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This paper studies the approximation capacity of ReLU neural networks with norm constraint on the weights. We prove upper and lower bounds on the approximation error of these networks for smooth function classes. The lower bound is derived through the Rademacher complexity of neural networks, which may be of independent interest. We apply these approximation bounds to analyze the convergences of regression using norm constrained neural networks and distribution estimation by GANs. In particular, we obtain convergence rates for over-parameterized neural networks. It is also shown that GANs can achieve optimal rate of learning probability distributions, when the discriminator is a properly chosen norm constrained neural network. [ABSTRACT FROM AUTHOR]

The signature of a path, introduced by K.T. Chen [10] in 1954, has been extensively studied in recent years. The fundamental 2010 paper [20] of Hambly and Lyons showed that the signature is an injective function on the space of continuous, finite-variation paths up to a general notion of reparameterisation called tree-like equivalence. This result has been extended to geometric rough paths by Boedihardjo et al. [5]. More recently, the approximation theory of the signature has been widely used in the literature in applications. The archetypal instance of these results, see e.g. [24] , guarantees uniform approximation, on compact sets, of a continuous function by a linear functional on the (extended) tensor algebra acting on the signature. In this paper we study in detail, and for the first time, the properties of three natural candidate topologies on the set of unparameterised paths, i.e. the tree-like equivalence classes. These are obtained by privileging different properties of the signature and are: (1) the product topology, obtained by equipping the range of the signature with the (subspace topology of the) product topology in the extended tensor algebra and then requiring S to be an embedding, (2) the quotient topology derived from the 1-variation topology on the underlying path space, and (3) the metric topology associated to d ([ γ ] , [ σ ]) : = | | γ ⁎ − σ ⁎ | | 1 using the (constant-speed) tree-reduced representatives γ ⁎ and σ ⁎ of the respective equivalence classes. We evaluate these spaces from the point of view of their suitability when it comes to studying (probability) measures on them. We prove that the respective collections of open sets are ordered by strict inclusion, (1) being the weakest and (3) the strongest. Our other conclusions can be summarised as follows. All three topological spaces are separable and Hausdorff, (1) being both metrisable and σ -compact, but not a Baire space and hence being neither Polish nor locally compact. The completion of (1), in any metric inducing the product topology, is the subspace G ⁎ of group-like elements. The quotient topology (2) is not metrisable and the metric d is not complete. We also discuss some open problems related to these spaces. We consider finally the implications of the selection of the topology for uniform approximation results involving the signature. A stereotypical model for a continuous function on (unparameterised) path space is the solution of a controlled differential equation. We thus prove, for a broad class of these equations, well-definedness and measurability of the (fixed-time) solution map with respect to the Borel sigma-algebra of each topology. Under stronger regularity assumptions, we further show continuity of this same map on explicit compact subsets of the product topology (1). We relate these results to the expected signature model of [24]. [ABSTRACT FROM AUTHOR]

The recovery of multivariate functions and estimating their integrals from finitely many samples is one of the central tasks in modern approximation theory. Marcinkiewicz–Zygmund inequalities provide answers to both the recovery and the quadrature aspect. In this paper, we put ourselves on the q -dimensional sphere S q , and investigate how well continuous L p -norms of polynomials f of maximum degree n on the sphere S q can be discretized by positively weighted L p -sum of finitely many samples, and discuss the distortion between the continuous and discrete quantities, the number and distribution of the (deterministic or randomly chosen) sample points ξ 1 , ... , ξ N on S q , the dimension q , and the degree n of the polynomials. [ABSTRACT FROM AUTHOR]

In this paper we initiate the study of a fundamental yet untapped random model of non-selfadjoint, bounded linear operators acting on a separable complex Hilbert space. We replace the weights w n = 1 in the classical unilateral shift T , defined as T e n = w n e n + 1 , where { e n } n = 1 ∞ form an orthonormal basis of a complex Hilbert space, by a sequence of i.i.d. random variables { X n } n = 1 ∞ ; that is, w n = X n. This paper answers basic questions concerning such a model. We propose that this model can be studied in comparison with the classical Hardy/Bergman/Dirichlet spaces in function-theoretic operator theory. We calculate the spectra and determine their fine structures (Section 3). We classify the samples up to four equivalence relationships (Section 4). We introduce a family of random Hardy spaces and determine the growth rate of the coefficients of analytic functions in these spaces (Section 5). We compare them with three types of classical operators (Section 6); this is achieved in the form of generalized von Neumann inequalities. The invariant subspaces are shown to admit arbitrarily large indices and their semi-invariant subspaces model arbitrary contractions almost surely. We discuss a Beurling-type theorem (Section 7). We determine various non-selfadjoint algebras generated by T (Section 8). Their dynamical properties are clarified (Section 9). Their iterated Aluthge transforms are shown to converge (Section 10). In summary, they provide a new random model from the viewpoint of probability theory, and they provide a new class of analytic functional Hilbert spaces from the viewpoint of operator theory. The technical novelty in this paper is that the methodology used draws from three (largely separate) sources: probability theory, functional Hilbert spaces, and the approximation theory of bounded operators. [ABSTRACT FROM AUTHOR]

Highlights • Nonlinear operators are used to smooth a hysteretic system with rate-independent hysteresis property. • It is applied the harmonic probing method on the equivalent system that carries the nonlinear smooth operator. • Analytical expressions to the higher-order frequency response functions are derived to evaluate a hysteretic system. • The analytical approach proposed can reproduce the hysteresis loop of a benchmark involving a Bouc-Wen oscillator for narrow-banded input conditions. Abstract This paper proposes a new way to approach hysteresis with the rate-independent property through an analytical response for the non-smooth hysteresis loop using frequency response approximations. The method consists of rewriting the loading and unloading loop using smooth operators and after applying a harmonic probing in the equivalent system to obtain the higher-order frequency response functions computed by Volterra series. The novelty of this paper lies on predicting analytically, through closed-form equations of the Volterra kernels, the output and the hysteresis loop from a non-smooth system. To illustrate the applicability of the proposed approach, a challenging benchmark with hysteretic damping, described by the Bouc-Wen model, is simulated through a numerical integration scheme. The hysteresis loops, as well as the outputs, are compared to the analytical approach proposed here. The results show that the Volterra model is able to predict the hysteretic outputs when the excitation amplitude is weak and the hysteresis draws a single loop in the restoring force × displacement plane. The higher-order FRFs are given as a function of the model parameters. This framework could turn into an alternative tool to perform nonlinear modal analysis on a hysteretic system. [ABSTRACT FROM AUTHOR]

