on the measure of totally real algebraic integers ii for a certain measure s2 a defined for totally real algebraic integers a 0 1 we find the four smallest values of q a the methods used involve linear programming and the results are verified using interval arithmetic

taylor vortex flow a dynamical system taylor vortex flows have been studied over a period of more than sixty years they occur between two concentric circular cylinders for example when the inner cylinder rotates and the outer is at rest and the flow takes on a toroidal vortex form with periodicity or quasi periodicity along the axis complex variations on this simple description occur when the angular speed is raised or when the length of the apparatus is reduced a rather special form of turbulence can occur and there may be some relation to chaos in the sense of ordinary differential equations as a dynamical system the experimental and theoretical evidence concerned with these phenomena is discussed

a generalization of the schur siegel smyth trace problem abstract let be a totally positive algebraic integer and define its absolute trace to be tr deg the trace of divided by the degree of elementary considerations show that the absolute trace is always at least one while it is plausible that for any 0 the absolute trace is at least 2 with only finitely many exceptions this is known as the schur siegel smyth trace problem our aim in this paper is to show that the schur siegel smyth trace problem can be considered as a special case of a more general problem

camera self calibration from bivariate polynomials derived from kruppa s equations this paper presents a new set of equations for the self calibration of a moving camera with constant intrinsic parameters unlike most existing methods that require solving equations in three or more unknowns the proposed equations are only bivariate in particular we show that the three scale factors appearing in the kruppa s equations that are due to a triplet of images are not independent but rather closely related this relationship is used to derive sextic bivariate polynomial equations and allow the recovery of the unknown scale factors using a homotopy continuation method once the scale factors are calculated an estimate of kruppa s coefficients can be linearly retrieved and then refined through a nonlinear least squares optimization procedure the results of our experiments conducted on simulated data as well as the three dimensional structure reconstruction using real images are also presented in the paper

on the spectrum of the zhang zagier height from recent work of zhang and of zagier we know that their height n is bounded away from 1 for every algebraic number a different from 0 1 1 2 3 2 the study of the related spectrum is especially interesting for it is linked to lehmer s problem and to a conjecture of bogomolov after recalling some definitions we show an improvement of the so called zhang zagier inequality to achieve this we need some algebraic numbers of small height so in the third section we describe an algorithm able to find them and we give an algebraic number with height 1 2875274z discovered in this way this search up to degree 64 suggests that the spectrum of n may have a limit point less than 1 292 we prove this fact in the fourth part

the largest liapunov exponent for random matrices and directed polymers in a random environment we study the largest liapunov exponent for products of random matrices the two classes of matrices considered are discrete d dimensional laplacians with random entries and symplectic matrices that arise in the study ofd dimensional lattices of coupled nonlinear oscillators we derive bounds on this exponent for all dimensions d and we show that ifd 3 and the randomness is not too strong one can obtain an explicit formula for the largest exponent in the thermodynamic limit our method is based on an equivalence between this problem and the problem of directed polymers in a random environment

globally convergent autocalibration using interval analysis we address the problem of autocalibration of a moving camera with unknown constant intrinsic parameters existing autocalibration techniques use numerical optimization algorithms whose convergence to the correct result cannot be guaranteed in general to address this problem we have developed a method where an interval branch and bound method is employed for numerical minimization thanks to the properties of interval analysis this method converges to the global solution with mathematical certainty and arbitrary accuracy and the only input information it requires from the user are a set of point correspondences and a search interval the cost function is based on the huang faugeras constraint of the essential matrix a recently proposed interval extension based on bernstein polynomial forms has been investigated to speed up the search for the solution finally experimental results are presented

sums of heights of algebraic numbers for at x f x tg x we consider the set at 0 h t q the polynomials f x g x are in z x with only mild restrictions and h is the weil height of we show that this set is dense in d for some effectively computable limit point d

