Symbolic computation of the Hartree-Fock energy from a chiral EFT three-nucleon interaction at NLO

We present the first of a two-part Mathematica notebook collection that implements a symbolic approach for the application of the density matrix expansion (DME) to the Hartree-Fock (HF) energy from a chiral effective field theory (EFT) three-nucleon interaction at N²LO. The final output from the notebooks is a Skyrme-like energy density functional that provides a quasi-local approximation to the non-local HF energy. In this paper, we discuss the derivation of the HF energy and its simplification in terms of the scalar/vector-isoscalar/isovector parts of the one-body density matrix. Furthermore, a set of steps is described and illustrated on how to extend the approach to other three-nucleon interactions.

Program summary

Program title: SymbHFNNNCatalogue identifier: AEGC_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGC_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 96 666No. of bytes in distributed program, including test data, etc.: 378 083Distribution format: tar.gzProgramming language: Mathematica 7.1Computer: Any computer running Mathematica 6.0 and later versionsOperating system: Windows Xp, Linux/UnixRAM: 256 MbClassification: 5, 17.16, 17.22Nature of problem: The calculation of the HF energy from the chiral EFT three-nucleon interaction at N²LO involves tremendous spin-isospin algebra. The problem is compounded by the need to eventually obtain a quasi-local approximation to the HF energy, which requires the HF energy to be expressed in terms of scalar/vector-isoscalar/isovector parts of the one-body density matrix. The Mathematica notebooks discussed in this paper solve the latter issue.Solution method: The HF energy from the chiral EFT three-nucleon interaction at N²LO is cast into a form suitable for an automatic simplification of the spin-isospin traces. Several Mathematica functions and symbolic manipulation techniques are used to obtain the result in terms of the scalar/vector-isoscalar/isovector parts of the one-body density matrix.Running time: Several hours     

Object-Oriented Symbolic Derivation and Automatic Programming of Finite Elements in Mechanics

Symbolic approaches to assist in the development of finite element formulations have been used since the late 1970s. Today, symbolic mathematical software such as Mathematica, Maple, etc., has proved to be helpful when testing formulations. In earlier work, the authors introduced a new way of integrating naturally symbolic concepts in numerical finite element codes, taking advantage of an objectoriented code organization. In this paper, we wish to prove on practical examples that the proposed approach is very attractive and promising today, leading to an alternative way of conceiving finite element codes. After presenting a state-of- the-art of symbolic approaches for finite element developments, we first give a practical application of symbolic developments (for discontinuous space-time formulations), and then examples of Computer Aided Software Engineering tools that can be introduced into such a finite element environment.     

Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions

A method for symbolically computing conservation laws of nonlinear partial differential equations (PDEs) in multiple space dimensions is presented in the language of variational calculus and linear algebra. The steps of the method are illustrated using the Zakharov–Kuznetsov and Kadomtsev–Petviashvili equations as examples.The method is algorithmic and has been implemented in Mathematica. The software package, ConservationLawsMD.m, can be used to symbolically compute and test conservation laws for polynomial PDEs that can be written as nonlinear evolution equations.The code ConservationLawsMD.m has been applied to multi-dimensional versions of the Sawada–Kotera, Camassa–Holm, Gardner, and Khokhlov–Zabolotskaya equations.     

Efficient symbolic computation of process expressions

This paper describes three optimization techniques for the eb³ process algebra. The optimizations are expressed in a new deterministic operational semantics which is shown to be trace-equivalent to a traditional non-deterministic operational semantics. Internal action transitions are eliminated by an efficient preruntime analysis of the structure of a process expression. Execution environments are used to optimize variable instantiation using lazy evaluation. Non-determinism is eliminated by returning a choice between possible transitions. This new operational semantics is implemented in the eb³pai process algebra interpreter to support the eb³ method. The goal of this method is to automate the development of information systems using, among other mechanisms, efficient symbolic computation of process expressions.     

Differential invariants of a Lie group action: Syzygies on a generating set

Given a group action, known by its infinitesimal generators, we exhibit a complete set of syzygies on a generating set of differential invariants. For that we elaborate on the reinterpretation of Cartan’s moving frame by Fels and Olver [Fels, M., Olver, P.J., 1999. Moving coframes. II. Regularization and theoretical foundations. Acta Appl. Math. 55 (2), 127–208]. This provides constructive tools for exploring algebras of differential invariants.     

