many clinical applications such as gastric juice motion in the small intestine when an endoscope is inserted through it. Therefore, we have studied the problem of peristaltic flow with heat and mass transfer through the gap between coaxial inclined tubes where the inner tube is rigid and the outer tube has sinusoidal wave traveling down its wall. The inner tube fulfilled the slip condition while the outer tube has a no-slip condition. The lubrication approximation theory is utilized to simplify the normalized equations. The perturbation procedure is employed to concede the results for the pressure gradient and velocity field, whereas exact outcomes are established for the energy and concentration fields. Frictional forces and pressure rise per wavelength are calculated numerically. The impacts of embedded parameters are portrayed through graphs. It is found that fluid velocity reduces by increasing the couple stress parameter however the inverse effect is noted for the slip parameter. Furthermore, better pumping is viewed in the vertical tube as compared to the horizontal tube. A validity of perturbation solutions for velocity distribution is made with the finite element method and an admirable comparison is also noticed with previously published results.

Magnetic properties of NdFe[formula omitted]Ti and YFe[formula omitted]Ti, from experiment and theory

NdFe11Ti and YFe11Ti serve as prototypes for rare-earth (RE) lean or REfree magnets with the ThMn12-type structure. Although NdFe11Ti has been studied for a long time the origin of its complex magnetism at low temperature is so far not well-understood. We present a comprehensive theoretical and experimental study of the magnetic properties of NdFe11Ti and RE-free YFe11Ti to elucidate the influence of the 4f electrons. The partially localized 4f electrons of Nd are the driving force behind the complex behavior of the magnetocrystalline anisotropy which changes from cone to uniaxial above 170 dK. The spontaneous magnetization and the five leading anisotropy constants were determined from high-quality single crystal samples over a wide temperature range using field dependencies of magnetization measured along the principle crystallographic directions. The experimental data are compared with density functional theory combined with a Hartree-Fock correction (+U) and an approximate dynamical mean-field theory.

Computational Study of Very High Spin Actinyl Peroxide Matryoshka Nanoclusters

AbstractGeneralised gradient approximation and hybrid density functional theory is used to study the geometries, electronic structures and spin state energetics of a series of high symmetry multi‐layer “Matryoshka” actinyl peroxide manganese nanoclusters “E@Mn12@An20n+” (An=U, Np, Pu; E=S, Se, Te, n=2; E=Br, n=3; E=Xe, n=4). These systems are all highly open shell, with up to 100 unpaired electrons in the case of the highest spin state plutonyl complexes. Very strong linear correlations are found in all cases between the number of unpaired electrons and the energies of the complexes, with the highest spin states being much the most stable. We confirm a key conclusion from our previous work in this area (Chemistry, A European Journal 24 (2018) 347) i. e. that the highest spin states of the chalcogen‐centred systems are the ground states only when the chalcogen is present; removal of the central group 16 element causes lower spin states to become the most stable. The compositions of the molecular orbitals associated with the changing spin states are studied in detail, and the relative energies of the highest and next highest spin states of the E‐free uranyl and plutonyl clusters rationalized on the basis of the characters of the orbitals whose occupation changes between the different spin states.

Stochastic resonance of a high-order-degradation bistable system and its application in fault diagnosis with variable speed condition

Rolling element bearings play a critical role in rotating machines with variable speed conditions, and their fault detection has attracted considerable interest from researchers in recent decades. In this paper, a high-order-degradation bistable system is proposed, and the corresponding stochastic resonance (SR) based on a new measurement index is investigated for fault detection with variable speed conditions. It successfully achieves fault detection without prior knowledge. Firstly, the output signal-to-noise ratio (SNR) of the high-order-degradation bistable stochastic resonance (HBSR) is theoretically analyzed using the approximate adiabatic theory. Secondly, a novel measurement index for determining the occurrence of stochastic resonance is proposed. Thirdly, the experimental data are used to validate the efficiency of the proposed nonlinear system and measurement index when the fault types and the actual fault order are unknown. Finally, the ground test data from a civil aircraft engine is also used to further validate the performance of the HBSR based on the novel measurement index.