Highlights • DSMC simulations of the Enceladus South polar plumes. • Parametrization of the outgassing flow parameters (number density, speed components, kinetic and rotational temperature) at 10 km above ground, where the flow becomes collisionless. • Parametric study of the outgassing flow parameters variations as a function of the vent exit conditions: vent radius, water mixture mass flow, water vapor/water ice mass ratio, water ice grain radius, water vapor and water ice exit speed, vent exit angle and flow temperature. • The DSMC results that have been run for this paper are available at an online repository. Abstract The field parameters – number density, velocity components and temperature – for the Enceladus geysers, and possibly similar jets from other bodies, such as Europa, Ceres or comets, are expensive to obtain using physically correct simulation methods, such as the Direct Simulation Monte Carlo (DSMC) approach. It would be very useful to be able to correctly reproduce all the different states that the flow is undergoing while expanding into vacuum, from a high density and collisional state near the surface to free-molecular and collisionless at high altitude without resorting to expensive DSMC simulations in every case. In this work, we consider a two-phase water vapor/grain mixture exiting a circular vent, assuming a uniform radius of the water ice grains, and study how the field parameters can be fitted at an altitude of 10 km, where the flow is collisionless. To do so, we define simple functional forms for each of these fitted parameters, and we study how their coefficients vary as a function of the vent exit parameters, i.e. the vent radius, the water mixture mass flow, the water vapor/water ice mass ratio, the water ice grain radius, the water vapor and water ice exit speed, the vent exit angle and flow temperature. We define polynomial approximations to model these variations. We show that all the vent parameters have nearly-independent influences on the radial profiles at 10 km, except for the water vapor and water ice exit speed, for which we considered cross-correlations. We finally show that the geyser field parameters can be reconstructed using our parametric study for variations of the vent parameters within the range considered here, and in a time frame of a few milliseconds. The results of the parametrizations presented in this study can now be used to propagate the geyser field parameters using computationally inexpensive free molecular/ballistic codes up to higher altitudes. The DSMC results that have been run for this paper are available at an online repository. [ABSTRACT FROM AUTHOR]

Abstract In this paper a semi-analytical solution for transverse vibrations induced by a moving oscillator is derived and validated. It is assumed that the oscillator is moving uniformly on an infinite beam, which may be subjected to a normal force, and is supported by a two-parameter visco-elastic foundation. Full evolution of deflection shapes is derived with the help of integral transforms and methods of contour integration. Analytical solution of the problem is presented in the Laplace domain. In the time domain, vibrations are given by a sum of truly steady-state part, induced harmonic part and transient vibration that has generally low importance and rapidly decreasing tendency. Except for the transient vibration, solution is expressed as a finite sum of analytical expressions (sum of residues), of which each has at most one parameter that has to be obtained numerically. With the help of iterative techniques proposed in this paper, these parameters can be easily determined with any precision. These parameters are named as the induced frequencies and as such can identify the onset of unstable behaviour. Thus, derivations in this paper allow to predict not only the onset of instability, but also its severity, which is important for mitigation measures. Transverse vibrations obtained for infinite beams are validated by analyses on long finite beams, exploiting a program written in Matlab software, that was previously validated by finite element software LS-DYNA. The effects of the normal force, Pasternak modulus, harmonic component of the vertical force and foundation damping are discussed. Highlights • New form of a semi-analytical solution for moving oscillator problem on infinite beams. • Analysis and determination and of induced frequencies. • Identification of the source of the transient part of the solution and its analysis. • Investigation of the critical frequency of the harmonic force. • Determination of the onset of instability and its severity. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we will investigate the approximations of inverse moments for double-indexed weighted sums of random variables under sub-linear expectations. The asymptotic approximations of inverse moments and the convergence rate of approximations are established under the meaning of upper expectation E ˆ and the lower expectation ε ˆ , respectively. Moreover, we carry out some simulations to verify the validity of our theoretical results. The results obtained in the paper improve and extend the corresponding ones of Yang et al. [24] from independent random variables in the classical probability space to END random variables in sub-linear expectation space. [ABSTRACT FROM AUTHOR]

Abstract Next generation, conservative finite element hydrodynamic and transport models are vital for accurate and efficient ocean, estuary and river simulation. Numerical models such as these have been developed for decades by the U.S. Army Corps of Engineers at the Engineering Research and Development Center (ERDC). This paper focuses on recently developed implicit, multi-dimensional finite element 3D shallow water and transport models included in the Adaptive Hydraulics (AdH) numerical suite. These AdH 3D models benefit from their adaptive meshing capabilities to resolve sharp solution gradients, such as those often encountered with baroclinic wedges traveling up an estuary channel. This paper presents the AdH 3D mathematical formulation and solution procedure used to solve the weak finite element equations for these models, along with results for several common verification cases and an AdH Galveston Bay validation study. A novel Streamline Upwind Petrov–Galerkin (SUPG) method for 3D shallow water models is described which reduces to the AdH 2D shallow water SUPG formulation under certain conditions. Careful attention is placed on ensuring discrete consistency in the equation set. [ABSTRACT FROM AUTHOR]

