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Nonstandard analysis has been put to use in a theorem-prover, where it assists in the analysis of formulae involving limits. The theorem-prover in question is used in the computer program Mathpert to ensure the correctness of calculations in calculus. Although non-standard analysis is widely viewed as non-constructive, it can alternately be viewed as a method of reducing logical manipulation (of formulae with quantifiers) to computation (with rewrite rules). We give a logical theory of nonstandard analysis which is implemented in Mathpert. We describe a procedure for “elimination of infinitesimals” (also implemented in Mathpert) and prove its correctness.

We show how the anticommutation relations for Fermi operators can be implemented with computer algebra using SymbolicC++. We describe applications to the Hubbard model. An important identity for Fermi operators is proved. Then, we test for higher order constants of motion for the Hubbard model. Finally, the matrix representation for the four point Hubbard model is calculated.

A SU(2) Gauge Theory with spherical symmetry over the Minkowski space-time is considered. The self-duality equation of the gauge fields are written and their solutions are obtained. Two exact solutions, one of which is statical and another of dynamical type are given. All the calculations are performed using an analytical program written in GRTensor computer algebra package, which runs on the MapleV platform.

A model of Poincaré gauge self-dual theory over the Minkowski space–time, endowed with spherical symmetry, is considered. The self-duality (S-D) conditions are imposed and two sets of the self-duality equations for the gauge fields are obtained: one for the torsion tensor and another for the curvature tensor. An analytical solution of the gauge field equations is also obtained. The Yang–Mills (Y–M) equations of the gauge fields are also derived. It is shown that these Y–M equations can be obtained from the S-D equations, a result which is generally true for any S-D gauge theory. Most of all calculations are performed using the GRTensorII package, running on the MapleV platform.

Muon reactivation coefficient are determined for muonic He for up to six (n = 1, 2, 3, …, 6) states of formation and at temperature T_p = 100 eV and for various relative ion densities. In the next decade it may be possible to explore new conditions for further energy gain in muon catalyzed fusion system, μCF, using nonuniform (temperature and density) plasma states. Here, we have considered a model for inhomogeneous μCF for mixtures of D/T and H/D/T. Using coupled dynamical equations it is shown that the neutrons yield per muon injection, Y_n (neutrons/muon), in the dt branch of an inhomogeneous H/D/T mixture is at least 2.24 times higher than similar homogeneous systems and this rate for a D/T mixture is 1.92. Also, we have compared the neutron yield in the dt branch of homogeneous D/T and H/D/T mixtures (temperature range T = 300–800 K, and density ϕ = 1 LHD). It is shown that Y_n(D/T)/Y_n(H/D/T) = 1.32, which is in good agreement with recently measured experimental values. In other words our calculations show that the addition of protonium to a D/T mixture leads to a significant decrease in the cycling rate for the physical conditions described herein.

A deSitter gauge theory of gravitation over a spherical symmetric Minkowski space–time is developed. The "passive" point of view is adapted, i.e., the space–time coordinates are not affected by group transformations; only the fields change under the action of the symmetry group. A particular ansatz for the gauge fields is chosen and the components of the strength tensor are computed. An analytical solution of Schwarzschild–deSitter type is obtained in the case of null torsion. It is concluded that the deSitter group can be considered as a "passive" gauge symmetry for gravitation.

Because of their complexity, all the calculations, inclusive of the integration of the field equations, are performed using an analytical program conceived in GRTensorII for MapleV. The program allows one to compute (without using a metric) the strength tensor , Riemann tensor , Ricci tensor , curvature scalar , field equations, and the integration of these equations.

It is shown that the cycling rate in the optimum tritium concentration in μCF and in the n=2, J=1, ν=0 state is 2.35 times higher than in n=1, J=ν=0 state. The methods to explore forced μCF of the n=2 state is discussed. Although the n=2 state shows to be more efficient in terms of its cycling rate, all suggested drivers seems to be useless with respect to the input energy requirement and short life-time of resonance states. It is shown that even using x-ray sources as a driver, and designing hybrid system, the suggested forced hybrid system energy gain is 13 and very good gain enhancement but still is far away to be of interest for practical applications with respect to very short life-time of resonance states.

A de-Sitter gauge theory of the gravitational field is developed using a spherical symmetric Minkowski space–time as base manifold. The gravitational field is described by gauge potentials and the mathematical structure of the underlying space–time is not affected by physical events. The field equations are written and their solutions without singularities are obtained by imposing some constraints on the invariants of the model. An example of such a solution is given and its dependence on the cosmological constant is studied. A comparison with results obtained in General Relativity theory is also presented.

Coupled first order IVPs are frequently used in many parts of engineering and sciences. In this article, we presented a code including three computer programs which are joint with the Matlab software to solve and plot the solutions of the first order coupled stiff or non-stiff IVPs. Some engineering and scientific problems related to IVPs are given and fuel depletion (production of the ²³⁹Pu isotope) in a Pressurized Water Nuclear Reactor (PWR) are computed by the present code.

