phase path tracing methods for solving the high order zeros of nonlinear systems abstract an adaptive phase path tracing method for solving high order zeros of nonlinear systems is developed in this paper a global convergence of the method is then given a numerical example is given

exact critical exponent for the shortest path scaling function in percolation it is shown that the critical exponentg1 related to pair connectiveness and shortest path or chemical distance scaling recently studied by porto et al dokholyan et al and grassberger can be found exactly in two dimensions by using a crossing probability result of cardy with the outcomeg1d 25 24 this prediction is consistent with existing simulation results

the convergence of euler like method for the simultaneous inclusion of polynomial zeros abstract in this paper we give the convergence analysis of the euler like iterative method for the simultaneous inclusion of all simple real or complex zeros of a polynomial the established initial conditions provide the safe convergence of the considered method and the fourth order of convergence these conditions are computationally verifiable which is of practical importance a procedure for the choice of initial inclusion disks is also given

exponential control of a rotational motion of a rigid body using quaternions a new developed technique is used to study the problem of the exponential stabilization of the rotational motion of a rigid body about a fixed axis in the body with the help of rotors in this technique the exponential asymptotic stability of this motion is proved the control moments leading to the rotational motion are obtained the conditions that ensure the exponential asymptotic stabilization of this motion are used to obtain the stabilizing control law as non linear functions of the phase coordinates of the system finally many special cases and analyzing the results of the studied problem are investigated

conformal invariance in two dimensional percolation the word percolation borrowed from the latin refers to the seeping or oozing of a liquid through a porous medium usually to be strained in this and related senses it has been in use since the seventeenth century it was introduced more recently into mathematics by s r broadbent and j m hammersley bh and is a branch of probability theory that is especially close to statistical mechanics broadbent and hammersley distinguish between two types of spreading of a fluid through a medium or between two aspects of the probabilistic models of such processes diffusion processes in which the random mechanism is ascribed to the fluid and percolation processes in which it is ascribed to the medium a percolation process typically depends on one or more probabilistic parameters for example if molecules of a gas are absorbed at the surface of a porous solid as in a gas mask then their ability to penetrate the solid depends on the sizes of the pores in it and their positions both conceived to be distributed in some random manner a simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner that could be determined by a periodic graph and to be open thus very large or closed thus smaller than the molecules with probabilities p and 1 p as p increases the probability of deeper penetration of the gas into the interior of the solid grows

the root and bell s disk iteration methods are of the same error propagation characteristics in the simultaneous determination of the zeros of a polynomial part ii round off error analysis by use of interval arithmetic in part i ikhile 2008 4 it was established that the root and bell s disk point iteration methods with or without correction term are of the same asymptotic error propagation characteristics in the simultaneous determination of the zeros of a polynomial this concluding part of the investigation is a study in round offs its propagation and its effects on convergence employing interval arithmetic means the purpose is to consequently draw attention on the effects of round off errors introduced from the point arithmetic part on the rate of convergence of the generalized root and bell s simultaneous interval iteration algorithms and its enhanced modifications introduced in part i for the numerical inclusion of all the zeros of a polynomial simultaneously the motivation for studying the effects of round off error propagation comes from the fact that the readily available computing devices at the moment are limited in precision more so that accuracy expected from some programming or computing environments or from these numerical methods are or can be machine dependent in fact a part of the finding is that round off propagation effects beyond a certain controllable order induces overwhelmingly delayed or even a severely retarded convergence speed which manifest glaringly as poor accuracy of these interval iteration methods in the computation of the zeros of a polynomial simultaneously however in this present consideration and even in the presence of overwhelming influence of round offs we give conditions under which convergence is still possible and derive the error round off relations along with the order r order of convergence of these methods with the results extended to similar interval iteration methods for computing the zeros of a polynomial simultaneously especially to bell s interval methods for refinement of zeros that form a cluster our findings are instructive and quite revealing and supported by evidence from numerical experiments the analysis is preferred in circular interval arithmetic

weierstrass formula and zero finding methods classical weierstrass formula 29 has been often the subject of investigation of many authors in this paper we give some further applications of this formula for finding the zeros of polynomials and analytic functions we are concerned with the problems of localization of polynomial zeros and the construction of iterative methods for the simultaneous approximation and inclusion of these zeros conditions for the safe convergence of weierstrass method depending only on initial approximations are given in particular we study polynomials with interval coefficients using an interval version of weierstrass method enclosures in the form of disks for the complex valued set containing all zeros of a polynomial with varying coefficients are obtained we also present weierstrass like algorithm for approximating simultaneously all zeros of a class of analytic functions in a given closed region to demonstrate the proposed algorithms three numerical examples are included