This paper considers an integrated framework for health measures prediction and optimal maintenance policy for mechanical systems subject to condition monitoring (CM) and random failure. We propose the proportional hazards model (PHM) to consider CM information as well as the age of the mechanical systems. Although the form of health prediction for the mechanical systems under periodic monitoring in the PHM with Markov chain was developed previously, the case of the continuous-state degradation process allowing possible degradation between the inspections still has not appeared. To this aim, the paper allows the use of Gamma process with non-constant degradation, which broadens the application area of PHM. A matrix-based approximation method is employed to compute health measures of the machine, such as condition reliability, mean residual life, residual life distribution. Based on the health measures, the optimal maintenance policy, which considers both hazard rate control limit and age control limit, is proposed and the optimization problem is formulated and solved in a semi-Markov decision process (SMDP) framework. The objective is to minimize the long-run expected average cost. The method is illustrated using two real data sets obtained from feed subsystem of a boring machine and GaAs lasers collected at regular time epochs, respectively. A comparison with other methods is given, which illustrates the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

The main results in this paper contribute to bringing to the fore novel underlying connections between the contemporary concepts and methods springing from greedy approximation theory with the well-established techniques of classical Banach spaces. We do that by showing that bounded-oscillation unconditional bases, introduced by Dilworth et al. in 2009 in the setting of their search for extraction principles of subsequences verifying partial forms of unconditionality, are the same as truncation quasi-greedy bases, a new breed of bases that appear naturally in the study of the performance of the thresholding greedy algorithm in Banach spaces. We use this identification to provide examples of bases that exhibit that bounded-oscillation unconditionality is a stronger condition than Elton's near unconditionality. We also take advantage of our arguments to provide examples that allow us to tell apart certain types of bases that verify either debilitated unconditionality conditions or weaker forms of quasi-greediness in the context of abstract approximation theory. [ABSTRACT FROM AUTHOR]

This paper is concerned with the nonlinear stability of traveling wave solutions for a conserved system of parabolic equations derived from a singular chemotaxis model describing the initiation of tumor angiogenesis. When the initial datum is a continuous small perturbation with zero integral from the spatially shifted traveling wave, the asymptotic stability of the large-amplitude (strong) traveling waves has been established in a series of works [29,34,35] by the second author with his collaborators. In this paper, we shall show that similar stability results indeed hold true for large and discontinuous initial data (i.e. the initial perturbation from the traveling wave could be discontinuous and has large oscillations) such as Riemann data with large jumps. To the best of our knowledge, this paper provides a first result on the asymptotic stability of large-amplitude traveling waves with large initial perturbation for a system of conservation laws, although similar results have been available for the scalar equations (cf. [8,42] ). We also extend existing results to the initial data with lower regularity. [ABSTRACT FROM AUTHOR]

It is well known that for any irrational rotation number α , the noncommutative torus A α must have representations π such that the generated von Neumann algebra π ( A α ) ″ is of type III. Therefore, it could be of interest to exhibit and investigate such kind of representations, together with the associated spectral triples whose twist of the Dirac operator and the corresponding derivation arises from the Tomita modular operator. In the present paper, we show that this program can be carried out, at least when α is a Liouville number satisfying a faster approximation property by rationals. In this case, we exhibit several type I I ∞ and II I λ , λ ∈ [ 0 , 1 ] , factor representations and modular spectral triples. The method developed in the present paper can be generalised to CCR algebras based on a locally compact abelian group equipped with a symplectic form. [ABSTRACT FROM AUTHOR]

In the present paper we introduce a notion of ( ω , t ) -convexity as a natural generalization of the notion of usual t -convexity, t -strongly convexity, approximate t -convexity, delta t -convexity and many other. The main result of this paper establishes the necessary and sufficient conditions under which an ( ω , t ) -convex map can be supported at a given point by an ( ω , t ) -affine support function. Several applications of this support theorem are presented. For instance, new characterizations of inner product spaces among normed spaces involving the notion of ( ω , t ) -convexity are given. [ABSTRACT FROM AUTHOR]

This study investigates the influence on the wheel-rail contact forces of the running speed and the shape and position of weld defects along the track. For this purpose, a vertical dynamic model in the space domain is used. The model is obtained from the transformation between the domains of frequency and space using a Rational Fraction Polynomials (RFP) method, which is modified with multiobjective genetic algorithms in order to improve the fitting of track receptance and to assist integration during simulations. This produces a precise model with short calculation times, which is essential to this study. The wheel-rail contact is modelled using a non-linear Hertz spring. The contact forces are studied for several types of characteristic welds. The way in which forces vary as a function of weld position and running speed is studied for each type of weld. This paper studies some of the factors that affect the maximum forces when the vehicle moves over a rail weld, such as weld geometry, parametric excitation and contact stiffness. It is found that the maximum force in the wheel-rail contact when the vehicle moves over a weld is not always proportional to the running speed. The paper explains why it is not proportional in specific welds. [ABSTRACT FROM AUTHOR]

As a generalization of cyclic codes, constacyclic codes is an important and interesting class of codes due to their nice algebraic structures and various applications in engineering. This paper is devoted to the study of the q -polynomial approach to constacyclic codes. Fundamental theory of this approach will be developed, and will be employed to construct some families of optimal and almost optimal codes in this paper. [ABSTRACT FROM AUTHOR]

This paper deals with the initial-boundary value problem (P) for the nonlinear diffusion equation ∂ u ∂ t + ( − Δ + 1 ) β ( u ) = g in Ω × ( 0 , T ) in a general domain Ω ⊂ R N with smooth bounded boundary, where N ∈ N , T > 0 and β is a single-valued maximal monotone function on R , e.g., β ( r ) = | r | q − 1 r ( q > 0 ) . The above equation represents a number of well-known models, e.g., porous media equation. Colli and Fukao [6] studied the above problem (P) and the approximate problem ( P ) ε , which consists of the Cahn–Hilliard system, with error estimates when Ω is a bounded domain, N = 2 or 3 and − Δ + 1 is replaced with −Δ. They considered one more approximation ( P ) ε , λ to solve ( P ) ε . They used compactness methods for abstract doubly nonlinear evolution equations to solve the second approximate problem (P) ε , λ . They established the error estimate saying that the solution of ( P ) ε converges to the solution of (P) as ε ↘ 0 . The present work asserts that one can solve the original problem (P) and the approximate problem ( P ) ε individually and directly even if Ω is a general domain. Moreover, this paper gives an error estimate between the solution of ( P ) ε and the solution of (P). [ABSTRACT FROM AUTHOR]

