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

matching procedure for the sixth painleve equation in the framework of the isomonodromy deformation method we present a constructive procedure to obtain the critical behaviour of painleve vi transcendents and solve the connection problem this procedure yields two and one parameter families of solutions including trigonometric and logarithmic behaviours and three classes of solutions with taylor expansion at a critical point

lyapunov based range identification for paracatadioptric systems many applications require the interpretation of the euclidean coordinates of features of a 3 d object through 2 d images in this paper the relative range and the euclidean coordinates of an object or camera undergoing general affine motion are determined for paracatadioptric imaging systems via a nonlinear observer the nonlinear observer asymptotically determines the range information provided that the motion parameters are known the observer is developed through a lyapunov based design and stability analysis and simulation results are provided that illustrate the performance of the state estimator the contributions of this paper are that the developed observer 1 can be applied to both affine and nonaffine systems 2 can be used for both the fixed and moving camera configurations 3 is continuous and 4 yields an asymptotic result

big line bundles over arithmetic varieties 5 equidistribution theory 26 5 1 a generic equidistribution theorem 27 5 2 equidistribution at infinite places 30 5 3 equidistribution at finite places 32 5 4 equidistribution of small subvarieties 35 5 5 equidistribution over algebraic dynamical systems 37 5 6 equidistribution over multiplicative groups 39 5 7 equidistribution theory 41

sofic groups and diophantine approximation we prove the algebraic eigenvalue conjecture of j dodziuk p linnell v mathai t schick and s yates see 2 for sofic groups moreover we give restrictions on the spectral measure of elements in the integral group ring finally we define integer operators and prove a quantization of the operator norm below 2 to the knowledge of the author there is no group known that is not sofic

a height gap theorem for finite subsets of mathrm gl _ d overline bbb q and nonamenable subgroups we introduce a conjugation invariant normalized heightb f on nite subsets of matrices f in gld q and describe its properties in particular we prove an analogue of the lehmer problem for this height by showing that b f whenever f generates a nonvirtually solvable subgroup of gld q where d 0 is an absolute constant this can be seen as a global adelic analog of the classical margulis lemma from hyperbolic geometry as an application we prove a uniform version of the classical burnside schur theorem on torsion linear groups in a companion paper we will apply these results to prove a strong uniform version of the tits alternative

equidistribution quantitative des points de petite hauteur sur la droite projective we introduce a new class of adelic heights on the projective line we estimate their essential minimum and prove a result of equidistribution at every place for points of small height with estimates on the speed of convergence to each rational function r in one variable and defined over a number field k is associated a normalized height on the algebraic closure of k we show that these dynamically defined heights are adelic in our sense and deduce from this equidistribution results for preimages of points under r at every place of k our approach follows that of bilu and relies on potential theory in the complex plane as well as in the berkovich space associated to the projective line over open image in new window for each prime p

central limit theorems from the roots of probability generating functions abstract for each n let x n 0 n be a random variable with mean n standard deviation n and let p n z k 0 n p x n k z k be its probability generating function we show that if none of the complex zeros of the polynomials p n z is contained in a neighborhood of 1 c and n n e for some e 0 then x n x n n n 1 is asymptotically normal as n that is it tends in distribution to a random variable z n 0 1 on the other hand we show that there exist sequences of random variables x n with n c log n for which p n z has no roots near 1 and x n is not asymptotically normal these results disprove a conjecture of pemantle and improve upon various results in the literature we go on to prove several other results connecting the location of the zeros of p n z and the distribution of the random variable x n

computing monodromy via continuation methods on random riemann surfaces we consider a riemann surface x defined by a polynomial f x y of degree d whose coefficients are chosen randomly hence we can suppose that x is smooth that the discriminant x of f has d d 1 simple roots and that 0 0 i e the corresponding fiber has d distinct points y1 yd when we lift a loop 0 c by a continuation method we get d paths in x connecting y1 yd hence defining a permutation of that set this is called monodromy here we present experimentations in maple to get statistics on the distribution of transpositions corresponding to loops around each point of multiplying families of neighbor transpositions we construct permutations and the subgroups of the symmetric group they generate this allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups assuming that these two conjectures are true we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions

on the absolute mahler measure of polynomials having all zeros in a sector let 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 we compute the greatest lower bound c of the absolute mahler measure i 1 d max 1 i 1 d of for belonging to nine subintervals of 0 2 3 in particular we show that c 2 1 12933793 from which it follows that any integer 1 and e i 3 all of whose conjugates have positive real part has absolute mahler measure at least c 2 this value is achieved for satisfying 1 0 2 where 0 1 3247 is the smallest pisot number the real root of 0 3 0 1

