Algebraic Expansion Research Articles

A Neat Embedding Theorem for Expansions of Cylindric Algebras

We generalize two classical results on cylindric algebra to certain expansions of cylindric algebras where the extra operations are defined via first order formulas. The first result is the Neat Embedding Theorem of Henkin and the second is Monk's classical non-finitizability result of the class of representable algebras. As a corollary we obtain known classical results of Johnson and Biro published in the Journal of Symbolic logic.

Force constants and transition intensities in the U ( ν + 1) model for molecular vibrational excitations

A connection between the unitary group approach U ( ν + 1) and the traditional description in configuration space of vibrational excitations is proposed. Local operators b ˆ i † ( b ˆ i ) satisfying the su (2) commutation relations are used to establish approximate algebraic expansions of the local coordinates and momenta. The use of the proposed relations allows to obtain an algebraic representation of traditional Hamiltonians in terms of the U ( ν + 1) model. This approach provides in natural form the connection between the spectroscopic parameters and force constants. Using the linear expansion of the coordinates in terms of the b ˆ i † ( b ˆ i ) operators, an approach to study local dipole transition intensities based on traditional descriptions is proposed. A closed general analytical expression for the local dipole operators is obtained. The analysis of the stretching vibrational excitations of arsine is taken as an example for the determination of both force constants and the description of dipole transition intensities.

Algebraic expansions for curvature-coupled scalar field models

A late time asymptotic perturbative analysis of curvature-coupled complex scalar field models with accelerated cosmological expansion is carried out on the level of formal power series expansions. For this, algebraic analogues of the Einstein scalar field equations in Gaussian coordinates for spacetime dimensions greater than two are postulated and formal solutions are constructed inductively and shown to be unique. The results obtained this way are found to be consistent with already known facts on the asymptotics of such models. In addition, the algebraic expansions are used to provide a prospect of the large time behaviour that might be expected of the considered models.

COMPUTABLY ENUMERABLE ALGEBRAS, THEIR EXPANSIONS, AND ISOMORPHISMS

Computably enumerable algebras are the ones whose positive atomic diagrams are computably enumerable. Computable algebras are the ones whose atomic diagrams are computable. In this paper we investigate computably enumerable algebras and provide several algebraic and computable theoretic distinctions of these algebras from the class of computable algebras. We give a characterization of computably enumerable but not computable algebras in terms of congruences and effective conjunctions of [Formula: see text]-sentences. Our characterization, for example, shows that computable conjunctions of negative atomic formulas true in a given c.e. algebra can be preserved in infinitely many of its homomorphic images. We also study questions on how expansions of algebras by finitely many new functions affect computable isomorphism types. In particular, we construct a c.e. algebra with unique computable isomorphism type but which has no finitely generated c.e. expansion.

End effects of heat conduction in circular cylinders of functionally graded materials and laminated composites

Heat conduction in circular cylinders of functionally graded materials and laminated composites is studied with emphasis on the end effects. By means of matrix algebra and eigenfunction expansion, the decay length that characterizes the end effects on the thermal filed is evaluated and the 2D solution as a useful approximation assessed.

Frenet-Based Algorithm for Trajectory Prediction

An algorithm for aircraft trajectory prediction is presented, that is based on a third-order accurate algebraic expansion of the aircraft center of gravity (e.g.) trajectory in the Frenet frame. This technique allows for an efficient evaluation of future e.g. positions in a time interval the extension of which depends on the aircraft maneuver state. The evaluation of the coefficients in the series expansion requires the knowledge of aircraft e.g. acceleration and its time derivative. Estimates of first and second derivatives of rate-gyro signals are also necessary when the accelerometers are not located in the aircraft e.g. A Kalman filtering technique with a fixed-lag smoother is used for simultaneously filtering the noise and estimating the signal derivatives. The algorithm is implemented on a F-16 fighter aircraft model performing a reverse-turn maneuver, where both control angles and control rates present relevant variations. The reported results show that the effect of a step variation of commanded load factor and/or roll angle on the aircraft position at following times can be determined in real-time and displayed to the pilot.