A method for experimentally modeling complex connections, such as “Line-Type” connection (LTC) and “Area-Type” connection (ATC), is proposed in this paper. Unlike traditional methods, instead of treating the junction forces as presumed approximate functions, a new strategy proposed in the present paper is to estimate them from experimentally measured accelerations. Along with them, the junction motion information is also estimated from the measured data. Based on this estimated information, the connection’s model is built. In this way, two potential disadvantages of traditional methods in modeling complex connections, i.e. the difficulty in finding appropriate functions for complex connections and the computation burden in identifying too many parameters through optimization process, can be avoided. In the proposed method, the LTC is modeled as a series of independent junction node pairs, and the ATC is modeled as a combination of a presumed Virtual Structure and a series of independent Virtual Node Pairs. Numerical and experimental results with the constructed example have verified the effectiveness of the proposed method in modeling both LTC and ATC. [ABSTRACT FROM AUTHOR]

Behavior of the entropy numbers of classes of multivariate functions with mixed smoothness is studied here. This problem has a long history and some fundamental problems in the area are still open. The main goal of this paper is to develop a new method of proving the upper bounds for the entropy numbers. This method is based on recent developments of nonlinear approximation, in particular, on greedy approximation. This method consists of the following two steps strategy. At the first step we obtain bounds of the best m -term approximations with respect to a dictionary. At the second step we use general inequalities relating the entropy numbers to the best m -term approximations. For the lower bounds we use the volume estimates method, which is a well known powerful method for proving the lower bounds for the entropy numbers. It was used in a number of previous papers. [ABSTRACT FROM AUTHOR]

The material in this paper has been divided into two main parts. In the first part we describe two optimization problems--one maximization and one minimization--related to a sharp trace inequality that was recently obtained by G. Auchmuty. In both problems the admissible set is the one comprising characteristic functions whose supports have a fixed measure. We prove the maximization to be solvable, whilst the minimization will turn out not to be solvable in general. We will also discuss the case of radial domains. In the second part of the paper, we study approximation and stability results regarding rearrangement optimization problems. First, we show that if a sequence of the generators of rearrangement classes converges, then the corresponding sequence of the optimal solutions will also converge. Second, a stability result regarding the Hausdorff distance between the weak closures of two rearrangement classes is presented. [ABSTRACT FROM AUTHOR]

Let f ∈ C [ − 1 , 1 ] and denote by E n ( f ) its degree of approximation by algebraic polynomials of degree < n . Assume that f changes its monotonicity, respectively, its convexity finitely many times, say s ≥ 2 times, in ( − 1 , 1 ) and we know that for q = 1 or q = 2 and some 1 < α ≤ 2 , such that q α ≠ 4 , we have E n ( f ) ≤ n − q α , n ≥ s + q + 1 . The purpose of this paper is to prove that the degree of comonotone, respectively, coconvex approximation, of f , by algebraic polynomials of degree < n , n ≥ N , is also ≤ c ( α , s ) n − q α , where the constant N depends only on the location of the extrema, respectively, inflection points in ( − 1 , 1 ) and on α . This answers, affirmatively, questions left open by the authors in papers with Kopotun (in Ukrainian Math. J.) and with Vlasiuk (see the list of references). [ABSTRACT FROM AUTHOR]

In this paper, we consider families of linear systems (linear ensembles) defined by matrix pairs (A (θ) , B (θ)) depending on a parameter θ ∈ P that is varying over a compact subset P of the complex plane. In particular, we investigate the existence of open-loop controls which are independent of the parameter θ ∈ P and steer a given family of initial states x 0 (θ) arbitrarily close to a desired family of terminal states f (θ) in finite time. Here, the maps θ ↦ x 0 (θ) and θ ↦ f (θ) are assumed to lie in a common appropriately chosen Banach space X n (P) of C n -valued functions. If this task is solvable for all initial and terminal states, the pair (A (θ) , B (θ)) is called (completely) ensemble controllable with respect to X n (P). Using a well-known infinite-dimensional version of the Kalman rank condition for systems on Banach spaces, we derive sufficient conditions for cascade and parallel connections of linear ensembles. Moreover, we prove an abstract decomposition theorem which results from a spectral splitting of the matrix family A (θ). Based on these findings as well as on cyclicity conditions for multiplication operators and approximation theory, we obtain necessary and sufficient conditions for ensemble controllability (reachability) with respect to the space of continuous functions and the space of L q -functions. In the last section, results on averaged controllability (reachability) for linear families (A (θ) , B (θ) , C (θ)) are presented. [ABSTRACT FROM AUTHOR]

In this paper we study d -variate approximation for weighted Korobov spaces in the worst-case setting. The considered algorithms use finitely many evaluations of arbitrary linear functionals. We give matching necessary and sufficient conditions for some notions of tractability in terms of two weight parameters. Our result is an affirmative answer to a problem which is left open in a recent paper of Kritzer, Pillichshammer and Woźniakowski. [ABSTRACT FROM AUTHOR]

Zonotopal algebra is the study of a family of pairs of dual vector spaces of multivariate polynomials that can be associated with a list of vectors X . It connects objects from combinatorics, geometry, and approximation theory. The origin of zonotopal algebra is the pair ( D ( X ) , P ( X ) ) , where D ( X ) denotes the Dahmen–Micchelli space that is spanned by the local pieces of the box spline and P ( X ) is a space spanned by products of linear forms. The first main result of this paper is the construction of a canonical basis for D ( X ) . We show that it is dual to the canonical basis for P ( X ) that is already known. The second main result of this paper is the construction of a new family of zonotopal spaces that is far more general than the ones that were recently studied by Ardila–Postnikov, Holtz–Ron, Holtz–Ron–Xu, Li–Ron, and others. We call the underlying combinatorial structure of those spaces forward exchange matroid. A forward exchange matroid is an ordered matroid together with a subset of its set of bases that satisfies a weak version of the basis exchange axiom. [ABSTRACT FROM AUTHOR]