influence of inlet and bulk noise on rayleigh benard convection with lateral flow spatiotemporal properties of convective fluctuations and of their correlations are investigated theoretically in the vicinity of the threshold for onset of convection in the presence of a lateral through flow using the full linearized equations of fluctuating hydrodynamics the effect of external forcing by inlet boundary conditions on the downstream evolution of convective fields is separated from the effect of internal bulk thermal forcing with the use of spatial laplace transformations they show how the spatial variation of fluctuations and of their correlations are governed by the six spatial characteristic exponents of the field equations

families of subcritical spirals in highly counter rotating taylor couette flow a comprehensive numerical exploration of secondary finite amplitude solutions in small gap taylor couette flow for high counter rotating reynolds numbers is provided using newton krylov methods embedded within arclength continuation schemes two different families of rotating waves have been identified short axial wavelength subcritical spirals ascribed to centrifugal mechanisms and large axial scale supercritical spirals and ribbons associated with shear dynamics in the outer linearly stable radial region this study is a first step taken in order to provide the inner structure of the skeleton of equilibria that may be responsible for the intermittent regime usually termed as spiral turbulence that has been reported by many experimentalists in the past

characterization of the spatial complex behavior and transition to chaos in flow systems abstract we introduce a spatial lyapunov exponent to characterize the complex behavior of non chaotic but convectively unstable flow sytems this complexity is of spatial type and is due to sensitivity to the boundary conditions we show that there exists a relation between the spatial complexity index we define and the comoving lyapunov exponents in such systems the transition to chaos i e the occurrence of a positive lyapunov exponent can manifest itself in two different ways in the first case from neither chaotic nor spatially complex behavior to chaos one observes the typical scenario i e as the system size grows up the spectrum of the lyapunov exponents gives rise to a density in the second case when the chaos develops from a convectively unstable situation one observes only a finite number of positive lyapunov exponents

the integer chebyshev problem the main problem of this chapter is to find a polynomial in z n of minimal supremum norm on an interval this is p1 and it is of a slightly different flavour than most of the other problems in this book in that there is no restriction on the size of the coefficients we now state p1 with greater precision

pattern selection in the absolutely unstable regime as a nonlinear eigenvalue problem taylor vortices in axial flow a unique pattern selection in the absolutely unstable regime of a driven nonlinear open flow system is analyzed the spatiotemporal structures of rotationally symmetric vortices that propagate downstream in the annulus of the rotating taylor couette system due to an externally imposed axial through flow are investigated for two different axial boundary conditions at the inlet and outlet detailed quantitative results for the oscillation frequency the axial profile of the wave number and the temporal fourier amplitudes of the propagating vortex patterns obtained by numerical simulations of the navier stokes equations are compared with results of the appropriate ginzburg landau amplitude equation approximation and also with experiments unlike the stationary patterns in systems without through flow the spatiotemporal structures of propagating vortices are independent of parameter history initial conditions and system length they do however depend on the axial boundary conditions in addition to the driving rate of the inner cylinder and the through flow rate our analysis of the amplitude equation shows that the pattern selection can be described by a nonlinear eigenvalue problem with the frequency being the eigenvalue the complex amplitude being the corresponding eigenfunction describes the axial structure of intensity and wave number small but characteristic differences in the structural dynamics between the navier stokes equations and the amplitude equation are mainly due to the different dispersion relations approaching the border between absolute and convective instability the eigenvalue problem becomes effectively linear and the selection mechanism approaches that of linear front propagation

globally optimal algorithms for stratified autocalibration we present practical algorithms for stratified autocalibration with theoretical guarantees of global optimality given a projective reconstruction we first upgrade it to affine by estimating the position of the plane at infinity the plane at infinity is computed by globally minimizing a least squares formulation of the modulus constraints in the second stage this affine reconstruction is upgraded to a metric one by globally minimizing the infinite homography relation to compute the dual image of the absolute conic diac the positive semidefiniteness of the diac is explicitly enforced as part of the optimization process rather than as a post processing step for each stage we construct and minimize tight convex relaxations of the highly non convex objective functions in a branch and bound optimization framework we exploit the inherent problem structure to restrict the search space for the diac and the plane at infinity to a small fixed number of branching dimensions independent of the number of views chirality constraints are incorporated into our convex relaxations to automatically select an initial region which is guaranteed to contain the global minimum experimental evidence of the accuracy speed and scalability of our algorithm is presented on synthetic and real data

