Second-order Extensions Research Articles

Second-order teleparallel gauge theory

In this work, we analyze second-order extensions of teleparallel theories of gravity as gauge theories for the translation group. We use Utiyama’s approach to gauge theories and show that it is possible to include second-order derivative terms in the Lagrangian of the gauge potential and preserve gauge invariance. Besides the usual field strength, a new object has to be introduced in order to preserve both gauge and diffeomorphism invariances. From this new object, we obtain a set of fourteen independent invariants which leads to equations that are linear in the fourth derivative of the tetrad field. We analyze a particular example with one of these invariants and evaluate the weak field limit, showing that the effective gravitational potential is a combination of Newton and Yukawa potentials.

Efficient conservative second-order central-upwind schemes for option-pricing problems

The conservative Kurganov–Tadmor (KT) scheme has been successfully applied to option-pricing problems by Germán I. Ramírez-Espinoza and Matthias Ehrhardt. These included the valuation of European, Asian and nonlinear options as Black–Scholes partial differential equations, written in the conservative form, by simply updating fluxes in the black box approach. In this paper, we describe an improvement of this idea through a fully vectorized algorithm of nonoscillatory slope limiters and the efficient use of time solvers. We also propose the application of second-order extensions of KT to option-pricing problems. Our test problems solve one-dimensional benchmark and convection-dominated European options as well as digital and butterfly options. These demonstrate the robustness and flexibility of the pricing methods and set a basis for complex problems. Further, the computation of option Greeks ensures the reliability of these methods. Numerical experiments are performed on barrier options, early exercisable American options and two-dimensional fixed and floating strike Asian options. To the authors’ knowledge, this is the first time American options have been priced by applying the early exercise condition on the semi-discrete formulation of central-upwind schemes. Our results show second-order, nonoscillatory and high-resolution properties of the schemes as well as computational efficiency.

Second-order extensions to nearly orthogonal-and-balanced (NOAB) mixed-factor experimental designs

ABSTRACTWhen simulation studies involve many quantitative (i.e., discrete and continuous) and qualitative (i.e., categorical) input factors with different numbers of levels for each, meta-models of simulation responses can benefit from the use of mixed-factor space-filling designs. The first-order nearly orthogonal-and-balanced (NOAB) design is a popular approach in these situations. This research develops second-order extensions for an existing construction method of NOAB designs, estimating the pairwise correlations between possible first-order and second-order terms. These extensions permit additional linear constraints in the mixed-integer linear programming (MILP) formulations previously developed for first-order NOAB designs. A case study is presented for NOAB designs of different sizes and construction approaches. The second-order MILP extensions show improvements in design performance measures for parameter estimation and prediction variance for an assumed second-order model as well as for model misspecification with respect to second-order terms for an assumed first-order model.

The highly excited-state manifold of guanine: calibration for nonlinear electronic spectroscopy simulations

A computational protocol based on the complete and restricted active space self-consistent field (CASSCF/RASSCF) methods and their second-order perturbation theory extensions (CASPT2/RASPT2) is employed to benchmark the highly excited-state manifold of the DNA/RNA canonical purine nucleobase guanine in vacuo. Several RASPT2 schemes are tested, displaying a steady convergence of electronic transition energies and dipole moments upon active space enlargement toward the reference values. The outcome allows calibrating and optimizing computational efforts by considering cheaper and more approximate RAS schemes that could enable the characterization of the excited-state manifolds of multi-chromophoric systems, such as DNA/RNA nucleobase dimers or multimers. Simulations of two-dimensional electronic spectra show similar trends to those observed on the other purine nucleobase adenine, deviating from this and other pyrimidine nucleobases in featuring its main excited-state absorption signal, embodied by sizable double HOMO to LUMO excitation contributions, in the UV probing window.

Second-order Accurate Ensemble Transform Particle Filters

Particle filters (also called sequential Monte Carlo methods) are widely used for state and parameter estimation problems in the context of nonlinear evolution equations. The recently proposed ensemble transform particle filter (ETPF) [S. Reich, SIAM J. Sci. Comput., 35, (2013), pp. A2013--A2014] replaces the resampling step of a standard particle filter by a linear transformation which allows for a hybridization of particle filters with ensemble Kalman filters and renders the resulting hybrid filters applicable to spatially extended systems. However, the linear transformation step is computationally expensive and leads to an underestimation of the ensemble spread for small and moderate ensemble sizes. Here we address both of these shortcomings by developing second-order accurate extensions of the ETPF. These extensions allow one in particular to replace the exact solution of a linear transport problem by its Sinkhorn approximation. It is also demonstrated that the nonlinear ensemble transform filter aris...