This paper studies the approximation of the Hilbert transform f ˜ = H f of continuous functions f with continuous conjugate f ˜ based on a finite number of samples. It is known that every sequence { H N f } N ∈ N which approximates f ˜ from samples of f diverges (weakly) with respect to the uniform norm. This paper conjectures that all of these approximation sequences even contain no convergent subsequence. A property which is termed strong divergence. The conjecture is supported by two results. First it is proven that the sequence of the sampled conjugate Fejér means diverges strongly. Second, it is shown that for every sample based approximation method { H N } N ∈ N there are functions f such that ‖ H N f ‖ ∞ exceeds any given bound for any given number of consecutive indices N . As an application, the later result is used to investigate a problem associated with a question of Ul’yanov on Fourier series which is related to the possibility to construct adaptive approximation methods to determine the Hilbert transform from sampled data. This paper shows that no such approximation method with a finite search horizon exists. [ABSTRACT FROM AUTHOR]

Most classical beam theories assume that the beam cross section remains a rigid surface under an arbitrary loading condition. However, in the absolute nodal coordinate formulation (ANCF) continuum-based beams, this assumption can be relaxed allowing for capturing deformation modes that couple the cross-section deformation and beam bending, torsion, and/or elongation. The deformation modes captured by ANCF finite elements depend on the interpolating polynomials used. The most widely used spatial ANCF beam element employs linear approximation in the transverse direction, thereby restricting the cross section deformation and leading to locking problems. The objective of this investigation is to examine the behavior of a higher order ANCF beam element that includes quadratic interpolation in the transverse directions. This higher order element allows capturing warping and non-uniform stretching distribution. Furthermore, this higher order element allows for increasing the degree of continuity at the element interface. It is shown in this paper that the higher order ANCF beam element can be used effectively to capture warping and eliminate Poisson locking that characterizes lower order ANCF finite elements. It is also shown that increasing the degree of continuity requires a special attention in order to have acceptable results. Because higher order elements can be more computationally expensive than the lower order elements, the use of reduced integration for evaluating the stress forces and the use of explicit and implicit numerical integrations to solve the nonlinear dynamic equations of motion are investigated in this paper. It is shown that the use of some of these integration methods can be very effective in reducing the CPU time without adversely affecting the solution accuracy. [ABSTRACT FROM AUTHOR]

Fractal interpolation function defined with the aid of iterated function system can be employed to show that any continuous real-valued function defined on a compact interval is a special case of a class of fractal functions (self-referential functions). Elements of the iterated function system can be selected appropriately so that the corresponding fractal function enjoys certain properties. In the first part of the paper, we associate a class of self-referential L p -functions with a prescribed L p -function. Further, we apply our construction of fractal functions in L p -spaces in some approximation problems, for instance, to derive fractal versions of the full Müntz theorems in L p -spaces. The second part of the paper is devoted to identify parameters so that the fractal functions affiliated to a given continuous function satisfy certain conditions, which in turn facilitate them to find applications in some one-sided uniform approximation problems. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the notion of Kantorovich-type sampling operators using the Fejér-type singular integral. By means of these operators we are able to reconstruct signals (functions) which are not necessarily continuous. Moreover, our generalization allows us to take the measurement error into account. The goal of this paper is to estimate the rate of approximation by the above operators via high-order modulus of smoothness. [ABSTRACT FROM AUTHOR]

In this paper strict monotonicity, lower and upper uniform monotonicities, uniform monotonicity as well as decreasing and increasing uniform monotonicities of Orlicz spaces equipped with the p -Amemiya norm are studied. Criteria for these six properties in the Orlicz spaces L Φ , p are given in the most general case of Orlicz function Φ and for all 1 ≤ p ≤ ∞ . Finally, some applications of the results to the best dominated approximation problems are presented. This paper is a continuation of the studies from [5–7] . [ABSTRACT FROM AUTHOR]

The construction of local reduced-order models via multiscale basis functions has been an area of active research. In this paper, we propose online multiscale basis functions which are constructed using the offline space and the current residual. Online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators. We derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error. This error decrease is independent of physical parameters, such as the contrast and multiple scales in the problem. The offline spaces are constructed using Generalized Multiscale Finite Element Methods (GMsFEM). We show that if one chooses a sufficient number of offline basis functions, one can guarantee that additional online multiscale basis functions will reduce the error independent of contrast. We note that the construction of online basis functions is motivated by the fact that the offline space construction does not take into account distant effects. Using the residual information, we can incorporate the distant information provided the offline approximation satisfies certain properties. In the paper, theoretical and numerical results are presented. Our numerical results show that if the offline space is sufficiently large (in terms of the dimension) such that the coarse space contains all multiscale spectral basis functions that correspond to small eigenvalues, then the error reduction by adding online multiscale basis function is independent of the contrast. We discuss various ways computing online multiscale basis functions which include a use of small dimensional offline spaces. [ABSTRACT FROM AUTHOR]

We study ideal cotorsion pairs associated to almost exact structures in extension closed subcategories of triangulated categories. This approach allows us to extend the recent ideal approximation theory developed by Fu, Herzog et al. for exact categories in the above mentioned context. In the last part of the paper we apply the theory in order to study projective classes (in particular localization or smashing subcategories) in compactly generated triangulated categories. [ABSTRACT FROM AUTHOR]