Investigation of the middle-energy region of the above-threshold ionization spectrum in a few-cycle laser field

We use the improved strong-field approximation (ISFA) theory to investigate the evolution of the energy spectra with the carrier-envelope phase (CEP) of a few-cycle laser pulse. The theoretical calculations are consistent with the experimental measurements performed by Lindner et al (2005 Phys. Rev. Lett. 95 040401) which can be attributed to the scattering electrons whose ionization rates are modulated by the CEP of the laser pulse. We further investigate the dynamic behavior of the relevant scattering electrons by a Wigner distribution-like function based on the ISFA. Our analysis indicates that the zeroth-return forward-scattering electrons that move directly to the parent ion after tunneling and scatter forward with the parent core play a key role in the energy range of 2U p ∼ 4U p and the backward-scattering electrons are responsible for the high-energy region. Meanwhile, it is found that the contribution of the second-return backward-scattering electron to the middle energy spectrum is also important in the few-cycle laser field case.

Blade coating analysis of carreau fluid model with Magnetohydrodynamics and slip effects

Coating is a process that improves and enhances life, quality and efficiency of the web. Common industrial adhesive application to board and paper, metal foils polymer coatings, photographic film emulsions, recording films, textile fabrics, paints, coating on food items, medicine, catalogues are common examples. There are many fluids that are used as coatings. This paper analyses blade coating using both plane as well as exponential geometries along with slip and magnetohydrodynamics effects for a Carreau fluid model. To study the blade coating phenomenon, slip is introduced at the blade surface and a magnetic field is applied normal to the fluid flow. The LAT (Lubrication approximation theory) has been introduced to convert governing partial differential equations into a simpler form and a numerical solution is obtained by using shooting method. The significant physical quantities like volumetric flow rate, pressure, velocity, load, coated film thickness and pressure gradient are discussed in detail. The effects of power law index, slip parameter, Weissenberg number and Hartmann number on these physical quantities are presented. The observations are depicted in tabular data and in the graphical form as well. The Weissenberg number and Hartmann number play a very important role in controlling the load and coated film thickness.

Parameter identification of contact model for tracked vehicles

As the main weapons in the ground combat, tracked vehicles have very powerful attack capabilities, can quickly and safely pass through a variety of complex road conditions. Meanwhile, considering the complex structure of a tracked vehicle and the huge cost of a real vehicle test, it is urgent to establish an accurate dynamic model and use technical means to conduct a dynamic simulation. In the process of driving and fighting, due to the working environment of the track system is extremely harsh, and there is a serious collision between the track shoes and various components, so it is very necessary to establish the contact models between track shoes and road wheels, sprockets, idlers, supporting rollers, respectively. In this paper, the authors build a virtual prototype model of a tracked vehicle based on the multi-body dynamics method and validate it through dynamics simulation. It is well known that two components in contact with each other begin to separate when the contact force is zero. However, the deformation still exists after separation. At this time, there are two situations: one is the restoration of the original state of the component, and the other is that the component undergoes irreversible plastic deformation. Traditional collision models often use approximate theories to describe collisions, which often have large errors in actual collisions. Therefore, in this paper, a novel optimization model is established to ensure that the actual recovery coefficient of two colliding components is close to the ideal recovery coefficient, and the damping coefficient of each continuous contact model is obtained based on the proposed optimization process. The test results show that the identified damping coefficient is closer to the actual situation and has excellent engineering application value.

Perturbation based analytical and numerical solutions of non-Newtonian differential equation during reverse roll coating process under lubrication approximation theory

Reverse roll coating process are utilized for the purpose of applying a coating to a substrate or web by utilising different rollers in order to get the required coated surface. As a result of this, reverse roll coating process have found applications in a wide variety of industries, such as those dealing with food and medicine, electronic components, optical and LCD devices, and optical products. The main objective of this paper is to develop the mathematical formulation for the coating of thin film for an incompressible isothermal viscoelastic fluid between two reversely rotating rolls. Through the use of an appropriate dimensionless parameter, Non-dimensional nonlinear ordinary differential equations (ODEs) are derived from governing partial differential equations (PDEs). LAT (lubrication approximation theory) simplifies the dimensionless equations of fluid motion. The expressions for velocity of flow, pressure gradient and flow rate is obtained analytically by using regular perturbation method, while numeric solution of some mechanical parameters such as power input, coating thickness, roll separation force and separation points are calculated. The numerical validation of the analytical solution of the fundamental model of nonlinear constitutive flow laws is done in the Maple environment using the numeric technique which is based on finite difference method. The influence of numerous non-Newtonian parameters such as velocities ratio and Weissenberg number on velocity profiles, pressure gradient, power input, roll separation force, separation points and coating thickness of a non-Newtonian Johnson–Segalman (JS) fluid are explored via graphically and in tabular form. The outcomes demonstrate that on increasing the Weisenberg number and velocities ratio, the coating thickness on web is decreases. For the numerous values of velocities ratio, it is important to note that separation points shifted towards the nip region. In addition, the non-Newtonain parameter have signifcant impact on power input and roll separating force. The maximum coating thickness that is 1.0927 has been observed at the value of separation poin 0.9620. Hence, these factors may help in an efficient coating process and improve the substrate life.

