Irredundant Triangular Decomposition

Triangular decomposition is a classic, widely used and well-developed way to represent algebraic varieties with many applications. In particular, there exist sharp degree bounds for a single triangular set in terms of intrinsic data of the variety it represents, and powerful randomized algorithms for computing triangular decompositions using Hensel lifting in the zero-dimensional case and for irreducible varieties. However, in the general case, most of the algorithms computing triangular decompositions produce embedded components, which makes it impossible to directly apply the intrinsic degree bounds. This, in turn, is an obstacle for efficiently applying Hensel lifting due to the higher degrees of the output polynomials and the lower probability of success. In this paper, we give an algorithm to compute an irredundant triangular decomposition of an arbitrary algebraic set $W$ defined by a set of polynomials in C[x_1, x_2, ..., x_n]. Using this irredundant triangular decomposition, we were able to give intrinsic degree bounds for the polynomials appearing in the triangular sets and apply Hensel lifting techniques. Our decomposition algorithm is randomized, and we analyze the probability of success

Fuchs versus Painlev\'e

We briefly recall the Fuchs-Painlev\'e elliptic representation of Painlev\'e VI. We then show that the polynomiality of the expressions of the correlation functions (and form factors) in terms of the complete elliptic integral of the first and second kind, $K$ and $E$, is a straight consequence of the fact that the differential operators corresponding to the entries of Toeplitz-like determinants, are equivalent to the second order operator $L_E$ which has $E$ as solution (or, for off-diagonal correlations to the direct sum of $L_E$ and $d/dt$). We show that this can be generalized, mutatis mutandis, to the anisotropic Ising model. The singled-out second order linear differential operator $L_E$ being replaced by an isomonodromic system of two third-order linear partial differential operators associated with $\Pi_1$, the Jacobi's form of the complete elliptic integral of the third kind (or equivalently two second order linear partial differential operators associated with Appell functions, where one of these operators can be seen as a deformation of $L_E$). We finally explore the generalizations, to the anisotropic Ising models, of the links we made, in two previous papers, between Painlev\'e non-linear ODE's, Fuchsian linear ODE's and elliptic curves. In particular the elliptic representation of Painlev\'e VI has to be generalized to an ``Appellian'' representation of Garnier systems.Comment: Dedicated to the : Special issue on Symmetries and Integrability of Difference Equations, SIDE VII meeting held in Melbourne during July 200

Muon Physics: A Pillar of the Standard Model

Since its discovery in the 1930s, the muon has played an important role in our quest to understand the sub-atomic theory of matter. The muon was the first second-generation standard-model particle to be discovered, and its decay has provided information on the (Vector -Axial Vector) structure of the weak interaction, the strength of the weak interaction, G_F, and the conservation of lepton number (flavor) in muon decay. The muon's anomalous magnetic moment has played an important role in restricting theories of physics beyond the standard standard model, where at present there is a 3.4 standard-deviation difference between the experiment and standard-model theory. Its capture on the atomic nucleus has provided valuable information on the modification of the weak current by the strong interaction which is complementary to that obtained from nuclear beta decay.Comment: 8 pages, 9 figures. Invited paper for the Journal of Physical Society in Japan (JPSJ), Special Topics Issue "Frontiers of Elementary Particle Physics, The Standard Model and beyond

On the Regularity Property of Differential Polynomials Modulo Regular Differential Chains

International audienceThis paper provides an algorithm which computes the normal form of a rational differential fraction modulo a regular differential chain if, and only if, this normal form exists. A regularity test for polynomials modulo regular chains is revisited in the nondifferential setting and lifted to differential algebra. A new characterization of regular chains is provided

Kochen-Specker Vectors

We give a constructive and exhaustive definition of Kochen-Specker (KS) vectors in a Hilbert space of any dimension as well as of all the remaining vectors of the space. KS vectors are elements of any set of orthonormal states, i.e., vectors in n-dim Hilbert space, H^n, n>3 to which it is impossible to assign 1s and 0s in such a way that no two mutually orthogonal vectors from the set are both assigned 1 and that not all mutually orthogonal vectors are assigned 0. Our constructive definition of such KS vectors is based on algorithms that generate MMP diagrams corresponding to blocks of orthogonal vectors in R^n, on algorithms that single out those diagrams on which algebraic 0-1 states cannot be defined, and on algorithms that solve nonlinear equations describing the orthogonalities of the vectors by means of statistically polynomially complex interval analysis and self-teaching programs. The algorithms are limited neither by the number of dimensions nor by the number of vectors. To demonstrate the power of the algorithms, all 4-dim KS vector systems containing up to 24 vectors were generated and described, all 3-dim vector systems containing up to 30 vectors were scanned, and several general properties of KS vectors were found.Comment: 19 pages, 6 figures, title changed, introduction thoroughly rewritten, n-dim rotation of KS vectors defined, original Kochen-Specker 192 (117) vector system translated into MMP diagram notation with a new graphical representation, results on Tkadlec's dual diagrams added, several other new results added, journal version: to be published in J. Phys. A, 38 (2005). Web page: http://m3k.grad.hr/pavici