on the universality of crossing probabilities in two dimensional percolation six percolation models in two dimensions are studied percolation by sites and by bonds on square hexagonal and triangular lattices rectangles of widtha and heightb are superimposed on the lattices and four functions representing the probabilities of certain crossings from one interval to another on the sides are measured numerically as functions of the ratioa b in the limits set by the sample size and by the conventions and on the range of the ratioa b measured the four functions coincide for the six models we conclude that the values of the four functions can be used as coordinates of the renormalization group fixed point

smooth orientation interpolation using parametric quintic polynomial based quaternion spline curve abstract in this paper a g 2 continuous quintic polynomial based unit quaternion interpolation spline curve with tension parameters is presented to interpolate a given sequence of solid orientations the curve in unit quaternion space s 3 is an extension of the quintic polynomial interpolation spline curve in euclidean space it preserves the interpolatory property and g 2 continuity meanwhile the unit quaternion interpolation spline curve possesses the local shape adjustability due to the presence of tension parameters the change of one tension parameter will only affect the adjacent two pieces of curves compared with the traditional b spline unit quaternion interpolation curve and v spline unit quaternion interpolation curve the proposed curve can automatically interpolate the given data points without solving the nonlinear system of equations over quaternions to obtain the control points which greatly improves the computational efficiency simulation results demonstrate the effectiveness of the proposed scheme

the arithmetic of the digital computer a new approach a new approach to the arithmetic of the digital computer is surveyed the methodology for defining and implementing floating point arithmetic is described shortcomings of elementary floating point arithmetic are revealed through sample problems the development of automatic computation with emphasis on the user control of errors is reviewed the limitations of conventional rule of thumb procedures for error control in scientific computation are demonstrated by means of examples computer arithmetic is extended so that the arithmetic operations in the linear spaces and their interval correspondents which are most commonly used in computation can be performed with maximum accuracy on digital computers a new fundamental computer operation the scalar product is introduced to develop this advanced computer arithmetic a process of automatic error control called validation which delivers high accuracy with guarantees for scientific computations is described validation of computations for a large class of nu

efficient numerical validation of solutions of nonlinear systems a new stopping criterion for newton s method is derived by combining the properties of the krawczyk operator and a corollary of the newton kantorovich theorem when this criterion is satisfied the authors use the last three newton iterates to compute an interval vector that is very likely to contain a solution of the given nonlinear system the existence of such a solution is tested using krawczyk s operator furthermore each element from this interval vector considered as an approximation to the solution has a relative error that is of the order of the machine precision extensive numerical testing has shown that the proposed method has very good practical performance

exceptional planes of percolation consider the standard continuous percolation in 4 and choose the parameters so that the induced percolation on a fixed two dimensional linear subspace is critical although two dimensional critical percolation dies we show that there are exceptional two dimensional linear subspaces in which percolation occurs

convergence of threshold estimates for two dimensional percolation using a recently introduced algorithm for simulating percolation in microcanonical fixed occupancy samples we study the convergence with increasing system size of a number of estimates for the percolation threshold for an open system with a square boundary specifically for site percolation on a square lattice we show that the convergence of the average probability estimate is described by a nontrivial correction to scaling exponent as predicted previously and measure the value of this exponent to be 0 90 ifmmode pm else textpm fi 0 02 for the median and cell to cell estimates of the percolation threshold we verify that convergence does not depend on this exponent having instead a slightly faster convergence with a trivial analytic leading exponent

critical exponents for the homology of fortuin kasteleyn clusters on a torus a fortuin kasteleyn cluster on a torus is said to be of type a b a b in z if it is possible to draw a curve belonging to the cluster that winds a times around the first cycle of the torus as it winds b times around the second even though the q potts models make sense only for q integers they can be included into a family of models parametrized by beta square root of q for which the fortuin kasteleyn clusters can be defined for any real beta 0 2 for this family we study the probability pi a b of a given type of clusters as a function of the torus modular parameter tau tau r itau i we compute the asymptotic behavior of some of these probabilities as the torus becomes infinitely thin for example the behavior of pi 1 0 is studied for tau i infinity exponents describing these behaviors are defined and related to weights h r s of the extended kac table for r and s integers but also half integers numerical simulations are also presented possible relationship with recent works and conformal loop ensembles is discussed

