Higher-order Part Research Articles

Edgeworth approximations for distributions of symmetric statistics

We study the distribution of a general class of asymptotically linear statistics which are symmetric functions of N independent observations. The distribution functions of these statistics are approximated by an Edgeworth expansion with a remainder of order o(N^{-1}). The Edgeworth expansion is based on Hoeffding’s decomposition which provides a stochastic expansion into a linear part, a quadratic part as well as smaller higher order parts. The validity of this Edgeworth expansion is proved under Cramér’s condition on the linear part, moment assumptions for all parts of the statistic and an optimal dimensionality requirement for the non linear part.

Spatial non-locality of electronic correlations beyond GW approximation

The question of spatial locality of electronic correlations beyond GW approximation is one of the central issues of the famous combination of GW and dynamical mean field theory, GW + DMFT. In this work, the above question is addressed directly (for the first time) by performing calculations with and without assumption of locality of the corresponding diagrams. For this purpose we use sc(GW + G3W2) approach where the higher order part (G3W2) is evaluated with fully momentum dependent Green’s function G and screened interaction W and with ‘local’ variant, where the single site approximation is assumed for both G and W. For all three materials studied in this work (NiO, α-Ce, LiFeAs), we have found the spatial non-locality effects to be strong. For NiO and LiFeAs they, in fact, are decisive for the proper evaluation of vertex corrections. The results of this study have direct impact on our understanding of approximations made in practical implementations of GW + DMFT method, where all diagrams beyond GW (DMFT part) are assumed to be local. Taking into account the fact that the first diagrams beyond GW represent the most important contribution also in GW + DMFT calculations, we conclude that the basic assumption of GW + DMFT, namely the locality of diagrams evaluated in the DMFT part, is not as good as it is believed to be.

The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: evolution of higher-order correlations demonstrated with Minkowski functionals

We probe the higher-order galaxy clustering in the final data release (DR12) of the Sloan Digital Sky Survey Baryon Oscillation Spectroscopic Survey (BOSS) using germ-grain Minkowski Functionals (MFs). Our data selection contains $979,430$ BOSS galaxies from both the northern and southern galactic caps over the redshift range $z=0.2 - 0.6$. We extract the higher-order part of the MFs, detecting the deviation from the purely Gaussian case with $\chi^2 \sim \mathcal{O}(10^3)$ on 24 degrees of freedom across the entire data selection. We measure significant redshift evolution in the higher-order functionals for the first time. We find $15-35\%$ growth, depending on functional and scale, between our redshift bins centered at $z=0.325$ and $z=0.525$. We show that the structure in higher order correlations grow faster than that in the two-point correlations, especially on small scales where the excess approaches a factor of $2$. We demonstrate how this trend is generalizable by finding good agreement of the data with a hierarchical model in which the higher orders grow faster than the lower order correlations. We find that the non-Gaussianity of the underlying dark matter field grows even faster than the one of the galaxies due to decreasing clustering bias. Our method can be adapted to study the redshift evolution of the three-point and higher functions individually.

The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: higher order correlations revealed by germ–grain Minkowski functionals

We probe the higher-order clustering of the galaxies in the final data release (DR12) of the Sloan Digital Sky Survey Baryon Oscillation Spectroscopic Survey (BOSS) using the method of germ-grain Minkowski Functionals (MFs). Our sample consists of 410,615 BOSS galaxies from the northern Galactic cap in the redshift range 0.450--0.595. We show the MFs to be sensitive to contributions up to the six-point correlation function for this data set. We ensure with a custom angular mask that the results are more independent of boundary effects than in previous analyses of this type. We extract the higher-order part of the MFs and quantify the difference to the case without higher-order correlations. The resulting $\chi^{2}$ value of over 10,000 for a modest number of degrees of freedom, O(200), indicates a 100-sigma deviation and demonstrates that we have a highly significant signal of the non-Gaussian contributions to the galaxy distribution. This statistical power can be useful in testing models with differing higher-order correlations. Comparing the galaxy data to the QPM and MultiDark-Patchy mocks, we find that the latter better describes the observed structure. From an order-by-order decomposition we expect that, for example, already a reduction of the amplitude of the MD-Patchy mock power spectrum by 5% would remove the remaining tension.

