Relative parametrization of linear multidimensional systems

In the last chapter of his book "The Algebraic Theory of Modular Systems " published in 1916, F. S. Macaulay developped specific techniques for dealing with " unmixed polynomial ideals " by introducing what he called " inverse systems ". The purpose of this paper is to extend such a point of view to differential modules defined by linear multidimensional systems, that is by linear systems of ordinary differential (OD) or partial differential (PD) equations of any order, with any number of independent variables, any number of unknowns and even with variable coefficients in a differential field. The first and main idea is to replace unmixed polynomial ideals by " pure differential modules ". The second idea is to notice that a module is 0-pure if and only if it is torsion-free and thus if and only if it admits an " absolute parametrization " by means of arbitrary potential like functions, or, equivalently, if it can be embedded into a free module by means of an " absolute localization ". The third idea is to refer to a difficult theorem of algebraic analysis saying that an r-pure module can be embedded into a module of projective dimension equal to r, that is a module admitting a projective resolution with exactly r operators. The fourth and final idea is to establish a link between the use of extension modules for such a purpose and specific formal properties of the underlying multidimensional system through the use of involution and a "relative localization " leading to a "relative parametrization ", that is to the use of potential-like functions satisfying a kind of "minimum differential constraint " limiting, in some sense, the number of independent variables appearing in these functions, in a way similar to the situation met in the Cartan-K\"ahler theorem of analysis. The paper is written in a rather effective self-contained way and we provide many explicit examples that should become test examples for a future use of computer algebra.Comment: Presented for publication in the Springer journal MSSP:Multidimensional Systems and Signal Processin

Towards a fully automated computation of RG-functions for the 3-dd O(N) vector model: Parametrizing amplitudes

Within the framework of field-theoretical description of second-order phase transitions via the 3-dimensional O(N) vector model, accurate predictions for critical exponents can be obtained from (resummation of) the perturbative series of Renormalization-Group functions, which are in turn derived --following Parisi's approach-- from the expansions of appropriate field correlators evaluated at zero external momenta. Such a technique was fully exploited 30 years ago in two seminal works of Baker, Nickel, Green and Meiron, which lead to the knowledge of the $\beta$-function up to the 6-loop level; they succeeded in obtaining a precise numerical evaluation of all needed Feynman amplitudes in momentum space by lowering the dimensionalities of each integration with a cleverly arranged set of computational simplifications. In fact, extending this computation is not straightforward, due both to the factorial proliferation of relevant diagrams and the increasing dimensionality of their associated integrals; in any case, this task can be reasonably carried on only in the framework of an automated environment. On the road towards the creation of such an environment, we here show how a strategy closely inspired by that of Nickel and coworkers can be stated in algorithmic form, and successfully implemented on the computer. As an application, we plot the minimized distributions of residual integrations for the sets of diagrams needed to obtain RG-functions to the full 7-loop level; they represent a good evaluation of the computational effort which will be required to improve the currently available estimates of critical exponents.Comment: 54 pages, 17 figures and 4 table

Two-particle interferometry for non-central heavy-ion collisions

In non-central heavy ion collisions, identical two particle Hanbury-Brown/Twiss (HBT) correlations C(K,q) depend on the azimuthal direction of the pair momentum K. We investigate the consequences for a harmonic analysis of the corresponding HBT radius parameters. Our discussion includes both, a model- independent analysis of these parameters in the Gaussian approximation, and the study of a class of hydrodynamical models which mimic essential geometrical and dynamical properties of peripheral heavy ion collisions. Also, we discuss the additional geometrical and dynamical information contained in the harmonic coefficients of these HBT radius parameters. The leading contribution of their first and second harmonics are found to satisfy simple constraints. This allows for a minimal, azimuthally sensitive parametrization of all first and second harmonic coefficients in terms of only two additional fit parameters. We determine to what extent these parameters can be extracted from experimental data despite finite multiplicity fluctuations and the resulting uncertainty in the reconstruction of the reaction plane.Comment: 14 pages, RevTeX, 7 eps-figures include

The equivalence of information-theoretic and likelihood-based methods for neural dimensionality reduction