distribution of algebraic numbers schur studied limits of the arithmetic means an of zeros for polynomials of degree n with integer coefficients and simple zeros in the closed unit disk if the leading coefficients are bounded schur proved that lim supn an 1 e 2 we show that an 0 and estimate the rate of convergence by generalizing the erd s turan theorem on the distribution of zeros as an application we show that integer polynomials have some unexpected restrictions of growth on the unit disk schur also studied problems on means of algebraic numbers on the real line when all conjugate algebraic numbers are positive the problem of finding the sharp lower bound for lim infn an was developed further by siegel and others we provide a solution of this problem for algebraic numbers equidistributed in subsets of the real line potential theoretic methods allow us to consider distribution of algebraic numbers in or near general sets in the complex plane we introduce the generalized mahler measure and use it to characterize asymptotic equidistribution of algebraic numbers in arbitrary compact sets of capacity one the quantitative aspects of this equidistribution are also analyzed in terms of the generalized mahler measure 1 schur s problems on means of algebraic numbers let e be a subset of the complex plane c consider the set of polynomials zn e of the exact degree n with integer coefficients and all zeros in e we denote the subset of zn e with simple zeros by zn e given m 0 we write pn anz zn e m if an m and pn zn e respectively pn zn e m if an m and pn zn e schur 45 4 8 studied the limit behavior of the arithmetic means of zeros for polynomials from zn e m as n where m 0 is an arbitrary fixed number his results may be summarized in the following statements let r 0 where r is the real line theorem a schur 45 satz ix given a polynomial pn z an n k 1 z k n define the arithmetic mean of squares of its zeros by sn n k 1 2 k n n if pn zn r m is any sequence of polynomials with degrees n then lim inf n sn e 1 6487 1 1 theorem b schur 45 satz xi for a polynomial pn z an n k 1 z k n define the arithmetic mean of its zeros by an n k 1 k n n if pn zn r m is any sequence of polynomials with degrees n then lim inf n an e 1 6487 1 2 1991 mathematics subject classification primary 11c08 secondary 11r04 26c10 30c15

catadioptric camera calibration using geometric invariants central catadioptric cameras are imaging devices that use mirrors to enhance the field of view while preserving a single effective viewpoint in this paper we propose a novel method for the calibration of central catadioptric cameras using geometric invariants lines and spheres in space are all projected into conics in the catadioptric image plane we prove that the projection of a line can provide three invariants whereas the projection of a sphere can only provide two from these invariants constraint equations for the intrinsic parameters of catadioptric camera are derived therefore there are two kinds of variants of this novel method the first one uses projections of lines and the second one uses projections of spheres in general the projections of two lines or three spheres are sufficient to achieve catadioptric camera calibration one important conclusion in this paper is that the method based on projections of spheres is more robust and has higher accuracy than that based on projections of lines the performances of our method are demonstrated by both the results of simulations and experiments with real images

calibration of ultrawide fisheye lens cameras by eigenvalue minimization we present a new technique for calibrating ultrawide fisheye lens cameras by imposing the constraint that collinear points be rectified to be collinear parallel lines to be parallel and orthogonal lines to be orthogonal exploiting the fact that line fitting reduces to an eigenvalue problem in 3d we do a rigorous perturbation analysis to obtain a practical calibration procedure doing experiments we point out that spurious solutions exist if collinearity and parallelism alone are imposed our technique has many desirable properties for example no metric information is required about the reference pattern or the camera position and separate stripe patterns can be displayed on a video screen to generate a virtual grid eliminating the grid point extraction processing

the distribution of galois orbits of points of small height in toric varieties we study the distribution of galois orbits of points of small height on proper toric varieties and its application to the bogomolov problem we introduce the notion of monocritical toric metrized divisor we prove that a toric metrized divisor d on a proper toric variety x over a global eld k is monocritical if and only if for every generic d small sequence of algebraic points of x and every place v of k the sequence of their galois orbits on the analytic space x an converges to a measure when this is the case the limit measure is a translate of the natural measure on the compact torus sitting in the principal orbit of x the key ingredient is the study of the v adic modulus distribution of ga lois orbits of generic d small sequences of algebraic points in particular we characterize all their cluster measures we generalize the bogomolov problem by asking when a subvariety of the principal orbit of a proper toric variety that has the same essential minimum than the ambient variety must be a translate of a subtorus we prove that the generalized bogomolov problem has a positive answer for monocritical toric metrized divisors and we give several examples of toric metrized divisors for which the bogomolov problem has a negative answer