Logarithmic divergences in the k-inflationary power spectra computed through the uniform approximation

We investigate a calculation method for solving the Mukhanov-Sasaki equation in slow-roll k-inflation based on the uniform approximation (UA) in conjunction with an expansion scheme for slow-roll parameters with respect to the number of e-folds about the so-called turning point. Earlier works on this method have so far gained some promising results derived from the approximating expressions for the power spectra among others, up to second order with respect to the Hubble and sound flow parameters, when compared to other semi-analytical approaches (e.g., Green's function and WKB methods). However, a closer inspection is suggestive that there is a problem when higher-order parts of the power spectra are considered; residual logarithmic divergences may come out that can render the prediction physically inconsistent. Looking at this possibility, we map out up to what order with respect to the mentioned parameters several physical quantities can be calculated before hitting a logarithmically divergent result. It turns out that the power spectra are limited up to second order, the tensor-to-scalar ratio up to third order, and the spectral indices and running converge to all orders. This indicates that the expansion scheme is incompatible with the working equations derived from UA for the power spectra but compatible with that of the spectral indices. For those quantities that involve logarithmically divergent terms in the higher-order parts, existing results in the literature for the convergent lower-order parts calculated in the equivalent fashion should be viewed with some caution; they do not rest on solid mathematical ground.

Adaptive discontinuous Galerkin methods on surfaces

We present a dual weighted residual-based a posteriori error estimate for a discontinuous Galerkin approximation of a surface partial differential equation. We restrict our analysis to a linear second-order elliptic problem posed on hypersurfaces in $$\mathbb {R}^{3}$$R3 which are implicitly represented as level sets of smooth functions. We show that the error in the energy norm may be split into a residual part and a higher order part. Upper and lower bounds for the resulting a posteriori error estimator are proven and we consider a number of challenging test problems to demonstrate the reliability and efficiency of the estimator. We also present a novel geometric driven refinement strategy for PDEs on surfaces which considerably improves the performance of the method on complex surfaces.

SEMIGLOBAL OUTPUT-FEEDBACK STABILIZATION FOR A CLASS OF FEEDFORWARD NONLINEAR SYSTEMS

This paper considers the semiglobal output-feedback stabilization for a class of feedforward nonlinear systems.Notably,the systems under investigation possess unmeasurable states dependent growth including linear and higher-order parts,which incorporate and extend those in the existing many literature.Based on a linear lower-gain observer,a linear output feedback controller is constructed by outputfeedback domination approach,which is structurally simple and easy to implement.By appropriately choosing the design parameter,it can be guaranteed that the zero solution of resulting closed-loop system is locally exponentially stable,and by starting from the given compact set containing the origin,all the solutions of the closed-loop system converge to the origin.A numerical example is provided to illustrate the effectiveness of the theoretical results.

Reducing Higher-Order Theorem Proving to a Sequence of SAT Problems

We describe a complete theorem proving procedure for higher-order logic that uses SAT-solving to do much of the heavy lifting. The theoretical basis for the procedure is a complete, cut-free, ground refutation calculus that incorporates a restriction on instantiations. The refined nature of the calculus makes it conceivable that one can search in the ground calculus itself, obtaining a complete procedure without resorting to meta-variables and a higher-order lifting lemma. Once one commits to searching in a ground calculus, a natural next step is to consider ground formulas as propositional literals and the rules of the calculus as propositional clauses relating the literals. With this view in mind, we describe a theorem proving procedure that primarily generates relevant formulas along with their corresponding propositional clauses. The procedure terminates when the set of propositional clauses is unsatisfiable. We prove soundness and completeness of the procedure. The procedure has been implemented in a new higher-order theorem prover, Satallax, which makes use of the SAT-solver MiniSat. We also describe the implementation and give several examples. Finally, we include experimental results of Satallax on the higher-order part of the TPTP library.

Global output feedback stabilization for a class of nonlinear planar systems with output-dependent growth rates

Abstract In this paper, we consider the global output feedback stabilization problem for a class of nonlinear planar systems under a more general growth condition. The nonlinearities in such a system are bounded by both lower-order and higher-order terms with an output-dependent growth rate. This problem is solved by integrating the feedback domination technique and a novel dual observer approach, where two individual observers are constructed in parallel with one estimating the linear part of the unmeasurable state and the other estimating the higher-order part. Moreover, the presented observers and the output feedback controller all have nonlinear gains. Simulations illustrate the effectiveness of the proposed stabilization scheme.