Open Wilson lines and group theory of noncommutative Yang-Mills theory in two dimensions

The correlation functions of open Wilson line operators in two-dimensional Yang-Mills theory on the noncommutative torus are computed exactly. The correlators are expressed in two equivalent forms. An instanton expansion involves only topological numbers of Heisenberg modules and enables extraction of the weak-coupling limit of the gauge theory. A dual algebraic expansion involves only group theoretic quantities, winding numbers and translational zero modes, and enables analysis of the strong-coupling limit of the gauge theory and the high-momentum behaviour of open Wilson lines. The dual expressions can be interpreted physically as exact sums over contributions from virtual electric dipole quanta.

Definable sets in boolean ordered o-minimal structures. II

AbstractLet (M, ≤,…) denote a Boolean ordered o-minimal structure. We prove that a Boolean subalgebra of M determined by an algebraically closed subset contains no dense atoms. We show that Boolean algebras with finitely many atoms do not admit proper expansions with o-minimal theory. The proof involves decomposition of any definable set into finitely many pairwise disjoint cells, i.e., definable sets of an especially simple nature. This leads to the conclusion that Boolean ordered structures with o-minimal theories are essentially bidefinable with Boolean algebras with finitely many atoms, expanded by naming constants. We also discuss the problem of existence of proper o-minimal expansions of Boolean algebras.

Beyond-all-orders effects in multiple-scales asymptotics: travelling-wave solutions to the Kuramoto-Sivashinsky equation

This paper concerns the possible `shock' patterns that can exist in the solution to a singularly perturbed, third-order nonlinear ordinary differential equation arising as the travelling-wave reduction of the Kuramoto-Sivashinsky equation. In particular, the existence (or otherwise) of oscillatory shocks and multiple shocks made up of combinations of oscillatory and monotonic shocks is examined, using an optimal truncation strategy to track crucial exponentially small terms lying beyond all orders of the (divergent) algebraic expansion. The results provide further understanding of numerical solutions previously obtained by others, as well as giving a methodology which is applicable to much broader classes of differential equations exhibiting multiscale phenomena; they also afford same new insight into the multi-scales technique.

Effect of combined triangularity and ellipticity on the stability limit of the ideal internal kink mode in a tokamak

The effect of combined triangular and elliptical shaping of the plasma cross section on the stability limit of the ideal, internal kink mode in a tokamak is analyzed by means of a computer algebra expansion of the magnetohydrodynamic equations. The work extends the result of a previous investigation of the effect of ellipticity alone on this mode in toroidal plasmas [Wahlberg, Phys. Plasmas 5, 1387 (1998)]. It is shown that, in contrast to the strongly destabilizing effect of (positive) ellipticity alone, the effect of combined positive ellipticity and positive triangularity is stabilizing. The effect is of the same importance as the effect of ellipticity alone if the triangularity of the q=1 surface is of the same order of magnitude as the inverse aspect ratio. When the safety factor at the magnetic axis is close to (but below) unity, the geometrical factor governing the stability of the ideal, internal kink mode is found to be the same as the corresponding factor in the Mercier criterion for shaped tokamaks.

Array Algebra Expansion of Matrix and Tensor Calculus: Part 1

Array algebra expands the foundations of linear and nonlinear estimation theories, differential and integral calculus, numerical analysis, and fast transform techniques. It originates from an extension of the two-dimensional Kronecker or tensor products and related operators of the traditional vector, matrix, and tensor calculus using the general theory of matrix inverses called inverses. A summary of the foundations of multilinear array algebra and loop inverse estimation is presented in part 1 of this paper. It is then expanded to include the latest developments in nonlinear estimation and applied mathematics using some unified matrix and tensor operators. The new operators are used in part 2 to derive the general theory of direct solution (one hyper iteration) techniques of rank-deficient nonlinear systems as an expansion of the loop inverse estimators and Q-surface tensor solution.