on integer chebyshev polynomials we are concerned with the problem of minimizing the supremum norm on o 1 of a nonzero polynomial of degree at most n with integer coefficients we use the structure of such polynomials to derive an efficient algorithm for computing them we give a table of these polynomials for degree up to 75 and use a value from this table to answer an open problem due to p borwein and t erdelyi and improve a lower bound due to flammang et al

on the linear independence measure of logarithms of rational numbers in this paper we give a general theorem on the linear independence measure of logarithms of rational numbers and in particular the linear independence measure of 1 log 2 log 3 log 5 and of 1 log 2 log 3 log 5 log 7 we also give a method to search for polynomials of smallest norm on a real interval a b which may be suitable for computing or improving the linear independence measure of logarithms of rational numbers

mesures et equidistribution sur les espaces de berkovich the proof by ullmo and zhang of bogomolov s conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for small points in the associated complex abelian variety we study the analogous equidistribution property at p adic places our results can be conveniently stated within the framework of the analytic spaces defined by berkovich the first one is valid in any dimension but is restricted to algebraic metrics the second one is valid for curves but allows for more general metrics in particular to the normalized heights with respect to dynamical systems

quantitative equidistribution for the solutions of systems of sparse polynomial equations for a system of laurent polynomials f 1 ldots f n in bbb c x 1 pm1 ldots x n pm1 whose coefficients are not too big with respect to its directional resultants we show that the solutions in the algebraic torus bbb c times n of the system of equations f 1 cdots f n 0 are approximately equidistributed near the unit polycircle this generalizes to the multivariate case a classical result due to erd os and tur an on the distribution of the arguments of the roots of a univariate polynomial we apply this result to bound the number of real roots of a system of laurent polynomials and to study the asymptotic distribution of the roots of systems of laurent polynomials over bbb z and of random systems of laurent polynomials over bbb c

equidistribution for torsion points of a drinfeld module we prove an equidistribution result for torsion points of drinfeld modules of generic characteristic we also show that similar equidistribution statements would provide proofs for the manin mumford and the bogomolov conjectures for drinfeld modules

positive line bundles on arithmetic varieties p 5 u integers is defined to count the volume of the lattice i l n of integral sections in the space j7 lon of real sections with supremum norm we want to prove that the leading term of this hilbert function as n f oo is given in terms of the height of x in as 1 this is known as a theorem of gillet and soule gs2 if x has a regular generic fiber beside this known result our proof uses hironaka s theorem on resolutions of singularities and minkowski s theorem on successive minima by hironaka s theorem we may construct

picard and chazy solutions to the painleve vi equation we study the solutions of a particular family of painleve vi equations with parameters beta gamma 0 delta frac 1 2 and 2 alpha 2 mu 1 2 for 2 mu in mathbb z we show that in the case of half integer mu all solutions can be written in terms of known functions and they are of two types a two parameter family of solutions found by picard and a new one parameter family of classical solutions which we call chazy solutions we give explicit formulae for them and completely determine their asymptotic behaviour near the singular points 0 1 infty and their nonlinear monodromy we study the structure of analytic continuation of the solutions to the pvi mu equation for any mu such that 2 mu in mathbb z as an application we classify all the algebraic solutions for mu half integer we show that they are in one to one correspondence with regular polygons or star polygons in the plane for mu integer we show that all algebraic solutions belong to a one parameter family of rational solutions

littlewood type problems on 0 1 we consider the problem of minimizing the uniform norm on 0 1 over non zero polynomials p of the form p x sum j 0 n a jx j quad text with a j le 1 a j in bbb c where the modulus of the first non zero coefficient is at least delta 0 essentially sharp bounds are given for this problem an interesting related result states that there are absolute constants c 1 0 and c 2 0 such that exp c 1 sqrt n le inf 0 ne p in cal f n p 0 1 le exp c 2 sqrt n for every n ge 2 where cal f n denotes the set of polynomials of degree at most n with coefficients from 1 0 1 this chebyshev type problem is closely related to the question of how many zeros a polynomial from the above classes can have at 1 we also give essentially sharp bounds for this problem em inter alia we prove that there is an absolute constant c 0 such that every polynomial p of the form p x sum j 0 n a j x j quad text with a j le 1 a 0 a n 1 a j in bbb c has at most c sqrt n real zeros this improves the old bound c sqrt n log n given by schur in 1933 as well as more recent related bounds of bombieri and vaaler and up to the constant c this is the best possible result all the analysis rests critically on the key estimate stating that there are absolute constants c 1 0 and c 2 0 such that f 0 c 1 a le exp c 2 a f 1 a 1 for every f in cal s and a in 0 1 where cal s denotes the collection of all analytic functions f on the open unit disk d z in bbb c z f z le frac 1 1 z quad text for z in d