on crossing event formulas in critical two dimensional percolation several formulas for crossing functions arising in the continuum limit of critical two dimensional percolation models are studied these include watts s formula for the horizontal vertical crossing probability and cardy s new formula for the expected number of crossing clusters it is shown that for lattices where conformal invariance holds they simplify when the spatial domain is taken to be the interior of an equilateral triangle the two crossing functions can be expressed in terms of an equianharmonic elliptic function with a triangular rotational symmetry this suggests that rigorous proofs of watts s formula and cardy s new formula will be easiest to construct if the underlying lattice is triangular the simplification in a triangular domain of schramm s bulk cardy s formula is also studied

rigorous sensitivity analysis for systems of linear and nonlinear equations methods are presented for performing a rigorous sensitivity analysis of numerical problems with independent noncorrelated data for general systems of linear and nonlinear equations the methods may serve for the following two purposes first to bound the dependency of the solution on changes in the input data in contrast to condition numbers a componentwise sensitivity analysis of the solution vector is performed second to estimate the true solution set for problems the input data of which are afflicted with tolerances the methods presented are very effective with the additional property that due to an automatic error control mechanism every computed result is guaranteed to be correct examples are given for linear systems demonstrating that the computed bounds are in general very sharp interesting comparisons to traditional condition numbers are given

toward a conformal field theory for schramm loewner evolutions we discuss the partition function point of view for chordal schramm loewner evolutions and their relationship with correlation functions in conformal field theory both are closely related to crossing probabilities and interfaces in critical models in two dimensional statistical mechanics we gather and supplement previous results with different perspectives point out remaining difficulties and suggest directions for future studies

percolation and conduction in restricted geometries the finite size scaling behaviour for percolation and conduction is studied in two dimensional triangular shaped random resistor networks at the percolation threshold the numerical simulations are performed using an efficient star triangle algorithm the percolation exponents linked to the critical behaviour at corners are in good agreement with the conformal results the conductivity exponent t is found to be independent of the shape of the system its value is very close to recent estimates for the surface and bulk conductivity exponents

on the convergence order of accelerated root iterations a gauss seidel procedure for accelerating the convergence of the generalized method of the root iterations type of the k 2 th order k n for finding polynomial complex zeros given in 7 is considered in this paper it is shown that ther order of convergence of the accelerated method is at leastk 1 n k where n k 1 is the unique positive root of the equation n k 1 0 andn is the degree of the polynomial the examples of algebraic equations in ordinary and circular arithmetic are given

epsilon inflation in verification algorithms epsilon inflation is often used in verification numerics to find an interval vector x e such that some interval function f maps x e into itself we recall algorithms which use epsilon inflation to verify this subset property f x e x e and we derive criteria which guarantee that already finitely many inflation steps are sufficient to prove it these criteria require f to be a p contraction we recall results on p contractions and we derive rules to verify them we also introduce local p contractions and we use this concept to guarantee again the subset property above when using epsilon inflation

stability of quaternionic linear systems the main goal of this paper is to characterize stability and bounded input bounded output bibo stability of quaternionic dynamical systems after defining the quaternion skew field algebraic properties of quaternionic polynomials such as divisibility and coprimeness are investigated having established these results the smith and the smith mcmillan forms of quaternionic matrices are introduced and studied finally all the tools that were developed are used to analyze stability of quaternionic linear systems in a behavioral framework

universality of the crossing probability for the potts model for q 1 2 3 4 the universality of the crossing probability pi hs of a system to percolate only in the horizontal direction was investigated numerically by a cluster monte carlo algorithm for the q state potts model for q 2 3 4 and for percolation q 1 we check the percolation through fortuin kasteleyn clusters near the critical point on the square lattice by using representation of the potts model as the correlated site bond percolation model it was shown that probability of a system to percolate only in the horizontal direction pi hs has the universal form pi hs a q q z for q 1 2 3 4 as a function of the scaling variable z b q l 1 nu q p p c q l zeta q here p 1 exp beta is the probability of a bond to be closed a q is the nonuniversal crossing amplitude b q is the nonuniversal metric factor nu q is the correlation length index and zeta q is the additional scaling index the universal function q x approximately equal exp z the nonuniversal scaling factors were found numerically