Configuration logics: Modeling architecture styles

We study a framework for the specification of architecture styles as families of architectures involving a common set of types of components and coordination mechanisms. The framework combines two logics: 1) interaction logics for the specification of architectures as generic coordination schemes involving a configuration of interactions between typed components; and 2) configuration logics for the specification of architecture styles as sets of interaction configurations. The presented results build on previous work on architecture modeling in BIP. We show how propositional interaction logic can be extended into a corresponding configuration logic by adding new operators on sets of interaction configurations. In addition to the usual set-theoretic operators, configuration logic is equipped with a coalescing operator + to express combination of configuration sets. We provide a complete axiomatization of propositional configuration logic as well as decision procedures for checking that an architecture satisfies given logical specifications. To allow genericity of specifications, we study first-order and second-order extensions of the propositional configuration logic. First-order logic formulas involve quantification over component variables. Second-order logic formulas involve additional quantification over sets of components. We provide several examples illustrating the application of the results to the characterization of various architecture styles. We also provide an experimental evaluation using the Maude rewriting system to implement the decision procedure for the propositional flavor of the logic.

Spectroscopic fingerprints of DNA/RNA pyrimidine nucleobases in third-order nonlinear electronic spectra

Accurate ab initio modeling of spectroscopic signals in nonlinear electronic spectra, such as bidimensional (2D) spectra, requires the computation of the electronic transitions induced by the incoming pump/probe pulses, resulting in a challenging calculation of many electronic excited states. A protocol is thus required to evaluate the variations of spectral properties, like transition energies and dipole moments, with the computational level, and to estimate the sensitivity of the spectra to these variations. Such a protocol is presented here within the framework of complete and restricted active space self-consistent field (CASSCF/RASSCF) theory and its second-order perturbation theory extensions (CASPT2/RASPT2). The electronic excited-state manifolds of pyrimidine nucleobases (thymine, uracil, and cytosine) are carefully characterized in vacuo employing high-level RAS(0,0|10,8|2,12)//SS-RASPT2 calculations. The results provide a reference data set that can be used for optimizing computational efforts and costs, as required for studying computationally more demanding multichromophoric systems (e.g., di- and oligonucleotides). The spectroscopic signatures of the 2D electronic spectrum of a perfectly stacked uracil–cytosine dimer model are characterized, and experimental setups are proposed that can resolve non-covalent interchromophoric interactions in canonical pyrimidine nucleobase-stacked dimers.

Modeling the high-energy electronic state manifold of adenine: Calibration for nonlinear electronic spectroscopy.

Pump-probe electronic spectroscopy using femtosecond laser pulses has evolved into a standard tool for tracking ultrafast excited state dynamics. Its two-dimensional (2D) counterpart is becoming an increasingly available and promising technique for resolving many of the limitations of pump-probe caused by spectral congestion. The ability to simulate pump-probe and 2D spectra from ab initio computations would allow one to link mechanistic observables like molecular motions and the making/breaking of chemical bonds to experimental observables like excited state lifetimes and quantum yields. From a theoretical standpoint, the characterization of the electronic transitions in the visible (Vis)/ultraviolet (UV), which are excited via the interaction of a molecular system with the incoming pump/probe pulses, translates into the determination of a computationally challenging number of excited states (going over 100) even for small/medium sized systems. A protocol is therefore required to evaluate the fluctuations of spectral properties like transition energies and dipole moments as a function of the computational parameters and to estimate the effect of these fluctuations on the transient spectral appearance. In the present contribution such a protocol is presented within the framework of complete and restricted active space self-consistent field theory and its second-order perturbation theory extensions. The electronic excited states of adenine have been carefully characterized through a previously presented computational recipe [Nenov et al., Comput. Theor. Chem. 1040-1041, 295-303 (2014)]. A wise reduction of the level of theory has then been performed in order to obtain a computationally less demanding approach that is still able to reproduce the characteristic features of the reference data. Foreseeing the potentiality of 2D electronic spectroscopy to track polynucleotide ground and excited state dynamics, and in particular its expected ability to provide conformational dependent fingerprints in dimeric systems, the performances of the selected reduced level of calculations have been tested in the construction of 2D electronic spectra for the in vacuo adenine monomer and the unstacked adenine homodimer, thereby exciting the Lb/La transitions with the pump pulse pair and probing in the Vis to near ultraviolet spectral window.