preperiodic points and unlikely intersections in this article we combine complex analytic and arithmetic tools to study the preperiodic points of one dimensional complex dynamical systems we show that for any fixed a b c and any integer d 2 the set of c c for which both a and b are preperiodic for z c is infinite if and only if a b this provides an affirmative answer to a question of zannier which itself arose from questions of masser concerning simultaneous torsion sections on families of elliptic curves using similar techniques we prove that if rational functions c z have infinitely many preperiodic points in common then all of their preperiodic points coincide and in particular the maps must have the same julia set this generalizes a theorem of mimar who established the same result assuming that and are defined over q the main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields with non archimedean berkovich spaces playing an essential role

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

the trace of totally positive algebraic integers for all totally positive algebraic numbers except a finite number of explicit exceptions the following inequality holds 1 d 1 d max 1 780022 1 66 1 where d is the degree of a and 0 1 d its conjugates this improves previous results of smyth flammang and rhin

algebraic integers whose conjugates all lie in an ellipse we find all 15909 algebraic integers whose conjugates all lie in an ellipse with two of them nonreal while the others lie in the real interval 1 2 this problem has applications to finding certain subgroups of sl 2 c we use explicit auxiliary functions related to the generalized integer transfinite diameter of compact subsets of c this gives good bounds for the coefficients of the minimal polynomial of a

from klein to painleve via fourier laplace and jimbo we describe a method for constructing explicit algebraic solutions to the sixth painleve equation generalising that of dubrovin and mazzocco there are basically two steps first we explain how to construct finite braid group orbits of triples of elements of rm sl 2 rm c out of triples of generators of three dimensional complex reflection groups this involves the fourier laplace transform for certain irregular connections then we adapt a result of jimbo to produce the painleve vi solutions in particular this solves a riemann hilbert problem explicitly each step is illustrated using the complex reflection group associated to klein s simple group of order 168 this leads to a new algebraic solution with seven branches we also prove that unlike the algebraic solutions of dubrovin and mazzocco and hitchin this solution is not equivalent to any solution coming from a finite subgroup of rm sl 2 rm c the results of this paper also yield a simple proof of a recent theorem of inaba iwasaki and saito on the action of okamoto s affine rm d 4 symmetry group as well as the correct connection formulae for generic painleve vi equations

radial multi focal tensors the 1d radial camera maps all points on a plane containing the principal axis onto the radial line which is the intersection of that plane and the image plane it is a sufficiently general model to express both central and non central cameras since the only assumption it makes is of known center of distortion in this paper we study the multi focal tensors arising out of 1d radial cameras there exist no two view constraints like the fundamental matrix for 1d radial cameras however the 3 view and 4 view cases are interesting for the 4 view case we have the radial quadrifocal tensor which has 15 d o f and 2 internal constraints for the 3 view case we have the radial trifocal tensor which has 7 d o f and no internal constraints under the assumption of a purely rotating central camera this can be used to do a non parametric estimation of the radial distortion of a 1d camera even in the case of a non rotating camera it can be used to do parametric estimation assuming a planar scene finally we examine the mixed trifocal tensor which models the case of two 1d radial cameras and one standard pin hole camera of the above radial multifocal tensors only the radial trifocal tensor is useful practically since it doesn t require any knowledge of the scene and is extremely robust we demonstrate results based on real images for this for the quadrifocal tensor too we present a way to do a metric reconstruction of the scene and to undistort the image given a sufficiently dense set of point correspondences we also show results on synthetic images however it must be noted that currently the quadrifocal and mixed trifocal tensors are useful only from a theoretical stand point

local expansion of painleve vi transcendents around a fixed singularity we study special solutions of the sixth painleve equation with generic values of parameters whose linear monodromy can be calculated explicitly we give two kinds of special solutions one is the meromorphic solutions at a fixed singularity and the other is the solutions expressed in the puiseux series at t 0 we give the respective character of these special solutions