conformal invariance of voronoi percolation it is proved that in the voronoi model for percolation in dimension 2 and 3 the crossing probabilities are asymptotically invariant under conformal change of metric

oded schramm s contributions to noise sensitivity we survey in this paper the main contributions of oded schramm related to noise sensitivity we will describe in particular his various works which focused on the spectral analysis of critical percolation and more generally of boolean functions his work on the shape fluctuations of first passage percolation and finally his contributions to the model of dynamical percolation

computer methodologies for comparison of computational efficiency of simultaneous methods for finding polynomial zeros abstract the study and analysis of iterative methods for the determination of zeros of algebraic polynomials the subject of this paper is one of the most actual tasks in the theory and practice our main goal is to examine and compare the quality of most frequently used methods for simultaneously finding all zeros of a polynomial by combining numerical experiments theoretical models and empirical cpu time methodology special attention is devoted to empirical average case model assuming that average cpu times are calculated performing the experiment i on several different computers ii running each method through many loops for a fixed example iii testing reasonably large number of polynomials of different structure body and iv the use of almost arbitrary initial approximations the empirical model ranks the tested methods on the base of the consumed cpu times needed for the fulfillment of the required accuracy of approximations under the same initial conditions our aim is to select a group of the most powerful simultaneous methods with globally convergent performance convergence behavior of two most powerful methods is illustrated by trajectories in the complex plane that consist of the point approximations starting with initial points and going towards the endpoints where the iterative process stops due to the termination criterion

interval analysis theory and applications we give an overview on applications of interval arithmetic among others we discuss verification methods for linear systems of equations nonlinear systems the algebraic eigenvalue problem initial value problems for odes and boundary value problems for elliptic pdes of second order we also consider the item software in this field and give some historical remarks

shape dependent universality in percolation abstract the shape dependent universality of the excess percolation cluster number and cross configuration probability on a torus is examined besides the aspect ratio of the torus the universality class depends upon the twist in the periodic boundary conditions which may be introduced when triangular lattices are used in simulations

bond and site percolation in three dimensions we simulate the bond and site percolation models on a simple cubic lattice with linear sizes up to l 512 and estimate the percolation thresholds to be p c mathrm bond 0 248 phantom rule 0 16em 0ex 811 phantom rule 0 16em 0ex 82 10 and p c mathrm site 0 311 phantom rule 0 16em 0ex 607 phantom rule 0 16em 0ex 7 2 by performing extensive simulations at these estimated critical points we then estimate the critical exponents 1 ensuremath nu 1 1410 15 ensuremath beta ensuremath nu 0 477 phantom rule 0 16em 0ex 05 15 the leading correction exponent y i ensuremath 1 2 2 and the shortest path exponent d mathrm min 1 3756 3 various universal amplitudes are also obtained including wrapping probabilities ratios associated with the cluster size distribution and the excess cluster number we observe that the leading finite size corrections in certain wrapping probabilities are governed by an exponent ensuremath approx ensuremath 2 rather than y i ensuremath approx ensuremath 1 2

efficient methods for the inclusion of polynomial zeros abstract using a suitable zero relation and the inclusion isotonicity property new interval iterative methods for the simultaneous inclusion of simple complex zeros of a polynomial are derived these methods produce disks in the complex plane that contain the polynomial zeros in each iteration providing in this manner an information about upper error bounds of approximations starting from the basic method of the fourth order two accelerated methods with newton s and halley s corrections having the order of convergence five and six respectively are constructed this increase of the convergence rate is obtained without any additional operations which means that the methods with corrections are very efficient the convergence analysis of the basic method and the methods with corrections is performed under computationally verifiable initial conditions which is of practical importance two numerical examples are presented to demonstrate the convergence behavior of the proposed interval methods

on the new fourth order methods for the simultaneous approximation of polynomial zeros a new iterative method of the fourth order for the simultaneous determination of polynomial zeros is proposed this method is based on a suitable zero relation derived from the fourth order method for a single zero belonging to the schroder basic sequence one of the most important problems in solving polynomial equations the construction of initial conditions that enable both guaranteed and fast convergence is studied in detail for the proposed method these conditions are computationally verifiable since they depend only on initial approximations the polynomial coefficients and the polynomial degree which is of practical importance the construction of improved methods in ordinary complex arithmetic and complex circular arithmetic is discussed finally numerical examples and the comparison with existing fourth order methods are given

