Extended GCD and Hermite normal form algorithms via lattice basis reduction

Extended gcd calculation has a long history and plays an important role in computational number theory and linear algebra. Recent results have shown that finding optimal multipliers in extended gcd calculations is difficult. We present an algorithm which uses lattice basis reduction to produce small integer multipliers x(1), ..., x(m) for the equation s = gcd (s(1), ..., s(m)) = x(1)s(1) + ... + x(m)s(m), where s1, ... , s(m) are given integers. The method generalises to produce small unimodular transformation matrices for computing the Hermite normal form of an integer matrix.

On some geometry and equivalence classes of normal form games

Equivalence classes of normal form games are defined using the geometryof correspondences of standard equilibiurm concepts like correlated, Nash,and robust equilibrium or risk dominance and rationalizability. Resultingequivalence classes are fully characterized and compared across differentequilibrium concepts for 2 x 2 games. It is argued that the procedure canlead to broad and game-theoretically meaningful distinctions of games aswell as to alternative ways of viewing and testing equilibrium concepts.Larger games are also briefly considered.

Stochastic evolution of rules for playing normal form games

The evolution of boundedly rational rules for playing normal form games is studied within stationary environments ofstochastically changing games. Rules are viewed as algorithms prescribing strategies for the different normal formgames that arise. It is shown that many of the folk results of evolutionary game theory typically obtained witha fixed game and fixed strategies carry over to the present case. The results are also related to recent experimentson rules and games.

Diagonalization and Jordan Normal Form – motivation through Maple

Following an introduction to the diagonalization of matrices, one of the more difficult topics for students to grasp in linear algebra is the concept of Jordan normal form. In this note, we show how the important notions of diagonalization and Jordan normal form can be introduced and developed through the use of the computer algebra package Maple®.

An application of lattice basis reduction to polynomial identities for algebraic structures

The authors` recent classification of trilinear operations includes, among other cases, a fourth family of operations with parameter q epsilon Q boolean OR {infinity}, and weakly commutative and weakly anticommutative operations. These operations satisfy polynomial identities in degree 3 and further identities in degree 5. For each operation, using the row canonical form of the expansion matrix E to find the identities in degree 5 gives extremely complicated results. We use lattice basis reduction to simplify these identities: we compute the Hermite normal form H of E(t), obtain a basis of the nullspace lattice from the last rows of a matrix U for which UE(t) = H, and then use the LLL algorithm to reduce the basis. (C) 2008 Elsevier Inc. All rights reserved.

Nonhomogeneous subalgebras of Lie and special Jordan superalgebras

We consider polynomial identities satisfied by nonhomogeneous subalgebras of Lie and special Jordan superalgebras: we ignore the grading and regard the superalgebra as an ordinary algebra. The Lie case has been studied by Volichenko and Baranov: they found identities in degrees 3, 4 and 5 which imply all the identities in degrees <= 6. We simplify their identities in degree 5, and show that there are no new identities in degree 7. The Jordan case has not previously been studied: we find identities in degrees 3, 4, 5 and 6 which imply all the identities in degrees <= 6, and demonstrate the existence of further new identities in degree 7. our proofs depend on computer algebra: we use the representation theory of the symmetric group, the Hermite normal form of an integer matrix, the LLL algorithm for lattice basis reduction, and the Chinese remainder theorem. (C) 2009 Elsevier Inc. All rights reserved.

Global periodicity conditions for maps and recurrences via Normal Forms

We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two and three dimensional classes of polynomial or rational maps. In particular we find the global periodic cases for several Lyness type recurrences

L'Escola Normal de Mestres en els seus origens i el reformisme liberal. 'La Escuela Normal de Maestros en sus orígenes y el reformismo liberal'.

Presenta un apéndice documental que recoge documentación oficial de la época analizada

Classical and quantal aspects of resonant normal forms

The Birkhoff-Gustavson normal form is employed to study separately chaos and resonances in a system with two degrees of freedom. In the integrable regime, tunnelling effects are appreciable when the nearest level spacings show oscillations. Tunnelling among states in the libration and rotation tori regions is also observed. The regularity of avoided crossings due to tunnelling indicates a collective effect and is associated with an isolated resonance. The spectral fluctuations also show a strong level correlation. The Husimi distribution, on the other hand, is insensitive to avoided crossings. An integrable approximation to the overlap of resonances is obtained and a theoretical description is given for an isolated cubic resonance plus a complex orbit. In the non-integrable regime chaos is stronger after overlapping and preferentially at low energies.

Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Mice deficient for prion protein exhibit normal neuronal excitability and synaptic transmission in the hippocampus.

We recorded in the CA1 region from hippocampal slices of prion protein (PrP) gene knockout mice to investigate whether the loss of the normal form of prion protein (PrPC) affects neuronal excitability as well as synaptic transmission in the central nervous system. No deficit in synaptic inhibition was found using field potential recordings because (i) responses induced by stimulation in stratum radiatum consisted of a single population spike in PrP gene knockout mice similar to that recorded from control mice and (ii) the plot of field excitatory postsynaptic potential slope versus the population spike amplitude showed no difference between the two groups of mice. Intracellular recordings also failed to detect any difference in cell excitability and the reversal potential for inhibitory postsynaptic potentials. Analysis of the kinetics of inhibitory postsynaptic current revealed no modification. Finally, we examined whether synaptic plasticity was altered and found no difference in long-term potentiation between control and PrP gene knockout mice. On the basis of our findings, we propose that the loss of the normal form of prion protein does not alter the physiology of the CA1 region of the hippocampus.

Some remarks on static-feedback linearization for time-varying systems

This work summarizes some results about static state feedback linearization for time-varying systems. Three different necessary and sufficient conditions are stated in this paper. The first condition is the one by [Sluis, W. M. (1993). A necessary condition for dynamic feedback linearization. Systems & Control Letters, 21, 277-283]. The second and the third are the generalizations of known results due respectively to [Aranda-Bricaire, E., Moog, C. H., Pomet, J. B. (1995). A linear algebraic framework for dynamic feedback linearization. IEEE Transactions on Automatic Control, 40, 127-132] and to [Jakubczyk, B., Respondek, W. (1980). On linearization of control systems. Bulletin del` Academie Polonaise des Sciences. Serie des Sciences Mathematiques, 28, 517-522]. The proofs of the second and third conditions are established by showing the equivalence between these three conditions. The results are re-stated in the infinite dimensional geometric approach of [Fliess, M., Levine J., Martin, P., Rouchon, P. (1999). A Lie-Backlund approach to equivalence and flatness of nonlinear systems. IEEE Transactions on Automatic Control, 44(5), 922-937]. (C) 2008 Elsevier Ltd. All rights reserved.

Integer matrix diagonalization

We consider algorithms for computing the Smith normal form of integer matrices. A variety of different strategies have been proposed, primarily aimed at avoiding the major obstacle that occurs in such computations-explosive growth in size of intermediate entries. We present a new algorithm with excellent performance. We investigate the complexity of such computations, indicating relationships with NP-complete problems. We also describe new heuristics which perform well in practice. Wie present experimental evidence which shows our algorithm outperforming previous methods. (C) 1997 Academic Press Limited.