In linear quantum optics we consider phase shifters, beam splitters, displacement operations, squeezing operations etc. The evolution can be described by unitary operators using Bose creation and annihilation operators. This evolution can be reduced to matrix multiplication using unitary matrices. We derive these evolutions for the different unitary operators. Finally a computer algebra implementation is provided.

The Maxima computer algebra system, the open-source successor to MACSYMA, the first general-purpose computer algebra system that was initially developed at the Massachusetts Institute of Technology in the late 1960s and later distributed by the United States Department of Energy, has some remarkable capabilities, some of which are implemented in the form of add-on, “share” packages that are distributed along with the core Maxima system. One such share package is itensor, for indicial tensor manipulation. One of the more remarkable features of itensor is functional differentiation. Through this, it is possible to use itensor to develop a Lagrangian field theory and derive the corresponding field equations. In this paper, we demonstrate this capability by deriving Maxwell’s equations from the Maxwell Lagrangian, and exploring the properties of the system, including current conservation.

A program to compute the gluon self-energy diagram in class III nonrelativistic gauges has been constructed. It succeeded to show that axial gauges do not exhibit nonlocal counterterms. The high complexity of the calculations, the need to be able to control intermediate steps of calculations and the necessity to work inside a limited memory of a few Mbytes led us to give it special features which may serve outside the specific application for which it has been designed.

The structure of the program and its salient working features are described. It is divided into four modules, which are almost independent except that they are linked by their respective output. Files containing intermediate results of calculations are automatically generated. The amount of algebraic manipulations has been greatly reduced thanks to a careful choice of the representation of the expression adapted to each stage of the calculation. The adaptative character of this program is discussed as well as the generalization it needs to cover other types of Feynman diagram calculations.

Quantum groups and quantum algebras play a central role in theoretical physics. We show that computer algebra is a helpful tool in the investigations of quantum groups. We give an implementation of the Kronecker product together with the Yang-Baxter equation. Furthermore the quantum algebra obtained from the Yang-Baxter equation is implemented. We apply the computer algebra package REDUCE.

It is shown that for a large class of non-holonomic quantum mechanical systems one can make the computation of BRST charge fully algorithmic. Two computer algebra programs written in the language of REDUCE are described. They are able to realize the complex calculations needed to determine the charge for general nonlinear algebras. Some interesting specific solutions are discussed.

Tensor calculation of suffix-contraction is carried out by a C-program. Tensors are represented graphically, and the algorithm makes use of the topology of the graphs. Classical and quantum gravity, in the weak-field perturbative approach, is a special interest. Examples of the leading order calculation of some general invariants such as R_μνλσR^μνλσ are given. Application to Weyl anomaly calculation is also discussed.

We show how SymbolicC++, a symbolic and numeric computer algebra system written in C++, can be used to find exact solutions of soliton equations in (2+1)-dimensions.

This paper gives a short review of the status and applications of computer algebra systems for calculations in relativity and gravitation.

In the present work, an extension of the scaled particle theory (ESPT) for fluid using computer algebra is developed to obtain an equation of state (EOS), for Lennard-Jones fluid. A suitable functional form for surface tension S(r,d,∊) is assumed with intermolecular separation r as a variable, given below:

A computationally efficient method is proposed for computing the simplest normal forms of vector fields. A simple, explicit recursive formula is obtained for general differential equations. The most important feature of the approach is to obtain the "simplest" formula which reduces the computation demand to minimum. At each order of the normal form computation, the formula generates a set of algebraic equations for computing the normal form and nonlinear transformation. Moreover, the new recursive method is not required for solving large matrix equations, instead it solves linear algebraic equations one by one. Thus the new method is computationally efficient. In addition, unlike the conventional normal form theory which uses separate nonlinear transformations at each order, this approach uses a consistent nonlinear transformation through all order computations. This enables one to obtain a convenient, one step transformation between the original system and the simplest normal form. The new method can treat general differential equations which are not necessarily assumed in a conventional normal form. The method is applied to consider Hopf and Bogdanov–Takens singularities, with examples to show the computation efficiency. Maple programs have been developed to provide an "automatic" procedure for applications.

We consider the problem of computing $R(c,a)$, the number of unlabeled graded lattices of rank $3$ that contain $c$ coatoms and $a$ atoms. More specifically, we do this when $c$ is fairly small, but $a$ may be large. For this task, we describe a computational method that combines constructive listing of basic cases and tools from enumerative combinatorics. With this method, we compute the exact values of $R(c,a)$ for $c\le 9$ and $a\le 1000$.

We also show that, for any fixed $c$, there exists a quasipolynomial in $a$ that matches with $R(c,a)$ for all $a$ above a small value. We explicitly determine these quasipolynomials for $c\le 7$, thus finding closed form expressions of $R(c,a)$ for $c\le 7$.