orbits of algebraic numbers with low heights we prove that the two smallest values of h h 1 1 h 1 1 are 0 and 0 4218 for any algebraic integer

two new points in the spectrum of the absolute mahler measure of totally positive algebraic integers for totally positive algebraic integers 0 1 of degree d we consider the set l of values of m 1 d where m is the mahler measure of c j smyth has found the four smallest values of l and conjectured that the fifth point is 2 cos 2 60 2 we prove that this is so and moreover we give the sixth point of l

on the linear stability of spiral flow between rotating cylinders a numerical study is made of the effects of both axisymmetric and non axisymmetric disturbances on the stability of spiral flow between rotating cylinders if we let omega 1 and omega 2 be the angular speeds of the inner and outer cylinders and r 1 and r 2 be their respective radii then for fixed values of eta r 1 r 2 and mu omega 2 omega 1 the onset of instability depends on both the taylor number t and the axial reynolds number r here r is based on the gap width between the cylinders and the average axial velocity of the basic flow while t is based on the average angular speeds of the cylinders using the compound matrix method we have computed the complete stability boundary in the r t plane for axisymmetric disturbances with eta 0 95 and mu 0 we find that for sufficiently high reynolds numbers there are two distinct axisymmetric modes corresponding to the usual shear and rotational instabilities we have also obtained the stability boundaries for non axisymmetric disturbances for r leqslant 6000 for eta 0 95 and 0 77 with mu 0 these last results are found to be in substantial agreement with the experimental observations of snyder 1962 1965 nagih 1972 and mavec 1973 in the low and moderate axial reynolds number regimes

stability and noise in taylor couette flow with through flow abstract we present results of experimental and numerical work on a taylor couette system with imposed axial flow when the base flow is convectively unstable macroscopic patterns of travelling taylor vortices are observed downstream of the inlet numerical integration of the stochastic complex ginzburg landau equation indicates that these patterns arise from the spatial amplification of microscopic noise the noise leads to an irregular phase in the vortex pattern in contrast the phase is very regular when the base flow is absolutely unstable

imperfect hopf bifurcation in spiral poiseuille flow we present the results of an experimental study on the transition to spiral vortices in flow between concentric counter rotating cylinders in the presence of an axial through flow i e in spiral poiseuille flow the experiments were performed in an apparatus having an aspect ratio ensuremath gamma l d 22 8 l axial length d gap width between cylinders and end plates enabling an in and outflow of mass as a result of an applied axial through flow the classical hopf bifurcation to spiral vortices spi splits up and a primary and secondary branch of down and upstream propagating spi respectively as well as a transient quasiperiodic flow appear downstream propagating spi resulting from the primary supercritical hopf bifurcation are either convectively or absolutely unstable the bifurcation structure observed in this open flow experiment is in qualitative agreement with predictions from theory of hopf bifurcation with broken reflection symmetry j d crawford and e knobloch nonlinearity 1 617 1988 and also in quantitative agreement with results from recent numerical calculations a pinter m l ucke and c hoffmann phys rev e 67 026318 2003 c hoffmann m l ucke and a pinter phys rev e 69 056309 2004

continuous stereo self calibration by camera parameter tracking this paper presents a consistent framework for continuous stereo self calibration based on a practical analysis of the sensitivity of stereo reconstruction to camera calibration uncertainties we identify important parameters for self calibration we evaluate different geometric constraints for estimation and tracking of these parameters bundle adjustment with reduced structure representation relating corresponding points in image sequences the epipolar constraint between stereo image pairs and trilinear constraints between image triplets continuous recursive calibration refinement is obtained with a robust adapted iterated extended kalman filter to achieve high accuracy physically relevant geometric optimization criteria are formulated in a gauss helmert type model the self calibration framework is tested on an active stereo system experiments with synthetic data as well as on natural indoor and outdoor imagery indicate that the different constraints are complementing each other and thus a method combining two of the above constraints is proposed while reduced order bundle adjustment gives by far the most accurate results and might suffice on its own in some environments the epipolar constraint yields instantaneous calibration that is not affected by independently moving objects in the scene hence it expedites and stabilizes the calibration process