Geometrically constrained stabilization of wave equations with Wentzell boundary conditions

Uniform stabilization of wave equation subject to second-order boundary conditions is considered in this article. Both dynamic (Wentzell) and static (with higher derivatives in space only) boundary conditions are discussed. In contrast to the classical wave equation where stabilization can be achieved by applying boundary velocity feedback, for a Wentzell-type problem boundary damping alone does not cause the energy to decay uniformly to zero. This is the case for both dynamic and static second-order conditions. In order to achieve uniform decay rates of the associated energy, it is necessary to dissipate part of the collar near the boundary. It will be shown how a combination of partially localized boundary feedback and partially localized collar feedback leads to uniform decay rates that are described by a nonlinear differential equation. This goal is attained by combining techniques used for stabilization of ‘unobserved’ Neumann conditions with differential geometry techniques effective for stabilization on compact manifolds. These lead to a construction of special non-radial multipliers which are geometry dependent and allow reconstruction of the high-order part of the potential energy from the damping that is supported only in a far-off region of the domain.

Non-Equilibrium Thermo-Field Dynamics for a Fourth-Order Hamiltonian

The non-equilibrium thermo-field dynamics proposed by Arimitsu and Umezawa are generalized to the case of a fourth-order unperturbed Hamiltonian which includes not only a second-order (quadratic) part but also a fourth-order part. Fujita’s analysis for effects of the initial particle correlation of a quantum gas is proved generally in terms of TFD. The forms of the quasi-particle operators for a semi-free boson field are derived. It is shown that the energies and life-times of the quasi-particles depend on the adiabatic boson-reservoir interaction which leads to the fourth-order part of the unperturbed Hamiltonian. The form of the two-point Green’s function for the semi-free boson field is evaluated. A form of the admittance for a boson system interacting with its heat reservoir, which includes effects of the initial correlation and memory, is derived using the TCLE method formulated in terms of the generalized non-equilibrium thermo-field dynamics. A calculation method of the higher-order parts of the admittance in powers of the boson-boson interaction is given. Furthermore, a calculation method of the perturbation expansions of the two-point Green’s function for the boson system is given. Subject Index: 130

On Nontrivial Solutions of Variational-Hemivariational Inequalities with Slowly Growing Principal Parts

This paper is concerned with the inclusion −\mathrm{div}(a(|∇u|)∇u) + ∂_u G(x,u) \ni 0 \quad \text{in } Ω, with Dirichlet boundary condition u = 0 on ∂Ω , in the case where the higher order part has slow growth and the lower order part is locally Lipschitz. By using a Mountain Pass theorem for variational-hemivariational inequalities without the Palais–Smale condition in Orlicz–Sobolev spaces, we show the existence of nontrivial solutions of the above inclusion.

Application of optimal polynomial controller to a benchmark problem

In this paper, we investigate the performance of optimal polynomial control for the vibration suppression of a benchmark problem; namely, the active tendon system. The optimal polynomial controller is a summation of polynomials of different orders, i.e., linear, cubic, quintic, etc., and the gain matrices for different parts of the controller are calculated easily by solving matrix Riccati and Lyapunov equations. A Kalman-Bucy estimator is designed for the on-line estimation of the states of the design model. Hence, the Linear Quadratic Gaussian (LQG) controller is a special case of the current polynomial controller in which the higher-order parts are zero. While the percentage of reduction for displacement response quantities remains constant for the LQG controller, it increases with respect to the earthquake intensity for the polynomial controller. Consequently, if the earthquake intensity exceeds the design one, the polynomial controller is capable of achieving a higher reduction for the displacement response at the expense of control efforts. Such a property is desirable for the protection of civil engineering structures because of the inherent stochastic nature of the earthquake.

Application of optimal polynomial controller to a benchmark problem

In this paper, we investigate the performance of optimal polynomial control for the vibration suppression of a benchmark problem; namely, the active tendon system. The optimal polynomial controller is a summation of polynomials of different orders, i.e., linear, cubic, quintic, etc., and the gain matrices for different parts of the controller are calculated easily by solving matrix Riccati and Lyapunov equations. A Kalman–Bucy estimator is designed for the on-line estimation of the states of the design model. Hence, the Linear Quadratic Gaussian (LQG) controller is a special case of the current polynomial controller in which the higher-order parts are zero. While the percentage of reduction for displacement response quantities remains constant for the LQG controller, it increases with respect to the earthquake intensity for the polynomial controller. Consequently, if the earthquake intensity exceeds the design one, the polynomial controller is capable of achieving a higher reduction for the displacement response at the expense of control efforts. Such a property is desirable for the protection of civil engineering structures because of the inherent stochastic nature of the earthquake. © 1998 John Wiley & Sons, Ltd.