Array Algebra Expansion of Matrix and Tensor Calculus: Part 2

Part 1 of this paper summarized some extended matrix and tensor operators of the multilinear array algebra and loop inverse estimation. It also introduced the compact shorthand tensor notations of array polynomials for expressing the Taylor expansion of a nonlinear function and its least squares normal equation. These matrix-like operators will be further expanded using the nonlinear tensor transforms of Blaha's Q-surface technique. They enable an analytical derivation of the direct (one hyper iteration) solution of nonlinear systems. New operators are found such as the nonlinear counterpart of the Lm- or general pseudoinverse and neural net operators. The increased pull-in range and rate of convergence of the new technique are applied in some standard examples of nonlinear equations. Some applications of the nonlinear array algebra involving literally billions of parameters are discussed, including the automated stereo mensuration of three-dimensionaldigital imaging and mapping systems.

Partial Reduction in the N-Body Planetary Problem using the Angular Momentum Integral

We present a new set of variables for the reduction of the planetary n-body problem, associated to the angular momentum integral, which can be of any use for perturbation theory. The construction of these variables is performed in two steps. A first reduction, called partial is based only on the fixed direction of the angular momentum. The reduction can then be completed using the norm of the angular momentum. In fact, the partial reduction presents many advantages. In particular, we keep some symmetries in the equations of motion (d'Alembert relations). Moreover, in the reduced secular system, we can construct a Birkhoff normal form at any order. Finally, the topology of this problem remains the same as for the non-reduced system, contrarily to Jacobi's reduction where a singularity is present for zero inclinations. For three bodies, these reductions can be done in a very simple way in Poincare's rectangular variables. In the general n-body case, the reduction can be performed up to a fixed degree in eccentricities and inclinations, using computer algebra expansions. As an example, we provide the truncated expressions for the change of variable in the 4-body case, obtained using the computer algebra system TRIP.

Realistic constitutive relations for metal plasticity

The process of establishing quantitative constitutive relations is illustrated on the example of strain hardening: a phenomenological analysis in parallel with the derivation of physics-based combinations of variables; both followed by the use a sufficient set of experimental data to firm up unknown parameters or even unknown functions; and the eventual return to an assessment of the more detailed physical mechanisms that are suggested by the observed behavior. It is emphasized that a physics-based algebraic expansion around an accessible standard state is more reliable than a grand overall equation.

Asymptotic expansions of Laplace-type integrals. III

We consider the asymptotic expansion for large λ of Laplace-type integrals of the form ∫ 0 ∞ ∫ 0 ∞ g(x,y) e −λf(x,y) dx dy for a wide class of amplitude functions g( x, y) and ‘polynomial’ (noninteger powers are permitted) phases f( x, y) possessing an isolated, though possibly degenerate, critical point at the origin. The resulting algebraic expansions valid in a certain sector of the complex λ plane are based on recent results obtained in Kaminski and Paris (Philos. Trans. Roy. Soc. London A 356 (1998) 583–623; 625–667) when g( x, y)≡1. The limitation of the validity of the algebraic expansion to this sector as certain coefficients in the phase function are allowed to take on complex values is due to the appearance of exponential contributions. This is examined in detail in the special case when the phase function corresponds to a single internal point in the associated Newton diagram. Numerical examples illustrating the accuracy of the expansions are discussed.

Numerical self-consistent field method for polyatomic molecules

A method for non-relativistic self-consistent field (SCF) electronic structure calculations for polyatomic molecules is described, which retains the linear combination of atomic orbitals ansatz for molecular orbitals (MO-LCAO), but replaces the usual algebraic expansion of atom-centred radial parts in terms of basis functions (usually some kind of Gauss-type functions) by a numerical representation on a set of radial grid points around each centre. The radial parts are optimized, according to the variation principle, until self-consistency is achieved. Even though Fourier integral transform techniques are used the method works completely in ordinary space. Intermediate quantities defined in momentum space are evaluated in closed form.