The paper presents an oversampling technique to overcome the loss incurred at the time of binarizing the images. As such there are very few techniques available in the literature to approximate the loss in the images. This technique is based on the Estimate wavelet transform (EWT) which fulfills the loss and makes the image smooth. Coiflet and Daubechies D8 wavelets are employed in EWT and performances are compared with the traditional Gaussian lowpass filter. All the three approaches are tested on seven distorted images, and observed that the Daubechies D8 and Coiflet wavelets using EWT are better able to approximate the image than Gaussian lowpass filter. Out of seven images two are presented in this paper. [ABSTRACT FROM AUTHOR]

The prevalent description of low capillary number two-phase flow in porous media relies on the independence of phase transport. An extended Darcy’s law with a saturation dependent effective permeability is used for each phase. The driving force for each phase is given by its pressure gradient and the body force. This diagonally dominant form neglects momentum transfer from one phase to the other. Numerical and analytical modeling in regular geometries have however shown that while this approximation is simple and acceptable in some cases, many practical problems require inclusion of momentum transfer across the interface. Its inclusion leads to a generalized form of extended Darcy’s law in which both the diagonal relative permeabilities and the off-diagonal terms depend not only on saturation but also on the viscosity ratio. Analogous to application of thermodynamics to dynamical systems, any of the extended forms of Darcy’s law assumes quasi-static interfaces of fluids for describing displacement problems. Despite the importance of the permeability coefficients in oil recovery, soil moisture transport, contaminant removal, etc., direct measurements to infer the magnitude of the off-diagonal coefficients have been lacking. The published data based on cocurrent and countercurrent displacement experiments are necessarily indirect. In this paper, we propose a null experiment to measure the off-diagonal term directly. For a given non-wetting phase pressure-gradient, the null method is based on measuring a counter pressure drop in the wetting phase required to maintain a zero flux. The ratio of the off-diagonal coefficient to the wetting phase diagonal coefficient (relative permeability) may then be determined. The apparatus is described in detail, along with the results obtained. We demonstrate the validity of the experimental results and conclude the paper by comparing experimental data to numerical simulation. [ABSTRACT FROM AUTHOR]

Kirchberg's Embedding Problem (KEP) asks whether every separable C ⁎ algebra embeds into an ultrapower of the Cuntz algebra O 2 . In this paper, we use model theory to show that this conjecture is equivalent to a local approximate nuclearity condition that we call the existence of good nuclear witnesses . In order to prove this result, we study general properties of existentially closed C ⁎ algebras. Along the way, we establish a connection between existentially closed C ⁎ algebras, the weak expectation property of Lance, and the local lifting property of Kirchberg. The paper concludes with a discussion of the model theory of O 2 . Several results in this last section are proven using some technical results concerning tubular embeddings, a notion first introduced by Jung for studying embeddings of tracial von Neumann algebras into the ultrapower of the hyperfinite II 1 factor. [ABSTRACT FROM AUTHOR]

Multidimensional Riemann solvers that have internal sub-structure in the strongly-interacting state have been formulated recently (D.S. Balsara (2012, 2014) [5,16] ). Any multidimensional Riemann solver operates at the grid vertices and takes as its input all the states from its surrounding elements. It yields as its output an approximation of the strongly interacting state, as well as the numerical fluxes. The multidimensional Riemann problem produces a self-similar strongly-interacting state which is the result of several one-dimensional Riemann problems interacting with each other. To compute this strongly interacting state and its higher order moments we propose the use of a Galerkin-type formulation to compute the strongly interacting state and its higher order moments in terms of similarity variables. The use of substructure in the Riemann problem reduces numerical dissipation and, therefore, allows a better preservation of flow structures, like contact and shear waves. In this second part of a series of papers we describe how this technique is extended to unstructured triangular meshes. All necessary details for a practical computer code implementation are discussed. In particular, we explicitly present all the issues related to computational geometry. Because these Riemann solvers are Mu ltidimensional and have S elf-similar strongly- I nteracting states that are obtained by C onsistency with the conservation law, we call them MuSIC Riemann solvers. (A video introduction to multidimensional Riemann solvers is available on http://www.nd.edu/~dbalsara/Numerical-PDE-Course .) The MuSIC framework is sufficiently general to handle general nonlinear systems of hyperbolic conservation laws in multiple space dimensions. It can also accommodate all self-similar one-dimensional Riemann solvers and subsequently produces a multidimensional version of the same. In this paper we focus on unstructured triangular meshes. As examples of different systems of conservation laws we consider the Euler equations of compressible gas dynamics as well as the equations of ideal magnetohydrodynamics (MHD). Several stringent test problems are solved for both PDE systems. [ABSTRACT FROM AUTHOR]

This paper concerns the stability of a class of Helmholtz problems in rectangular domains. A well known application is the electromagnetic scattering from a rectangular cavity embedded in an infinite ground plane. Error analysis of numerical methods for cavity problems relies heavily on the stability estimates. However, it is extremely difficult to derive an optimal stability bound with the explicit dependency on wave numbers. In this paper a high-order finite element approximation is proposed for calculating the stability bound. Numerical experiments show that the stability depends strongly on wave numbers in extreme case and it is almost independent on the wave numbers in an average sense. Our numerical results also help to understand the stability of the multi-frequency inverse problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight w ( x ) = x α e − x − t / x , x ∈ ( 0 , ∞ ) , t > 0 and α > 0 . When the matrix size n → ∞ , we obtain an asymptotic formula for the Hankel determinants, valid uniformly for t ∈ ( 0 , d ] , d > 0 fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large- n asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift–Zhou nonlinear steepest descent method. [ABSTRACT FROM AUTHOR]