Stimulus dimensionality-reduction methods in neuroscience seek to identify a low-dimensional space of stimulus features that affect a neuron's probability of spiking. One popular method, known as maximally informative dimensions (MID), uses an information-theoretic quantity known as "single-spike information" to identify this space. Here we examine MID from a model-based perspective. We show that MID is a maximum-likelihood estimator for the parameters of a linear-nonlinear-Poisson (LNP) model, and that the empirical single-spike information corresponds to the normalized log-likelihood under a Poisson model. This equivalence implies that MID does not necessarily find maximally informative stimulus dimensions when spiking is not well described as Poisson. We provide several examples to illustrate this shortcoming, and derive a lower bound on the information lost when spiking is Bernoulli in discrete time bins. To overcome this limitation, we introduce model-based dimensionality reduction methods for neurons with non-Poisson firing statistics, and show that they can be framed equivalently in likelihood-based or information-theoretic terms. Finally, we show how to overcome practical limitations on the number of stimulus dimensions that MID can estimate by constraining the form of the non-parametric nonlinearity in an LNP model. We illustrate these methods with simulations and data from primate visual cortex

Airy, Beltrami, Maxwell, Morera, Einstein and Lanczos potentials revisited

The main purpose of this paper is to revisit the well known potentials, called stress functions, needed in order to study the parametrizations of the stress equations, respectively provided by G.B. Airy (1863) for 2-dimensional elasticity, then by E. Beltrami (1892), J.C. Maxwell (1870) and G. Morera (1892) for 3-dimensional elasticity, finally by A. Einstein (1915) for 4-dimensional elasticity, both with a variational procedure introduced by C. Lanczos (1949,1962) in order to relate potentials to Lagrange multipliers. Using the methods of Algebraic Analysis, namely mixing differential geometry with homological algebra and combining the double duality test involved with the Spencer cohomology, we shall be able to extend these results to an arbitrary situation with an arbitrary dimension n. We shall also explain why double duality is perfectly adapted to variational calculus with differential constraints as a way to eliminate the corresponding Lagrange multipliers. For example, the canonical parametrization of the stress equations is just described by the formal adjoint of the n2(n2 -- 1)/12 components of the linearized Riemann tensor considered as a linear second order differential operator but the minimum number of potentials needed in elasticity theory is equal to n(n -- 1)/2 for any minimal parametrization. Meanwhile, we can provide all the above results without even using indices for writing down explicit formulas in the way it is done in any textbook today. The example of relativistic continuum mechanics with n = 4 is provided in order to prove that it could be strictly impossible to obtain such results without using the above methods. We also revisit the possibility (Maxwell equations of electromag- netism) or the impossibility (Einstein equations of gravitation) to obtain canonical or minimal parametrizations for various other equations of physics. It is nevertheless important to notice that, when n and the algorithms presented are known, most of the calculations can be achieved by using computers for the corresponding symbolic computations. Finally, though the paper is mathematically oriented as it aims providing new insights towards the mathematical foundations of elasticity theory and mathematical physics, it is written in a rather self-contained way

Two-dimensional nuclear inertia : analytical relationships

The components of the nuclear inertia tensor, functions of the separation distance R and of the radius of the light fragment R2, BRR(R,R2), BRR2(R,R2), and BR2R2(R,R2) are calculated within the Werner-Wheeler approximation, by using the parametrization of two intersected symmetric or asymmetric spheres. Analytical relationships are derived. When projected to a path R2=R2(R), the reduced mass is obtained at the touching point. The two one-dimensional parametrizations with R2=const, and the volume V2=const previously studied, are found to be particular cases of the present more general approach. Illustrations for the cold fission, cluster radioactivity, and α decay of 252Cf are given

Pion Interferemetry from p+p to Au+Au in STAR

The geometric substructure of the particle-emitting source has been characterized via two-particle interferometry by the STAR collaboration for all energies and colliding systems at RHIC. We present systematic studies of charged pion interferometry. The collective nature of the source is revealed through the $m_T$ dependence of HBT radii for all particle types. Preliminary results suggest a scaling in the pion HBT radii with overall system size, as central Au+Au collisions are compared to peripheral collisions as well as with Cu+Cu and even with d+Au and p+p collisions, naively suggesting comparable flow strength in all systems. To probe this issue in greater detail, multidimensional correlation functions are studied using a spherical decomposition method. This allows clear identification of source anisotropy and, for the light systems, the presence of significant long-range non-femtoscopic correlations.Comment: Proceedings for WPCF, Kromeriz, Czech Republic, August 200