Symbolic integration of a product of two spherical Bessel functions with an additional exponential and polynomial factor

We present a Mathematica package that performs the symbolic calculation of integrals of the form

(1)     

Computation with finite stochastic chemical reaction networks

A highly desired part of the synthetic biology toolbox is an embedded chemical microcontroller, capable of autonomously following a logic program specified by a set of instructions, and interacting with its cellular environment. Strategies for incorporating logic in aqueous chemistry have focused primarily on implementing components, such as logic gates, that are composed into larger circuits, with each logic gate in the circuit corresponding to one or more molecular species. With this paradigm, designing and producing new molecular species is necessary to perform larger computations. An alternative approach begins by noticing that chemical systems on the small scale are fundamentally discrete and stochastic. In particular, the exact molecular counts of each molecular species present, is an intrinsically available form of information. This might appear to be a very weak form of information, perhaps quite difficult for computations to utilize. Indeed, it has been shown that error-free Turing universal computation is impossible in this setting. Nevertheless, we show a design of a chemical computer that achieves fast and reliable Turing-universal computation using molecular counts. Our scheme uses only a small number of different molecular species to do computation of arbitrary complexity. The total probability of error of the computation can be made arbitrarily small (but not zero) by adjusting the initial molecular counts of certain species. While physical implementations would be difficult, these results demonstrate that molecular counts can be a useful form of information for small molecular systems such as those operating within cellular environments.

The finite cell method for nearly incompressible finite strain plasticity problems with complex geometries

In this paper, the performance of the Finite Cell Method is studied for nearly incompressible finite strain plasticity problems. The Finite Cell Method is a combination of the fictitious domain approach with the high-order Finite Element Method. It provides easy mesh generation capabilities for highly complex geometries; moreover, this method offers high convergence rates, the possibility to overcome locking and robustness against high mesh distortions. The performance of this method is numerically investigated based on computations of benchmark and applied problems. The results are also verified with the $h$- and $p$-version Finite Element Method. It is demonstrated that the Finite Cell Method is an appropriate simulation tool for large plastic deformations of structures with complex geometries and microstructured materials, such as porous and cellular metals that are made up of ductile materials obeying nearly incompressible $J_2$ theory of plasticity.     

Development of the meshless finite volume particle method with exact and efficient calculation of interparticle area

The Finite Volume Particle Method (FVPM) is a meshless method based on a definition of interparticle area which is closely analogous to cell face area in the classical finite volume method. In previous work, the interparticle area has been computed by numerical integration, which is a source of error and is extremely expensive. We show that if the particle weight or kernel function is defined as a discontinuous top-hat function, the particle interaction vectors may be evaluated exactly and efficiently. The new formulation reduces overall computational time by a factor between 6.4 and 8.2. In numerical experiments on a viscous flow with an analytical solution, the method converges under all conditions. Significantly, in contrast with standard FVPM and SPH, error depends on particle size but not on particle overlap (as long as the computational domain is completely covered by particles). The new method is shown to be superior to standard FVPM for shock tube flow and inviscid steady transonic flow. In benchmarking on a viscous multiphase flow application, FVPM with exact interparticle area is shown to be competitive with a mesh-based volume-of-fluid solver in terms of computational time required to resolve the structure of an interface.     

Symbolic computation of robot models for geometric parameters identification with singularity analysis

The computation of the geometrical parameters identification model of a robot arm leads to an intrinsic extended Jacobian matrix. This matrix is often singular and its manual computation is tedious and may lead to error. A computer assisted procedure is developed to compute this matrix symbolically and to check its singularity. The program GPIM is developed in Pascal and works on PC's for this purpose. Jacobian singularity is checked using its symbolic expressions with the aid of a proposed table for robot arm parameters to obtain a minimum set of identified parameters. This program can be used for simple chain robot arms having revolute (R) and/or prismatic (P) joints including the case of consecutive near parallel axes. TH8 robot arm of type RPPRRR having two pairs of consecutive near parallel axes is studied to show the program efficiency and how singularity is eliminated.Nomenclature a _i Denavit-Hartenberg parameter. - C _i ,C _i andC ₁₂ cos _i , cos _i and cos ( ₁ + ₂). - J ₀, J Basic Jacobian and Jacobian matrices. - J _a , J _r , J , J and J Sub-Jacobian matrices. - J _0a , J _0r , J ₀, J ₀ and J ₀ Intrinsic sub-Jacobian matrices. - m Work space dimensions. - n Number of moving links of the robot arm. - p Number of pairs of consecutive near parallel axes. - P _i,i+1 and P _li [3×1] position vectors. - Q _pi , Q _ai , Q _ri , Q _i , Q _i and Q _i [4×4] differential operator matrices. - r _i Denavit-Hartenberg parameter. - R _l , R _i , R _i+1 and R _n+1 Base, links i–1, i and n coordinate frames. - R _i,i+1 and R _1i [3×3] orientation matrices. - S _i , S _i and S ₁₂ sin _i , sin _i and sin ₁ + ₂. - T _i,i+1 and T _1i [4×4] homogeneous transformation matrices. - X [m×1] robot arm operational coordinates. - 1st columns of the orientation matrices, and associated tensors. - 2nd columns of the orientation matrices, and associated tensors. - 3rd columns of the orientation matrices, and associated tensors. - _i Denavit-Hartenberg parameter. - _i Twist angle parameter. - X Changes in robot arm operational coordinates. - , ls and rr Changes in robot arm geometrical parameters, and their least square and ridge regression estimated changes. - _i Denavit-Hartenberg parameter. - , ^c and ⁿ Robot geometrical parameters and their correct and nominal values. - pi, ai, ri, i, i and i [4×4] differential operator matrices. - and ^–1 [3×3] conversion matrix and its inverse.     