Differential-Algebraic Integrability Analysis of the Generalized Riemann Type and Korteweg-de Vries Hydrodynamical Equations

A differential-algebraic approach to studying the Lax type integrability of the generalized Riemann type hydrodynamic equations at N = 3; 4 is devised. The approach is also applied to studying the Lax type integrability of the well known Korteweg-de Vries dynamical system.Comment: 11 page

Enhancing lepton flavour violation in the supersymmetric inverse seesaw beyond the dipole contribution

In minimal supersymmetric models the $Z$-penguin usually provides sub-dominant contributions to charged lepton flavour violating observables. In this study, we consider the supersymmetric inverse seesaw in which the non-minimal particle content allows for dominant contributions of the $Z$-penguin to several lepton flavour violating observables. In particular, and due to the low-scale (TeV) seesaw, the penguin contribution to, for instance, \Br(\mu \to 3e) and $\mu-e$ conversion in nuclei, allows to render some of these observables within future sensitivity reach. Moreover, we show that in this framework, the $Z$-penguin exhibits the same non-decoupling behaviour which had previously been identified in flavour violating Higgs decays in the Minimal Supersymmetric Standard Model.Comment: 29 pages, 9 figures, 4 tables; v2: minor corrections, version to appear in JHE

Precise Measurement of the Pi+ -> Pi0 e+ nu Branching Ratio

Using a large acceptance calorimeter and a stopped pion beam we have made a precise measurement of the rare Pi+ -> Pi0 e+ Nu,(pi_beta) decay branching ratio. We have evaluated the branching ratio by normalizing the number of observed pi_beta decays to the number of observed Pi+ -> e+ Nu, (pi_{e2}) decays. We find the value of Gamma(Pi+ -> Pi0 e+ Nu)/Gamma(total) = [1.036 +/- 0.004(stat.) +/- 0.004(syst.) +/- 0.003(pi_{e2})] x 10^{-8}$, where the first uncertainty is statistical, the second systematic, and the third is the pi_{e2} branching ratio uncertainty. Our result agrees well with the Standard Model prediction.Comment: 4 pages, 5 figures, 1 table, revtex4; changed content; updated analysi

Examining leptogenesis with lepton flavor violation and the dark matter abundance

Within a supersymmetric (SUSY) type-I seesaw framework with flavor-blind universal boundary conditions, we study the consequences of requiring that the observed baryon asymmetry of the Universe be explained by either thermal or non-thermal leptogenesis. In the former case, we find that the parameter space is very constrained. In the bulk and stop-coannihilation regions of mSUGRA parameter space (that are consistent with the measured dark matter abundance), lepton flavor-violating (LFV) processes are accessible at MEG and future experiments. However, the very high reheat temperature of the Universe needed after inflation (of about 10^{12} GeV) leads to a severe gravitino problem, which disfavors either thermal leptogenesis or neutralino dark matter. Non-thermal leptogenesis in the preheating phase from SUSY flat directions relaxes the gravitino problem by lowering the required reheat temperature. The baryon asymmetry can then be explained while preserving neutralino dark matter, and for the bulk or stop-coannihilation regions LFV processes should be observed in current or future experiments.Comment: 20 pages, 5 figures, 1 tabl

Minimal flavour violation extensions of the seesaw

We analyze the most natural formulations of the minimal lepton flavour violation hypothesis compatible with a type-I seesaw structure with three heavy singlet neutrinos N, and satisfying the requirement of being predictive, in the sense that all LFV effects can be expressed in terms of low energy observables. We find a new interesting realization based on the flavour group $SU(3)_e\times SU(3)_{\ell+N}$ (being $e$ and $\ell$ respectively the SU(2) singlet and doublet leptons). An intriguing feature of this realization is that, in the normal hierarchy scenario for neutrino masses, it allows for sizeable enhancements of $\mu \to e$ transitions with respect to LFV processes involving the $\tau$ lepton. We also discuss how the symmetries of the type-I seesaw allow for a strong suppression of the N mass scale with respect to the scale of lepton number breaking, without implying a similar suppression for possible mechanisms of N productionComment: 14 pages, 6 figure