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

root statistics of random polynomials with bounded mahler measure the mahler measure of a polynomial is a measure of complexity formed by taking the modulus of the leading coecient times the modulus of the product of its roots outside the unit circle the roots of a real degree n polynomial chosen uniformly from the set of polynomials of mahler measure at most 1 yields a pfaan point process on the complex plane when n is large with probability tending to 1 the roots tend to the unit circle and we investigate the asymptotics of the scaled kernel in a neighborhood of a point on the unit circle when this point is away from the real axis on which there is a positive probability of nding a root the scaled process degenerates to a determinantal point process with the same local statistics i e scalar kernel as the limiting process formed from the roots of complex polynomials chosen uniformly from the set of polynomials of mahler measure at most 1 three new matrix kernels appear in a neighborhood of 1 which encode information about the correlations between real roots between complex roots and between real and complex roots away from the unit circle the kernels converge to new limiting kernels which imply among other things that the expected number of roots in any open subset of c disjoint from the unit circle converges to a positive number we also give ensembles with identical statistics drawn from two dimensional electrostatics with potential theoretic weights and normal matrices chosen with regard to their topological entropy as actions on euclidean space

generalized camera calibration including fish eye lenses a method is described for accurately calibrating cameras including radial lens distortion by using known points such as those measured from a calibration fixture both the intrinsic and extrinsic parameters are calibrated in a single least squares adjustment but provision is made for including old values of the intrinsic parameters in the adjustment the distortion terms are relative to the optical axis which is included in the model so that it does not have to be orthogonal to the image sensor plane these distortion terms represent corrections to the basic lens model which is a generalization that includes the perspective projection and the ideal fish eye lens as special cases the position of the entrance pupil point as a function of off axis angle also is included in the model the complete camera model including all of these effects often is called cahvore a way of adding decentering distortion also is described a priori standard deviations can be used to apply weight to given initial approximations which can be zero for the distortion terms for the difference between the optical axis and the perpendicular to the sensor plane and for the terms representing movement of the entrance pupil so that the solution for these is well determined when there is insufficient information in the calibration data for the other parameters initial approximations needed for the nonlinear least squares adjustment are obtained in a simple manner from the calibration data and other known information weight can be given to these also if desired outliers among the calibration points that disagree excessively with the other data are removed by means of automatic editing based on analysis of the residuals the use of the camera model also is described including partial derivatives for propagating both from object space to image space and vice versa these methods were used to calibrate the cameras on the mars exploration rovers

strong limit multiplicity for arithmetic hyperbolic surfaces and 3 manifolds we show that every sequence of torsion free arithmetic congruence lattices in mathrm pgl 2 mathbb r or mathrm pgl 2 mathbb c satisfies a strong quantitative version of the limit multiplicity property we deduce that for r 0 in certain range growing linearly in the degree of the invariant trace field the volume of the r thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic 3 manifold m is of order at most mathrm vol m 11 12 as an application we prove gelander s conjecture on homotopy type of arithmetic hyperbolic 3 manifolds we show that there are constants a b such that every such manifold m is homotopy equivalent to a simplicial complex with at most a mathrm vol m vertices all of degrees bounded by b

catadioptric projective geometry catadioptric sensors are devices which utilize mirrors and lenses to form a projection onto the image plane of a camera central catadioptric sensors are the class of these devices having a single effective viewpoint in this paper we propose a unifying model for the projective geometry induced by these devices and we study its properties as well as its practical implications we show that a central catadioptric projection is equivalent to a two step mapping via the sphere the second step is equivalent to a stereographic projection in the case of parabolic mirrors conventional lens based perspective cameras are also central catadioptric devices with a virtual planar mirror and are thus covered by the unifying model we prove that for each catadioptric projection there exists a dual catadioptric projection based on the duality between points and line images conics it turns out that planar and parabolic mirrors build a dual catadioptric projection pair as a practical example we describe a procedure to estimate focal length and image center from a single view of lines in arbitrary position for a parabolic catadioptric system

integer transfinite diameter and polynomials with small mahler measure in this work we show how suitable generalizations of the integer transfinite diameter of some compact sets in c give very good bounds for coefficients of polynomials with small mahler measure by this way we give the list of all monic irreducible primitive polynomials of z x of degree at most 36 with mahler measure less than 1 324 and of degree 38 and 40 with mahler measure less than 1 31