Solubility Limits in Lennard-Jones Mixtures: Effects of Disparate Molecule Geometries.

In order to better understand general effects of the size and energy disparities between macromolecules and solvent molecules in solution, especially for macromolecular constructs self-assembled from smaller molecules, we use the first- and second-order exact bridge diagram extensions of the HNC integral equation theory to investigate single-component, binary, ternary, and quaternary mixtures of Lennard-Jones fluids. For pure fluids, we find that the HNCH3 bridge function integral equation (i.e., exact to third order in density) is necessary to quantitatively predict the pure gas and pure liquid sides of the coexistence region of the phase diagram of the Lennard-Jones fluid. For the mixtures, we find that the HNCH2 bridge function integral equation is sufficient to qualitatively predict solubility in the binary, ternary, and quaternary mixtures, up to the nominal solubility limit. The results, as limiting cases, should be useful to several problems, including accurate phase diagram predictions for complex mixtures, design of self-assembling nanostructures via solvent controls, and the solvent contributions to the conformational behavior of macromolecules in complex fluids.

Intensive tool-practice and skillfulness facilitate the extension of body representations in humans

The brain׳s representation of the body can be extended to include objects that are not originally part of the body. Various studies have found both extremely rapid extensions that occur as soon as an object is held, as well as extremely slow extensions that require weeks of training. Due to species and methodological differences, it is unclear whether the studies were probing different representations, or revealing multiple aspects of the same representation. Here, we present evidence that objects (cotton balls) held by a tool (chopsticks) are rapidly integrated into the body representation, as indexed by fading of the cotton balls (or ‘second-order extensions’) from a positive afterimage. Skillfulness with chopsticks was predictive of more rapid integration of the second-order cotton balls held by this tool. We also found that extensive training over a period of weeks augmented the level of integration. Together, our findings demonstrate integration of second-order objects held by tools, and reveal that the body representation probed by positive afterimages is subject to both rapid and slow processes of adaptive change.

Hessian Schatten-Norm Regularization for Linear Inverse Problems

We introduce a novel family of invariant, convex, and non-quadratic functionals that we employ to derive regularized solutions of ill-posed linear inverse imaging problems. The proposed regularizers involve the Schatten norms of the Hessian matrix, which are computed at every pixel of the image. They can be viewed as second-order extensions of the popular total-variation (TV) semi-norm since they satisfy the same invariance properties. Meanwhile, by taking advantage of second-order derivatives, they avoid the staircase effect, a common artifact of TV-based reconstructions, and perform well for a wide range of applications. To solve the corresponding optimization problems, we propose an algorithm that is based on a primal-dual formulation. A fundamental ingredient of this algorithm is the projection of matrices onto Schatten norm balls of arbitrary radius. This operation is performed efficiently based on a direct link we provide between vector projections onto lq norm balls and matrix projections onto Schatten norm balls. Finally, we demonstrate the effectiveness of the proposed methods through experimental results on several inverse imaging problems with real and simulated data.

Holonomic extension of connections and symmetrization of jets

Taking into account motivations from the geometrization of mathematical physics, we discuss the symmetrization of higher order jets and extension of connections. First we introduce an extension of a general connection Γ on a fibered manifold Y → M into an r -th order holonomic connection and we classify all second-order holonomic extensions of Γ. Then we study the general problem of symmetrization of jets. We introduce symmetrizations of higher-order semiholonomic and nonholonomic prolongations and we describe all symmetrizations of an r -th order semiholonomic prolongation j ¯ r .