The dispersion equation for the axisymmetric modes of viscothermal acoustic wave propagation in uniform hard-walled circular ducts containing a quiescent perfect gas is classical. This has been extended to cover the non-axisymmetric modes and real fluids in contemporary studies. The fundamental axisymmetric mode has been the subject of a large number of studies proposing approximate solutions and the characteristics of the propagation constants for narrow and wide ducts with or without mean flow is well understood. In contrast, there are only few publications on the higher order modes and the current knowledge about their propagation characteristics is rather poor. On the other hand, there is a void of papers in the literature on the effect of the mean flow on the quiescent modes of propagation. The present paper aims to contribute to the filling of these gaps to some extent. The classical theory is re-considered with a view to cover all modes of acoustic propagation in circular ducts carrying a real fluid moving axially with a uniform subsonic velocity. The analysis reveals a new branch of propagation constants for the axisymmetric modes, which appears to have escaped attention hitherto. The solution of the governing wave equation is expressed in a modal transfer matrix form in frequency domain and numerical results are presented to show the effects over wide ranges of frequency, viscosity and mean flow parameters on the propagation constants. The theoretical formulation allows for the duct walls to have finite impedance, but no numerical results are presented for lined ducts or ducts carrying a sheared mean flow. [ABSTRACT FROM AUTHOR]

Constructing neural networks for function approximation is a classical and longstanding topic in approximation theory. In this paper, we aim at constructing deep neural networks with three hidden layers using a sigmoidal activation function to approximate smooth and sparse functions. Specifically, we prove that the constructed deep nets with controllable magnitude of free parameters can reach the optimal approximation rate in approximating both smooth and sparse functions. In particular, we prove that neural networks with three hidden layers can avoid the phenomenon of saturation, i.e., the phenomenon that for some neural network architectures, the approximation rate stops improving for functions of very high smoothness. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we approximate the semi-classical Schrödinger equation in the presence of electromagnetic field by the Hagedorn wave packets approach. By operator splitting, the Hamiltonian is divided into the modified part and the residual part. The modified Hamiltonian, which is the main new idea of this paper, is chosen by the fact that Hagedorn wave packets are localized both in space and momentum so that a crucial correction term is added to the truncated Hamiltonian, and is treated by evolving the parameters associated with the Hagedorn wave packets. The residual part is treated by a Galerkin approximation. We prove that, with the modified Hamiltonian only, the Hagedorn wave packets dynamics give the asymptotic solution with error , where ε is the scaled Planck constant. We also prove that, the Galerkin approximation for the residual Hamiltonian can reduce the approximation error to , where k depends on the number of Hagedorn wave packets added to the dynamics. This approach is easy to implement, and can be naturally extended to the multidimensional cases. Unlike the high order Gaussian beam method, in which the non-constant cut-off function is necessary and some extra error is introduced, the Hagedorn wave packets approach gives a practical way to improve accuracy even when ε is not very small. [Copyright &y& Elsevier]

Abstract: In this paper we present a new meshless method to solve well-known problems in physics and engineering with either constant or variable material properties in 2D space. Harmonic and bi-harmonic problems are considered in this paper. The method constructs a set of basis functions, called Equilibrated Basis Functions (EqBFs), through a weighted residual integration over a fictitious domain embedding the main one. The bases can satisfy the governing partial differential equations approximately. Chebyshev polynomials of the first kind are employed to construct the EqBFs and exponential functions are used as the weights in the integrals. The parameters of the solution are arranged so that all the integrals can be decomposed into much simpler 1D ones over a normalized intervals. This reduces the computational efforts significantly. Either the EqBFs or the results of the integration process may be stored for further use. The validity of the results is examined through an extended patch test. A set of physical sample problems with variable/non-variable material properties; as the potential flow over a cylinder, steady-state heat conduction problems in an anisotropic inhomogeneous functionally graded material, potential and stokes flow through a channel of finite width obstructed by periodic cylinders, and the bending of thin elastic plates having constant or variable thickness are solved to demonstrate the capabilities of the method. As a preliminary study, we show that the method may effectively be used in a domain decomposition approach. [Copyright &y& Elsevier]

This paper presents a new methodology to analyze aeroelastic stability in a continuous range of flight envelope with varying parameter of velocity and altitude. The focus of the paper is to demonstrate that linear matrix inequalities can be used to evaluate the aeroelastic stability in a region of flight envelope instead of a single point, like classical methods. The proposed methodology can also be used to study if a system remains stable during an arbitrary motion from one point to another in the flight envelope, i.e., when the problem becomes time-variant. The main idea is to represent the system as a polytopic differential inclusion system using rational function approximation to write the model in time domain. The theory is outlined and simulations are carried out on the benchmark AGARD 445.6 wing to demonstrate the method. The classical pk-method is used for comparing results and validating the approach. It is shown that this method is efficient to identify stability regions in the flight envelope. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to study the free lateral responses of vertically translating media with variable length, velocity and tension, subject to general initial conditions. The translating media are modeled as taut strings with fixed boundaries. The problem can be used as a simple model to describe the lateral vibrations of an elevator cable, for which the length changes linearly in time, or for which the length changes harmonically about a constant mean length. In this paper an initial-boundary value problem for a linear, axially moving string equation is formulated. In the given model a rigid body is attached to the lower end of the string, and the suspension of this rigid body against the guide rails is assumed to be rigid. For linearly length variations it is assumed that the axial velocity of the string is small compared to nominal wave velocity and the string mass is small compared to car mass, and for the harmonically length variations small oscillation amplitudes are assumed and it is also assumed that the string mass is small compared to the total mass of the string and the car. A multiple-timescales perturbation method is used to construct formal asymptotic approximations of the solutions to show the complicated dynamical behavior of the string. For the linearly varying length analytic approximations of the exact solution are compared with numerical solution. For the harmonically varying length it will be shown that Galerkin׳s truncation method cannot be applied in all cases to obtain approximations valid on long timescales. [Copyright &y& Elsevier]

Abstract: The paper considers impulsive control systems, where the evolution equation depends linearly on the time derivatives of the control function. The basic theory of “graph completions” is here extended to control functions in feedback form. In the case where the control is piecewise smooth, with a jump along a hypersurface Σ in the t–x space, results are proved on the existence and uniqueness of solutions, and on their approximation by means of smooth feedbacks. The paper is concluded with a couple of examples, concerning the feedback control of mechanical systems by means of active constraints. [Copyright &y& Elsevier]