spatial and temporal resonances in a periodically forced hydrodynamic system abstract the occurrence of a naimark sacker bifurcation in a periodically forced spatially extended system is analyzed with floquet theory the basic periodic state of the flow in an annular region consists of the superposition of circular couette flow driven by the constant rotation of the inner cylinder and annular stokes flow driven by the harmonic oscillation of the inner cylinder in the axial direction windows in parameter space where onset of instability is via naimark sacker correspond to forcing frequencies close to the natural frequencies of the unforced system when the forcing amplitude is sufficiently large in these windows not only have strong temporal resonances been identified but also competition and resonances between various spatial modes that are simultaneously excited take place an excitation diagram mapping out these windows and loci of resonance points is produced providing a guide for future explorations into the nonlinear regime by either experimental or computational techniques

on the absolute mahler measure of polynomials having all zeros in a sector ii kklet be an algebraic integer of degree d not 0 or a root of unity all of whose conjugates i are confined to a sector arg z in the paper on the absolute mahler measure of polynomials having all zeros in a sector g rhin and c smyth compute the greatest lower bound c of the absolute mahler measure d i 1 max 1 i 1 d of for belonging to nine subintervals of 0 2 3 in this paper we improve the result to thirteen subintervals of 0 and extend some existing subintervals

self calibration and metric reconstruction inspite of varying and unknown intrinsic camera parameters in this paper the theoretical and practical feasibility of self calibration in the presence of varying intrinsic camera parameters is under investigation the paper s main contribution is to propose a self calibration method which efficiently deals with all kinds of constraints on the intrinsic camera parameters within this framework a practical method is proposed which can retrieve metric reconstruction from image sequences obtained with uncalibrated zooming focusing cameras the feasibility of the approach is illustrated on real and synthetic examples besides this a theoretical proof is given which shows that the absence of skew in the image plane is sufficient to allow for self calibration a counting argument is developed which depending on the set of constraints gives the minimum sequence length for self calibration and a method to detect critical motion sequences is proposed

characterisation of intermittency in chaotic systems the authors discuss the characterisation of intermittency in chaotic dynamical systems by means of the time fluctuations of the response to a slight perturbation on the trajectory a set of exponents is introduced which generalise the maximum lyapunov characteristic exponent the link with the statistical mechanics formalism is emphasised and they show that the exponents are connected to a free energy formally defined for chaotic systems they perform some analytical computations in simple cases and give a few numerical examples

constrained structure and motion from multiple uncalibrated views of a piecewise planar scene this paper is about multi view modeling of a rigid scene we merge the traditional approaches of reconstructing image extractable features and of modeling via user provided geometry we use features to obtain a first guess for structure and motion fit geometric primitives correct the structure so that reconstructed features lie exactly on geometric primitives and optimize both structure and motion in a bundle adjustment manner while enforcing the underlying constraints we specialize this general scheme to the point features and the plane geometric primitives the underlying geometric relationships are described by multi coplanarity constraints we propose a minimal parameterization of the structure enforcing these constraints and use it to devise the corresponding maximum likelihood estimator the recovered primitives are then textured from the input images the result is an accurate and photorealistic model experimental results using simulated data confirm that the accuracy of the model using the constrained methods is of clearly superior quality compared to that of traditional methods and that our approach performs better than existing ones for various scene configurations in addition we observe that the method still performs better in a number of configurations when the observed surfaces are not exactly planar we also validate our method using real images