A Godunov-type method in Lagrangian coordinates for computing linearly-perturbed planar-symmetric flows of gas dynamics

Linear stability studies of complex flows require that efficient numerical methods be devised for predicting growth rates of multi-dimensional perturbations. For one-dimensional (1D) basic flows – i.e. of planar, cylindrical or spherical symmetry – a general numerical approach is viable which consists in solving simultaneously the one-dimensional equations of gas dynamics and their linearized forms for three-dimensional perturbations. Extensions of artificial viscosity methods have thus been used in the past. More recently [Equations aux dérivées partielles et applications, articles dédiés à J.-L. Lions, 1998], Godunov-type schemes for single-fluid flows of gas dynamics and magnetohydrodynamics have been proposed. Pursuing this effort, we introduce, within the Lagrangian perturbation approach, a class of Godunov-type schemes which is well suited for solving multi-material problems of gas dynamics. These schemes are developed here for the planar-symmetric case and comprise two second-order extensions. The numerical capabilities of these methods are illustrated by computations of Richtmyer–Meshkov instabilities occurring at a single material interface. A systematic comparison of numerically computed growth rates with results of the linear theory for the Richtmyer–Meshkov instability is provided.

Planning in the Fluent Calculus Using Binary Decision Diagrams

BDDplan was created to perform certain reasoning processes in the fluent calculus, a flexible framework for reasoning about action and change based on first-order logic with equality (plus some second-order extensions in some cases). The reasoning is done by mapping the problems into propositional logic, which, in turn, can be implemented as operations on binary decision diagrams (BDDs).

Discussion

This article demonstrates a latent growth curve methodology for analyzing longitudinal data of adolescent substance use. Hypotheses concerning the form of growth in alcohol, cigarette, and marijuana use, and covariates influencing the form of growth, were tested. Participants were male and female adolescents (n=664) assessed at three time points. A common trajectory existed across the developmental period with significant increases in all three substances. Second-order multivariate extensions of the basic latent growth modeling framework suggested that associations among the individual differences parameters, representing growth or change in the various substance use behaviors, could be adequately modeled by a higher-order substance use construct. Inept parental monitoring, parent-child conflict, peer deviance, academic failure, gender, and age, were significant predictors of initial levels and the trajectory of substance use.

Contributions of the social context to the development of adolescent substance use: a multivariate latent growth modeling approach

This article demonstrates a latent growth curve methodology for analyzing longitudinal data of adolescent substance use. Hypotheses concerning the form of growth in alcohol, cigarette, and marijuana use, and covariates influencing the form of growth, were tested. Participants were male and female adolescents ( n=664) assessed at three time points. A common trajectory existed across the developmental period with significant increases in all three substances. Second-order multivariate extensions of the basic latent growth modeling framework suggested that associations among the individual differences parameters, representing growth or change in the various substance use behaviors, could be adequately modeled by a higher-order substance use construct. Inept parental monitoring, parent-child conflict, peer deviance, academic failure, gender, and age, were significant predictors of initial levels and the trajectory of substance use.

Empirical Evaluation of Innovations in Interval Branch and Bound Algorithms for Nonlinear Systems

Interval branch and bound algorithms for finding all roots use a combination of a computational existence/uniqueness procedure and a tessellation process (generalized bisection). Such algorithms identify, with mathematical rigor, a set of boxes that contains unique roots and a second set within which all remaining roots must lie. Though each root is contained in a box in oneof the sets, the second set may have several boxes in clusters near a single root. Thus, the output is of higher quality if there are relatively more boxes in the first set. In contrast to previously implemented similar techniques, a box expansion technique in this paper, based on using an approximate root finder, $\epsilon$-inflation, and exact set complementation, decreases the size of the second set, increases the size of the first set, and never loses roots. In addition to the expansion technique, use of second-order extensions to eliminate small boxes that do not contain roots, and interval slopes versus interval derivative matrices are studied. These items are evaluated empirically on a significant test problem set, within a Fortran-90 environment designed for such purposes. The results are compared with previous results and show that careful incorporation of the techniques yields both quantitatively and qualitatively superior computer codes.

First- and Second-Order Flux Difference Splitting Schemes for Dam-Break Problem

The first-order flux difference splitting scheme and its second-order extensions are investigated for their applicability to dam-break problems. Roe's first-order explicit scheme is first formulate...

A second-order Godunov-type scheme for compressible fluid dynamics

A second-order accurate scheme for the integration in time of the conservation laws of compressible fluid dynamics is presented. Two related versions are proposed, one Lagrangian and the second direct Eulerian. They both share the common ingredient which is a full analytic solution for the time derivatives of flow quantities at a jump discontinuity, assuming initial nonvanishing slopes on both sides. While this solution is an extension of the solution to the classical Riemann problem, the resulting schemes are second-order extensions of Godunov's methods. In both cases, they are very simple to implement in computer codes. Several numerical examples are shown, where the only additional mechanism is a simple monotonicity algorithm.