Factorizable subgroups of the circle group

A subgroup H of the circle group T is said to be a-characterized if there exists a strictly increasing sequence of positive integers (un)n∈N0, with un|un+1 for all n∈N0, such that H consists precisely of those elements x∈T with unx→0 in T. These subgroups appeared in the study of trigonometric series in harmonic analysis, as well as in Diophantine approximation, dynamical systems and ergodic theory. The aim of the paper is to show that any a-characterized subgroup of T can be presented as the sum of two of its proper a-characterized subgroups.

Numerical analysis of heat transfer and magnetohydrodynamic flow of viscoelastic Jeffery fluid during forward roll coating process

AbstractThe steady‐state laminar nonisothermal, incompressible viscoelastic fluid flows with the Jeffrey model between two counterrotating (same direction) rolls are studied analytically. The Jeffrey model is reduced to the Newtonian model after some appropriate modifications. The new dimensionless governing equations are acquired through suitable nondimensional values. The lubrication approximation theory is employed for the simplification of emerging equations. The exact expressions for pressure gradient, velocity, temperature, and flow rate are accomplished whereas some quantities of interest such as pressure, coating thickness, power transmitted by rolls to the fluid, and roll separation force are computed numerically by using a numerical method. The influence of emerging parameters such as the ratio of relaxation time to the retardation time , magnetohydrodynamic (MHD) parameter and Brinkman number on velocity profile, temperature profile, pressure gradient, pressure distribution, coating thickness, power input, and force of roll separation are presented numerically in a tabular way and specific outcomes are demonstrated graphically. The MHD acts as a regulatory parameter for various quantities of important interest, such as velocity, coating thickness, and temperature. It is worth noting that the coating thickness decreases as modified capillaries number is increased.

Optimal Approximate Choice Designs for a Two-step Coffee Choice, Taste and Choice Again Experiment

AbstractThis work deals with consumers' preferences about coffee. Firstly, a choice experiment is performed on a sample of potential consumers. Following this, a sensory test involving the tasting of two varieties of coffee is carried out with the respondents, after which the same choice experiment is supplied to them again. An innovative approach for building heterogeneous choice designs is specifically developed for the case-study, based on approximate design theory and compound design criterion. Panel Mixed Logit models are used, thereby allowing for the inclusion of correlation among consumers' responses; choice-sets are supplied to a proportion of respondents according to optimal weights. The estimation results of the Panel Mixed Logit model are satisfactory, confirming the validity of the proposed approach.

A novel two-dimensional exponential potential bi-stable stochastic resonance system and its application in bearing fault diagnosis

It is difficult for classical stochastic resonance systems to extract weak signals under strong noise environment, therefore a novel two-dimensional exponential potential bi-stable stochastic resonance system (NTBSR) is proposed. First, the equivalent potential function, the mean first-pass time (MFPT) and the output signal-to-noise ratio (SNR) of NTBSR are derived under the adiabatic approximation theory. At the same time, the influence of different system parameters on them is explored. Then, NTBSR, the one-dimensional bi-stable stochastic resonance system (OBSR) and the two-dimensional classical bi-stable stochastic resonance system (TCBSR) are respectively simulated numerically, based on the fourth-order Runge–Kutta algorithm. It is found that the output SNR of NTBSR is the best. Finally, the NTBSR is applied to the fault signal diagnosis of different types of bearings, and the parameters are optimized through the adaptive genetic algorithm (AGA). The test results are compared with wavelet transform method, and TCBSR. The detection results on two sets of bearing fault data show that the NTBSR system has better effects on the enhancement and detection of bearing fault signals, and it is verified that the stochastic resonance method is superior to the traditional wavelet transform method in terms of signal detection and noise utilization. This provides good theoretical support and application value for practical engineering application.