traub gander s family for the simultaneous determination of multiple zeros of polynomials abstract by combining traub gander s family of third order for finding a multiple zero and suitable corrective approximations od schroder s and halley s type a new family of iterative methods for the simultaneous approximation of multiple zeros of algebraic polynomials is proposed taking various forms of a function involved in the iterative formula a number of different simultaneous methods can be obtained it is proved that the order of convergence is 4 5 or 6 depending of the type of employed corrective approximations two numerical examples are given to demonstrate the convergence properties of the proposed family of simultaneous methods displayed trajectories of the sequences of approximations point to global characteristics of the proposed family of iterative methods

some improved inclusion methods for polynomial roots with weierstrass corrections one decade ago the third order method without derivatives for the simultaneous inclusion of simple zeros of a polynomial was proposed in 1 following nourein s idea 2 some modifications of this method with the increased convergence are proposed the acceleration of convergence is attained by using weierstrass corrections without additional calculations which provides a high computational efficiency of the modified methods it is proved that their r orders of convergence are asymptotically greater than 3 5 the presented interval methods are realized in circular complex arithmetic

the critical probability for random voronoi percolation in the plane is 1 2 we study percolation in the following random environment let z be a poisson process of constant intensity on 2 and form the voronoi tessellation of 2 with respect to z colour each voronoi cell black with probability p independently of the other cells we show that the critical probability is 1 2 more precisely if p 1 2 then the union of the black cells contains an infinite component with probability 1 while if p 1 2 then the distribution of the size of the component of black cells containing a given point decays exponentially these results are analogous to kesten s results for bond percolation in 2 the result corresponding to harris theorem for bond percolation in 2 is known zvavitch noted that one of the many proofs of this result can easily be adapted to the random voronoi setting for kesten s results none of the existing proofs seems to adapt the methods used here also give a new and very simple proof of kesten s theorem for 2 we hope they will be applicable in other contexts as well

numerical results for crossing spanning and wrapping in two dimensional percolation using a recently developed method to simulate percolation on large clusters of distributed machines 1 we have numerically calculated crossing spanning and wrapping probabilities in two dimensional site and bond percolation with exceptional accuracy our results are fully consistent with predictions from conformal field theory we present many new results that await theoretical explanation particularly for wrapping clusters on a cylinder we therefore provide possibly the most up to date reference for theoreticians working on crossing spanning and wrapping probabilities in two dimensional percolation

high order euler like method for the inclusion of polynomial zeros abstract improved iterative method of euler s type for the simultaneous inclusion of polynomial zeros is considered to accelerate the convergence of the basic method of the fourth order we applied borsch supan s correction it is proved that the r order of convergence of the improved euler like method is six the convergence analysis is derived under computationally verifiable initial conditions the proposed algorithm possesses great computational efficiency since the increase of the convergence rate from 4 to 6 is obtained with negligible number of additional calculations in order to demonstrate convergence properties of the suggested method two numerical examples are given

a class of simultaneous methods for the zeros of analytic functions abstract a family of iterative methods for simultaneously approximating simple zeros of analytic functions inside a simple smooth closed contour in the complex plane is presented the order of convergence of the considered methods in m 2 m 1 2 where m is the order of the highest derivative of analytic function appearing in the iterative formula a special attention is paid to the total step and single step methods with newton s and halley s corrections because of their high computational efficiency numerical examples are also included

the cholesky method for interval data abstract we apply the well known cholesky method to bound the solutions of linear systems with symmetric matrices and right hand sides both of which are varying within given intervals we derive criteria to guarantee the feasibility and the optimality of the method furthermore we discuss some general properties

on cardy s formula for the critical crossing probability in 2d percolation cardy s formula for the probability r of crossing a rectangular critical percolation system in the vertical direction with free boundaries on the two sides is written explicitly in terms of the aspect ratio r heightlwidth the first three terms are given by n 24 33i r 4 2 e r 3 e 7n 13 e 3zr 3 i which with n 1 nv i r m sufficient to determine x to high accuracy for all r