Boundary value problems for some classes of higher-order equations that are unsolved with respect to the highest derivative

UDC 517.95 Introduction. In the study of the properties of solutions to third-order equations that are unsolved with respect to the time variable (pseudoparabolic equations), in the author's article (1) a new approach was proposed making it possible for us to prove existence theorems for regular solutions, the degeneracy of the higher-order part notwithstanding. In the present article, the approach of (1) will be further developed in application to various new classes of higher-order equations that are unsolved with respect to the derivative related to a distinguished variable. Moreover, the higher-order part of the equations under study is assumed to be degenerate as regards the spatial variables. We will study boundary value problems for pseudoparabolic, quasielliptic and, "partially hyperbolic" equations of the form OQTTt

A finite element formulation with stabilization matrix for geometrically non-linear shells

An assumed strain finite element formulation with a stabilization matrix is developed for analysis of geometrically non-linear problems of isotropic and laminated composite shells. The present formulation utilizes the degenerate solid shell concept and assumes an independent strain as well as displacement. The assumed independent strain field is divided into a lower order part and a higher order part. Subsequently, the lower order part is set equal to the displacement-dependent strain evaluated at the lower order integration points and the remaining higher order part leads to a stabilization matrix. The strains and the determinant of the Jacobian matrix are assumed to vary linearly in the thickness direction. This assumption allows analytical integration through thickness, independent of the number of plies. A nine-node element with a judiciously chosen set of higher order assumed strain field is developed. Numerical tests involving isotropic and composite shells undergoing large deflections demonstrate the validity of the present formulation.

Theorems as meaningful cultural artifacts: Making the world additive

Mathematical theorems are cultural artifacts and may be interpreted much as works of art, literature, and tool-and-craft are interpreted. The Fundamental Theorem of the Calculus, the Central Limit Theorem of Statistics, and the Statistical Continuum Limit of field theories, all show how the world may be put together through the “arithmetic addition” of suitably prescribed parts (velocities, variances, and renormalizations and scaled blocks, respectively). In the limit — of smoothness, statistical independence, and large N — higher-order parts, such as accelerations, are, for the most, part irrelevant, affirming that, in the end, most of the world's particulars may be averaged over (a very un-Scriptural point of view). (We work out all of this in technical detail, including a nice geometric picture of stochastic integration, and a method of calculating the variance of the sum of dependent random variables using renormalization group ideas.) These fundamental theorems affirm a culture that is additive, ahistorical, Cartesian, and continuist, sharing in what might be called a species of modern culture. We understand mathematical results as useful because, like many other such artifacts, they have been adapted to fit the world, and the world has been adapted to fit their capacities. Such cultural interpretation is in effect motivation for the mathematics, and might well be offered to students as a way of helping them understand what is going on at the blackboard. Philosophy of mathematics might want to pay more attention to the history and detailed technical features of sophisticated mathematics, as a balance to the usual concerns that arise in formalist or even Platonist positions.

A sixteen node shell element with a matrix stabilization scheme

A sixteen node shell element is developed using a matrix stabilization scheme based on the Hellinger-Reissner principle with independent strain. Initially the assumed independent strain is divided into a lower order part and a higher order part. The stiffness matrix corresponding to the lower order assumed strain is equivalent to the stiffness matrix of the assumed displacement model element with the reduced integration scheme. The spurious kinematic modes of the element are suppressed by introducing a stabilization matrix associated with a judiciously chosen set of higher order assumed strain fields. Numerical results show that this element is free of locking even for very thin plates and shells.

A new efficient mixed formulation for thin shell finite element models

AbstractA nine node shell element is developed by a new and more efficient mixed formulation. The new shell element formulation is based on the Hellinger–Reissner principle with independent strain and the concept of a degenerate solid shell. The new formulation is made more efficient in terms of computing time than the conventional mixed formulation by dividing the assumed strain fields into a lower order part and a higher order part. Numerical results demonstrate that the present nine node element is free of locking even for very thin plates and shells and is also kinematically stable. In fact the stiffness matrix associated with the higher order assumed strain plays the role of a stabilization matrix.

A new efficient approach to the formulation of mixed finite element models for structural analysis

AbstractA new mixed finite element formulation is developed based on the Hellinger‐Reissner principle with independent strain. By dividing the assumed strain into its lower order and higher order parts, the new formulation can be made much more efficient than the conventional mixed formulation. In addition the present new approach provides an alternative way of introducing a stabilization matrix to suppress undesirable kinematic modes.