Abstract: Zero-order ultrasensitivity (ZOU) is a long known and interesting phenomenon in enzyme networks. Here, a substrate is reversibly modified by two antagonistic enzymes (a ‘push–pull' system) and the fraction in modified state undergoes a sharp switching from near-zero to near-unity at a critical value of the ratio of the enzyme concentrations, under saturation conditions. ZOU and its extensions have been studied for several decades now, ever since the seminal paper of Goldbeter and Koshland (1981); however, a complete probabilistic treatment, important for the study of fluctuations in finite populations, is still lacking. In this paper, we study ZOU using a modular approach, akin to the total quasi-steady state approximation (tQSSA). This approach leads to a set of Fokker–Planck (drift–diffusion) equations for the probability distributions of the intermediate enzyme-bound complexes, as well as the modified/unmodified fractions of substrate molecules. We obtain explicit expressions for various average fractions and their fluctuations in the linear noise approximation (LNA). The emergence of a ‘critical point’ for the switching transition is rigorously established. New analytical results are derived for the average and variance of the fractional substrate concentration in various chemical states in the near-critical regime. For the total fraction in the modified state, the variance is shown to be a maximum near the critical point and decays algebraically away from it, similar to a second-order phase transition. The new analytical results are compared with existing ones as well as detailed numerical simulations using a Gillespie algorithm. [Copyright &y& Elsevier]

Abstract: In a recent paper, we showed that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. The main consequence of this theorem is that every regular tournament on n vertices can be decomposed into edge-disjoint Hamilton cycles, whenever n is sufficiently large. This verified a conjecture of Kelly from 1968. In this paper, we derive a number of further consequences of our result on robust outexpanders, the main ones are the following: [(i)] an undirected analogue of our result on robust outexpanders; [(ii)] best possible bounds on the size of an optimal packing of edge-disjoint Hamilton cycles in a graph of minimum degree δ for a large range of values for δ. [(iii)] a similar result for digraphs of given minimum semidegree; [(iv)] an approximate version of a conjecture of Nash-Williams on Hamilton decompositions of dense regular graphs; [(v)] a verification of the ‘very dense’ case of a conjecture of Frieze and Krivelevich on packing edge-disjoint Hamilton cycles in random graphs; [(vi)] a proof of a conjecture of Erdős on the size of an optimal packing of edge-disjoint Hamilton cycles in a random tournament. [Copyright &y& Elsevier]

Abstract: Modal analysis is an effective method for monitoring a dam's health. Modal parameters can be identified from the measured vibration response of a dam under ambient excitation, such as that from an earthquake. In this paper, we first use the state space model to analyze the vibration of a dam under ambient support excitation and conclude that the nature excitation technique (NExT) can be used to the measured absolute acceleration response of the dam under band-limited stochastic support excitation to obtain its impulse response. To overcome some of the limitations of the traditional modal identification method for a structure under ambient excitation, we propose a modal parameter identification method based on the Hankel matrix joint approximate diagonalization (HJAD) technique. In this method, the Hankel matrix is defined as the covariance matrix of a vector, which is composed of measured acceleration responses, and their time-lagged data. This modal parameter identification method can be regarded as an improvement to the traditional time domain method because it introduces the joint approximate diagonalization (JAD) technique into the original method. On the other hand, the method can be regarded as an improvement to the SOBI-based modal identification method because it uses the Hankel matrix instead of the covariance matrix of response to perform the JAD. Therefore, this method combines the advantages of the two existing modal identification methods and overcomes some of their limitations. Compared with the SOBI-based modal parameter identification method, the implementation of the method presented in this paper is very convenient because we need only to add time-lagged response data to the analysis. The numerical analysis results show that the proposed modal parameter identification method based on HJAD technology not only has the advantage of a traditional blind modal parameter identification algorithm, but can also overcome the limitation of not being able to estimate more active modes than the number of available sensors. According to the satisfactory performance of this method in the analysis of strong-motion earthquake observation data for a gravity dam, the modal parameter identification method proposed in this paper has the potential for application in water conservancy engineering. [Copyright &y& Elsevier]

Abstract: This paper presents a development of the direction splitting algorithm for flows in complex geometries proposed in [2] to the case of flows containing rigid particles. The main novelty of this method is that the grid is fit to the time-dependent domain at each time step which allows for an optimal spatial approximation of the Navier–Stokes equations. In general, this is a very hard task for multidimensional problems. However, it is significantly simplified if the momentum equations are discretized by a direction splitting scheme. Such a scheme requires only solving a set of one-dimensional problems, and the grid can be easily fit to the boundary in the direction of each of these problems. Here we use a MAC discretization stencil but the same idea can be applied to other discretizations. The equations of motion of each particle are discretized explicitly and the computed particle velocities are imposed as Dirichlet boundary conditions for the Navier–Stokes equations on the adapted grid. The pressure is extended within the particles in a fictitious domain fashion, but the divergence stencil is fit to the particle boundaries. In this paper, the direction splitting algorithm with boundary fitting is compared to pure fictitious domain methods. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a system of backward stochastic differential equations (BSDEs) with regime switching. The BSDEs are coupled by a finite-state Markov chain. The underlying Markov chain is assumed to have a two-time scale (or weak and strong interactions) structure. Namely, the states of the Markov chain can be divided into a number of groups so that the chain jumps rapidly within a group and slowly between the groups. It is shown in this paper that the original BSDE system can be approximated by a limit system in which the states in each group are aggregated out and replaced by a single state. In particular, it is proved that the solution of the original BSDE system converges weakly under the Meyer–Zheng topology as the fast jump rate goes to infinity. The limit process is a solution of aggregated BSDEs which can be determined by the corresponding martingale problem. The results are applied to a set of partial differential equations and used to validate their convergence to the corresponding limit system. Finally, a numerical example is given to demonstrate the approximation results. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re-used for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse-grid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method. [Copyright &y& Elsevier]