monodromy of certain painleve vi transcendents and reflection groups we study the global analytic properties of the solutions of a particular family of painleve vi equations with the parameters 0 1 2 and 2 2 1 2 with arbitrary 2 we introduce a class of solutions having critical behaviour of algebraic type and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space the analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group we show that the finite orbits of this action correspond to the algebraic solutions of our painleve vi equation and use this result to classify all of them we prove that the algebraic solutions of our painleve vi equation are in one to one correspondence with the regular polyhedra or star polyhedra in the three dimensional space

a trace bound for positive definite connected integer symmetric matrices abstract let a be a connected integer symmetric matrix i e a a i j m n z for some n a a t and the underlying graph vertices corresponding to rows with vertex i joined to vertex j if a i j 0 is connected we show that if all the eigenvalues of a are strictly positive then tr a 2 n 1 there are two striking corollaries first the analogue of the schur siegel smyth trace problem is solved for characteristic polynomials of connected integer symmetric matrices second we find new examples of totally real separable irreducible monic integer polynomials that are not minimal polynomials of integer symmetric matrices

rational solutions of the painleve vi equation in this paper we classify all values of the parameters and of the painleve vi equation such that there are rational solutions we give a formula for them up to the birational canonical transformations and the symmetries of the painleve vi equation

structure from motion with wide circular field of view cameras this paper presents a method for fully automatic and robust estimation of two view geometry autocalibration and 3d metric reconstruction from point correspondences in images taken by cameras with wide circular field of view we focus on cameras which have more than 180deg field of view and for which the standard perspective camera model is not sufficient e g the cameras equipped with circular fish eye lenses nikon fc e8 183deg sigma 8 mm f4 ex 180deg or with curved conical mirrors we assume a circular field of view and axially symmetric image projection to autocalibrate the cameras many wide field of view cameras can still be modeled by the central projection followed by a nonlinear image mapping examples are the above mentioned fish eye lenses and properly assembled catadioptric cameras with conical mirrors we show that epipolar geometry of these cameras can be estimated from a small number of correspondences by solving a polynomial eigenvalue problem this allows the use of efficient ransac robust estimation to find the image projection model the epipolar geometry and the selection of true point correspondences from tentative correspondences contaminated by mismatches real catadioptric cameras are often slightly noncentral we show that the proposed autocalibration with approximate central models is usually good enough to get correct point correspondences which can be used with accurate noncentral models in a bundle adjustment to obtain accurate 3d scene reconstruction noncentral camera models are dealt with and results are shown for catadioptric cameras with parabolic and spherical mirrors

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

nonmetric calibration of camera lens distortion differential methods and robust estimation this paper addresses the problem of calibrating camera lens distortion which can be significant in medium to wide angle lenses our approach is based on the analysis of distorted images of straight lines we derive new distortion measures that can be optimized using nonlinear search techniques to find the best distortion parameters that straighten these lines unlike the other existing approaches we also provide fast closed form solutions to the distortion coefficients we prove that including both the distortion center and the decentering coefficients in the nonlinear optimization step may lead to instability of the estimation algorithm our approach provides a way to get around this and at the same time it reduces the search space of the calibration problem without sacrificing the accuracy and produces more stable and noise robust results in addition while almost all existing nonmetric distortion calibration methods needs user involvement in one form or another we present a robust approach to distortion calibration based on the least median of squares estimator our approach is thus able to proceed in a fully automatic manner while being less sensitive to erroneous input data such as image curves that are mistakenly considered projections of three dimensional linear segments experiments to evaluate the performance of this approach on synthetic and real data are reported

on tau functions for orthogonal polynomials and matrix models let v be a real polynomial of even degree and let rho be the equilibrium probability measure for v with support s to that v x greeater than or equal to int 2log x y rho dy c for some constant c with equality on s then s is the union of finitely many boundd intervals with endpoints delta j and rho is given by an algebraic weight w x on s then the system of orthogonal polynomials for w gives rise to a system of differential equations known as the schlesinger equations this paper identifies the tau function of this system with the hankel determinant det int x j k rho dx the solutions of the magnus schlesinger equation are realised by a linear system which is used to compute the tau functions in terms of a gelfand levitan equation the tau function is associated with a potential q and a scattering problem for the schrodinger equation with potential q the paper describes cases where this is integrable in terms of the nonlinear fourier transform