On neat reducts of algebras of logic

SC α, CA α, QA α and QEA α stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras, and quasipolyadic equality algebras of dimension α, respectively. Generalizing a result of Nemeti on cylindric algebras, we show that for K ∈ {SC, CA, QA, QEA} and ordinals α < β, the class Nr α K β of α-dimensional neat reducts of β-dimensional K algebras, though closed under taking homomorphic images and products, is not closed under forming subalgebras (i.e. is not a variety) if and only if α > 1. From this it easily follows that for 1 < α < β, the operation of forming α-neat reducts of algebras in K β does not commute with forming subalgebras, a notion to be made precise. We give a contrasting result concerning Halmos' polyadic algebras (with and without equality). For such algebras, we show that the class of infinite dimensional neat reducts forms a variety. We comment on the status of the property of neat reducts commuting with forming subalgebras for various reducts of polyadic algebras that are also expansions of cylindric-like algebras. We try to draw a borderline between reducts that have this property and reducts that do not. Following research initiated by Pigozzi, we also emphasize the strong tie that links the (apparently non-related) property of neat reducts commuting with forming subalgebras with proving amalgamation results in cylindric-like algebras of relations. We show that, like amalgamation, neat reducts commuting with forming subalgebras is another algebraic expression of definability and, accordingly, is also strongly related to the well-known metalogical properties of Craig, Beth and Robinson in the corresponding logics.

Intermultiplet interactions in normal and local mode molecules in the algebraic force-field expansion approach

The algebraic force-field expansion recently proposed [T. Sako, K. Yamanouchi, and F. Iachello, Chem. Phys. Lett. 299, 35 (1999)] is applied to fit the experimental vibrational term values of H2O and SO2 in the electronic ground X̃ 1A1 state. The comparison of results of least-squares fits by the algebraic force-field expansion with those by the conventional force-field expansion shows that the convergence of the algebraic model is much faster than that of the conventional model and this rapid convergence becomes more significant when the Hamiltonian is expressed in local coordinates rather than in normal coordinates. It is also demonstrated that coordinate-space vibrational wave functions can be constructed directly through the experimental-level energy fit by the algebraic Hamiltonian expansion. From the nodal patterns of the vibrational wave functions constructed with the optimized Hamiltonian parameters of SO2, the bifurcation of the wave functions characteristic of the local-mode doublet states are identified in a vibrationally highly excited energy region. It is shown that the local-mode structure of the normal-mode limit molecule SO2 has the same origin as that for the local-mode limit molecule H2O.

Algebraic force-field Hamiltonian expansion approach to linear polyatomic molecules

The algebraic force-field Hamiltonian expansion recently proposed for vibrations of bent triatomic molecules [T. Sako, K. Yamanouchi, and F. Iachello, Chem. Phys. Lett. 299, 35 (1999)] is extended to linear polyatomic molecules by introducing operators constructed from the unitary algebra U(3). A comparison with the conventional force-field method is made by fitting the experimental vibrational-level energies of CO2. The results show that the algebraic Hamiltonian in local coordinates reproduces the level-energies with a smaller root-mean-square deviation and much faster convergence.

Algebraic force-field Hamiltonian expansion for a strongly anharmonic potential: application to the [formula omitted] state of SO 2

A new, algebraically expanded Hamiltonian is proposed to describe vibrations of strongly anharmonic polyatomic molecules. This Hamiltonian is used to study the vibrational-level energies of the C ̃ state of SO 2. A comparison with the conventional force-field method is made. Results of fitting the experimental vibrational-level energies show that (i) the algebraic Hamiltonian expansion reproduces the experimental level energies with a nearly equal root-mean-square deviation of the force-field expansion but with fewer parameters and that (ii) convergence for the suggested algebraic Hamiltonian expansion is much faster than that of the force-field expansion.