a theory of self calibration of a moving camera there is a close connection between the calibration of a single camera and the epipolar transformation obtained when the camera undergoes a displacement the epipolar transformation imposes two algebraic constraints on the camera calibration if two epipolar transformations arising from different camera displacements are available then the compatible camera calibrations are parameterized by an algebraic curve of genus four the curve can be represented either by a space curve of degree seven contained in the intersection of two cubic surfaces or by a curve of degree six in the dual of the image plane the curve in the dual plane has one singular point of order three and three singular points of order two if three epipolar transformations are available then two curves of degree six can be obtained in the dual plane such that one of the real intersections of the two yields the correct camera calibration the two curves have a common singular point of order three experimental results are given to demonstrate the feasibility of camera calibration based on the epipolar transformation the real intersections of the two dual curves are found by locating the zeros of a function defined on the interval 0 2 the intersection yielding the correct camera calibration is picked out by referring back to the three epipolar transformations

influence of boundaries on pattern selection in through flow abstract the problem of pattern selection in absolutely unstable open flow systems is investigated by considering the example of rayleigh benard convection the spatio temporal structure of convection rolls propagating downstream in an externally imposed flow is determined for six different inlet outlet boundary conditions results are obtained by numerical simulations of the navier stokes equations and by comparison with the corresponding ginzburg landau amplitude equation a unique selection process is observed being a function of the control parameters and the boundary conditions but independent of history and system length the problem can be formulated in terms of a nonlinear eigen boundary value problem where the frequency of the propagating pattern is the eigenvalue

a case against kruppa s equations for camera self calibration we consider the self calibration problem for perspective cameras and especially the classical kruppa equation approach it is known that for several common types of camera motion self calibration is degenerate which manifests itself through the existence of ambiguous solutions the author previously 1997 1999 studied these critical motion sequences and showed their importance for practical applications here we reveal a type of camera motion that is not critical for the generic self calibration problem but for which the kruppa equation approach fails this is the case if the optical centers of all cameras lie on a sphere and if the optical axes pass through the sphere s center a very natural situation for 3d object modeling from images results of simulated experiments demonstrate the instability of numerical self calibration algorithms in near degenerate configurations

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

chronotopic lyapunov analysis i a detailed characterization of 1d systems instabilities in 1d spatially extended systems are studied with the aid of both temporal and spatial lyapunov exponents a suitable representation of the spectra allows a compact description of all the possible disturbances in tangent space the analysis is carried out for chaotic and periodic spatiotemporal patterns singularities of the spectra and localization properties of the associated lyapunov vectors are discussed

planar structure from motion with affine camera models closed form solutions ambiguities and degeneracy analysis planar structure from motion sfm is the problem of reconstructing a planar object or surface from a set of 2d images using motion information the problem is well understood with the perspective camera model and can be solved with homography decomposition hd however when the structure is small and or viewed far from the camera the perspective effects diminish and in the limit the projections become affine in these situations hd fails because the problem itself becomes ill posed we propose a stable alternative using affine camera models these have been used extensively to reconstruct non planar structures however a general accurate and closed form method for planar structures has been missing the problem is fundamentally different with planar structures because the types of affine camera models one can use are more restricted and it is inherently more ambiguous and non linear we provide a closed form method for the orthographic camera model that solves the general problem three or more views with three or more correspondences and missing correspondences and returns all metric structure solutions and corresponding camera poses the method does not require initialisation and optimises an objective function that is very similar to the reprojection error in fact there is no clear benefit in refining its solutions with bundle adjustment which is a remarkable result we also present a new theoretical analysis that deepens our understanding of the problem the main result is the necessary and sufficient geometric conditions for the problem to be degenerate with the orthographic camera we also show there can exist up to two solutions for metric structure with four or more views previously it was assumed to be unique and we give the necessary and sufficient geometric conditions for disambiguation other theoretical results include showing that in the case of three images the optimal reconstruction with respect to reprojection error can usually be found in closed form and additional prior knowledge needed to solve with non orthographic affine cameras