the logarithmic asymptotics of the sixth painleve equation we compute the monodromy group associated with the solutions of the sixth painleve equation with a logarithmic asymptotic behavior at a critical point and we characterize the asymptotic behavior in terms of the monodromy itself

canonical heights transfinite diameters and polynomial dynamics let z be a polynomial of degree at least 2 with coefficients in a number field k iterating gives rise to a dynamical system and a corresponding canonical height function as defined by call and silverman we prove a simple product formula relating the transfinite diameters of the filled julia sets of over various completions of k and we apply this formula to give a generalization of bilu s equidistribution theorem for sequences of points whose canonical heights tend to zero

a dynamical pairing between two rational maps this is the publisher s final pdf the published article is copyrighted by american mathematical society and can be found at http www ams org home page

automatic line extraction in uncalibrated omnidirectional cameras with revolution symmetry revolution symmetry is a realistic assumption for modelling the majority of catadioptric and dioptric cameras in central systems it can be described by a projection model based on radially symmetric distortion in these systems straight lines are projected on curves called line images these curves have in general more than two degrees of freedom and their shape strongly depends on the particular camera configuration therefore the existing line extraction methods for this kind of omnidirectional cameras require the camera calibration by contrast with the perspective case where the calibration is not involved in the shape of the projected line image however this drawback can be considered as an advantage because the shape of the line images can be used for self calibration in this paper we present a novel method to extract line images in uncalibrated omnidirectional images which is valid for radially symmetric central systems in this method we propose using the plumb line constraint to find closed form solutions for different types of camera systems dioptric or catadioptric the inputs of the proposed method are points belonging to the line images and their intensity gradient the gradient information allows to reduce the number of points needed in the minimal solution improving the result and the robustness of the estimation the scheme is used in a line image extraction algorithm to obtain lines from uncalibrated omnidirectional images without any assumption about the scene the algorithm is evaluated with synthetic and real images showing good performance the results of this work have been implemented in an open source matlab toolbox for evaluation and research purposes

camera calibration with distortion models and accuracy evaluation a camera model that accounts for major sources of camera distortion namely radial decentering and thin prism distortions is presented the proposed calibration procedure consists of two steps 1 the calibration parameters are estimated using a closed form solution based on a distribution free camera model and 2 the parameters estimated in the first step are improved iteratively through a nonlinear optimization taking into account camera distortions according to minimum variance estimation the objective function to be minimized is the mean square discrepancy between the observed image points and their inferred image projections computed with the estimated calibration parameters the authors introduce a type of measure that can be used to directly evaluate the performance of calibration and compare calibrations among different systems the validity and performance of the calibration procedure are tested with both synthetic data and real images taken by tele and wide angle lenses

computing totally positive algebraic integers of small trace we construct minimal polynomials of totally positive algebraic integers of small absolute trace by consideration of their reductions modulo auxiliary polynomials many new examples of such polynomials of minimal absolute trace for given degree are found the computations are pushed to degrees that previously were unattainable and one consequence is that the new examples form the majority of all those known as an application we produce a new bound for the schur siegel smyth trace problem

the monic integer transfinite diameter we study the problem of finding nonconstant monic integer polynomials normalized by their degree with small supremum on an interval i the monic integer transfinite diameter t m i is defined as the infimum of all such supremums we show that if i has length 1 then t m i we make three general conjectures relating to the value of t m i for intervals i of length less than 4 we also conjecture a value for t m 0 b where 0 b 1 we give some partial results as well as computational evidence to support these conjectures we define functions l t and l t which measure properties of the lengths of intervals i with t m i on either side of t upper and lower bounds are given for these functions we also consider the problem of determining t m i when i is a farey interval we prove that a conjecture of borwein pinner and pritsker concerning this value is true for an infinite family of farey intervals

distribution of integral division points on the algebraic torus let k be a number field with algebraic closure overline k and let s be a finite set of places of k containing all the infinite ones let it gamma 0 be a finitely generated subgroup of mathbb g textup m overline k and let it gamma subset mathbb g textup m overline k be the division group attached to it gamma 0 here is an illustration of what we will prove in this article fix a proper closed subinterval i of 0 infty and a nonzero effective divisor d on mathbb g textup m which is not the translate of any torsion divisor on the algebraic torus mathbb g textup m by any point of it gamma with height belonging to i then we prove a statement which easily implies that the set of integral division points on mathbb g textup m with height near i i e the set of points of it gamma with standard absolute logarithmic weil height in j which are s integral on mathbb g textup m relative to d is finite for some fixed subinterval j of 0 infty properly containing i we propose a conjecture on the nondensity of integral division points on semi abelian varieties with prescribed height values which generalizes some previously known conjectures as well as this finiteness result for mathbb g textup m finally we also propose an analogous version for a dynamical system on mathbb p 1