Secure computation with horizontally partitioned data using adaptive regression splines

When several data owners possess data on different records but the same variables, known as horizontally partitioned data, the owners can improve statistical inferences by sharing their data with each other. Often, however, the owners are unwilling or unable to share because the data are confidential or proprietary. Secure computation protocols enable the owners to compute parameter estimates for some statistical models, including linear regressions, without sharing individual records’ data. A drawback to these techniques is that the model must be specified in advance of initiating the protocol, and the usual exploratory strategies for determining good-fitting models have limited usefulness since the individual records are not shared. In this paper, we present a protocol for secure adaptive regression splines that allows for flexible, semi-automatic regression modeling. This reduces the risk of model mis-specification inherent in secure computation settings. We illustrate the protocol with air pollution data.     

仿真排队系统的统计计算研究

通过C 语言来仿真排队系统,该系统的数学模型为M／M／m/模型。然后用统计计算的方法来分析和研究仿真得到的数据,从而得出该系统的统计分布规律。     

Special purpose symbolic computation software with application to the optimization of composite laminates

Symbolic computation software is developed in the C language for the transformation of coordinate axes, failure analysis and the calculation of design sensitivities. These computations arise in the design optimization studies of structures made of fibre composite materials. The symbolic computations are integrated into an optimization algorithm resulting in a combined symbolic and numerical approach to determine the optimal design. The results are illustrated by considering the minimum thickness design of a laminate under in-plane loads. For this problem, the special purpose symbolic computation software handles matrices whose entries are series of double trigonometric functions, and determines the component functions of the design objective.     

Self-regulating finite-difference methods for the computation of reacting flows with nonlinear kinetics

This paper deals with the difficult problem of predicting the concentration (and temperature) fields of reacting flows in which the chemistry is modeled by a multicomponent set ofnonlinear reactions. A novel variable split-operator method is presented, which splits each individual chemical reaction at each point and for each step or iteration in such a way as to guarantee the nonpositiveness of the eigenvalues of the chemical Jacobian matrix. This helps to ensure stability/convergence of the proposed iterative scheme. The possibility of constructing accelerated schemes of this nature is also explored. Finally, computed results illustrate the usefulness and reliability of the proposed methods and provide evidence concerning their relative merits.     

A finite element approach for analyzing the effect of cushion type and thickness on pressure ulcer

Immobilization or continuous sitting with a constant posture for a long time in office environment causes sub-dermal tissue damage. Tissue damage occurs because of continuous prolonged compression of buttock soft tissue under body weight. It is treated as the end result of cell death or tissue deformation which ultimately gives rise to pressure ulcer. Although the sub-dermal tissue refers to the tissue beneath the skin, it is the tissue just below the bony portion undergoes maximum deformation because the muscle endures a continuous high stress. In this work, a simple but practical numerical approach has been proposed to estimate the maximum stress beneath the bony structure (ischial tuberosity). The model is validated with experimental data from the literature. Effect of properties of cushion material, loading angle and thickness of cushion has been analyzed. The cushion properties and thickness that can reduce maximum stress at ischial tuberosity have been demonstrated. The effect of sitting posture on maximum stress at ischial tuberosity has also been shown.     

FEMvrml: An interactive virtual environment for visualization of finite element simulation results

Finite element simulations produce information that is proportional to the size of the numerical model. For large models, the data produced is voluminous and can therefore be challenging to deal with and interpret. This paper discusses the use of virtual reality to manage this information and presents FEMvrml, an interactive virtual environment within which users can interact with and explore the simulation results. The structure and methodology behind FEMvrml and the algorithms employed are described along with application examples that show the effectiveness and capabilities of the system.     

Optimal buffer allocation in finite closed networks with multiple servers

In this study, we present an optimal buffer allocation procedure for closed queueing networks with finite buffers. The performance measures are evaluated using the expanded mean value analysis, and the solution procedure is incorporated into a nonlinear optimization scheme to arrive at the sub-optimal buffer space vector. The effectiveness of the method is demonstrated through several numerical experiments. Discussions on convergence and computational complexity are also included.     

Equivalencies between open and closed queueing networks with finite buffers

Equivalencies between open and closed models of queueing networks with finite buffers are investigated. Interarrival times in open networks and service times are assumed to be exponentially distributed. Using these equivalencies, bounds on the throughput of open networks are established.     

Lower bounds for the non-linear complexity of algebraic computation trees with integer inputs

Andrew Yao proved some lower bounds for algebraic computation trees with integer inputs. In his key result he proved bounds on the number of components of the leaf space of a homogeneous decision tree derived from a computation tree. In this paper we present a shorter and more conceptual proof. We introduce the concept of aregulated tree as a generalization of a regular tree which has the advantage of allowing the same lower bounds on the non-linear portion of the complexity. The proof is an application of a result of Ben-Or.