small height and infinite nonabelian extensions let e be an elliptic curve defined over q without complex multiplication the field f generated over q by all torsion points of e is an infinite nonabelian galois extension of the rationals which has unbounded wild ramification above all primes we prove that the absolute logarithmic weil height of an element of f is either zero or bounded from below by a positive constant depending only on e we also show that the neron tate height has a similar gap on e f and use this to determine the structure of the group e f

robust metric calibration of non linear camera lens distortion camera lens distortion is crucial to obtain the best performance cameral model up to now different techniques exist which try to minimize the calibration error using different lens distortion models or computing them in different ways some compute lens distortion camera parameters in the camera calibration process together with the intrinsic and extrinsic ones others isolate the lens distortion calibration without using any template and basing the calibration on the deformation in the image of some features of the objects in the scene like straight lines or circles these lens distortion techniques which do not use any calibration template can be unstable if a complete camera lens distortion model is computed they are named non metric calibration or self calibration methods traditionally a camera has been always best calibrated if metric calibration is done instead of self calibration this paper proposes a metric calibration technique which computes the camera lens distortion isolated from the camera calibration process under stable conditions independently of the computed lens distortion model or the number of parameters to make it easier to resolve this metric technique uses the same calibration template that will be used afterwards for the calibration process therefore the best performance of the camera lens distortion calibration process is achieved which is transferred directly to the camera calibration process

monic integer chebyshev problem we study the problem of minimizing the supremum norm by monic polynomials with integer coefficients let mn z denote the monic polynomials of degree n with integer coefficients a monic integer chebyshev polynomial mn mn z satisfies mn e infpn mn z pn e and the monic integer chebyshev constant is then defined by tm e limn mn e1 n this is the obvious analogue of the more usual integer chebyshev constant that has been much studied we compute tm e for various sets including all finite sets of rationals and make the following conjecture which we prove in many cases conjecture suppose a2 b2 a1 b1 is an interval whose endpoints are consecutive farey fractions this is characterized by a1b2 a2b1 1 then tm a2 b2 a1 b1 max 1 b1 1 b2 this should be contrasted with the nonmonic integer chebyshev constant case where the only intervals for which the constant is exactly computed are intervals of length 4 or greater

nonmetric calibration of wide angle lenses and polycameras images taken with wide angle cameras tend to have severe distortions which pull points towards the optical center this paper proposes a simple method for recovering the distortion parameters without the use of any calibration objects since distortions cause straight lines in the scene to appear as curves in the image our algorithm seeks to find the distortion parameters that map the image curves to straight lines the user selects a small set of points along the image curves recovery of the distortion parameters is formulated as the minimization of an objective function which is designed to explicitly account for noise in the selected image points experimental results are presented for synthetic data as well as real images we also present the idea of a polycamera which is defined as a tightly packed camera cluster possible configurations are proposed to capture very large fields of view such camera clusters tend to have a nonsingle viewpoint we therefore provide analysis of what we call the minimum working distance for such clusters finally we present results for a polycamera consisting of four wide angle sensors having a minimum working distance of about 4 m on undistorting the acquired images using our proposed technique we create real time high resolution panoramas

true multi image alignment and its application to mosaicing and lens distortion correction multiple images of a scene are related through 2d 3d view transformations and linear and nonlinear camera transformations we present an algorithm for true multi image alignment that does not rely on the measurements of a reference image being distortion free the algorithm is developed to specifically align and mosaic images using parametric transformations in the presence of lens distortion when lens distortion is present none of the images can be assumed to be ideal in our formulation all the images are modeled as intensity measurements represented in their respective coordinate systems each of which is related to an ideal coordinate system through an interior camera transformation and an exterior view transformation the goal of the accompanying algorithm is to compute an image in the ideal coordinate system while solving for the transformations that relate the ideal system with each of the data images key advantages of the technique presented in this paper are i no reliance on one distortion free image ii ability to register images and compute coordinate transformations even when the multiple images are of an extended scene with no overlap between the first and last frame of the sequence and iii ability to handle linear and nonlinear transformations within the same framework results of applying the algorithm are presented for the correction of lens distortion and creation of video mosaics