robust attitude stabilization of spacecraft using nonlinear quaternion feedback this paper considers the problem of three axis attitude stabilization of a rigid spacecraft a nonlinear control law which uses the feedback of the unit quaternion and the measured angular velocities is proposed and is shown to provide global asymptotic stability the control law does not require the knowledge of the system parameters and is therefore robust to modeling errors the significance of the control law is that it can be used for large angle maneuvers with guaranteed stability

improved halley like methods for the inclusion of polynomial zeros improved iterative methods of halley s type for the simultaneous inclusion of all simple complex zeros of a polynomial are proposed the presented convergence analysis which uses the concept of the r order of convergence of mutually dependent sequences shows that the convergence rate of the basic fourth order method is increased to 5 and 6 using newton s and halley s corrections respectively the proposed inclusion methods possess a high computational efficiency since the increase of the convergence is attained without additional calculations this advantage together with very fast convergence make that the presented methods are ranking among the most powerful inclusion methods for polynomial zeros in order to demonstrate convergence properties of the proposed methods two numerical examples are given

improved methods for the simultaneous inclusion of multiple polynomial zeros using a new fixed point relation the interval methods for the simultaneous inclusion of complex multiple zeros in circular complex arithmetic are constructed using the concept of the r order of convergence of mutually dependent sequences we present the convergence analysis for the total step and the single step methods with schroder s and halley like corrections under computationally verifiable initial conditions the suggested algorithms possess a great computational efficiency since the increase of the convergence rate is attained without additional calculations two numerical examples are given to demonstrate convergence characteristics of the proposed method

length scales and power laws in the two dimensional forest fire model abstract we re examine a two dimensional forest fire model via monte carlo simulations and show the existence of two length scales with different critical exponents associated with clusters and with the usual two point correlation function of trees we check resp improve previously obtained values for other critical exponents and perform a first investigation of the critical behaviour of the slowest relaxational mode we also investigate the possibility of describing the critical point in terms of a distribution of the global density we find that some qualitative features such as a temporal oscillation and a power law of the cluster size distribution can nicely be obtained from such a model that discards the spatial structure

on an iterative method for simultaneous inclusion of polynomial complex zeros abstract starting from disjoint discs which contain polynomial complex zeros the iterative interval method of the third order for the simultaneous finding inclusive discs for complex zeros is formulated the lagrangean interpolation formula and complex circular arithmetic are used the convergence theorem and the conditions for convergence are considered the proposed method has been applied for solving an algebraic equation

a hybrid method for a polynomial complex zero abstract using classical weierstrass result concerning polynomial zeros in a suitable interval form and some iterative method in floating point arithmetic a hybrid method for finding an isolated simple complex zero of a polynomial is established this method possesses a high computational efficiency and the capability of automatic determination of error bounds of the approximate solution a computationally verifiable test for the existence of a polynomial complex zeros in a given disk is also presented

a family of root finding methods with accelerated convergence a parametric family of iterative methods for the simultaneous determination of simple complex zeros of a polynomial is considered the convergence of the basic method of the fourth order is accelerated using newton s and halley s corrections thus generating total step methods of orders five and six further improvements are obtained by applying the gauss seidel approach accelerated convergence of all proposed methods is attained at the cost of a negligible number of additional operations detailed convergence analysis and two numerical examples are given

w extended fusion algebra of critical percolation two dimensional critical percolation is the member lm 2 3 of the infinite series of yang baxter integrable logarithmic minimal models lm p p we consider the continuum scaling limit of this lattice model as a rational logarithmic conformal field theory with extended w w2 3 symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture we find that the representation content of the ensuing closed fusion algebra contains 26 w indecomposable representations with eight rank1 representations 14 rank 2 representations and four rank 3 representations weidentifytheserepresentationswithsuitablelimitsofyang baxterintegrable boundary conditions on the lattice and obtain their associated w extended characters the latter decompose as finite non negative sums of w irreducible characters of which 13 are required implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit cayley table the closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry

time varying exponential stabilization of a rigid spacecraft with two control torques rigid body models with two controls cannot be locally asymptotically stabilized by continuous feedbacks which are functions of the state only this impossibility no longer holds when the feedback is also a function of time and time varying asymptotically stabilizing feedbacks have already been proposed however due to the smoothness of the feedbacks the convergence rate is only polynomial in this paper exponential convergence is obtained by considering time varying feedbacks which are only continuous

a class of generalized b spline quaternion curves unit quaternion curves have gained considerable attention in the fields of robot control and computer animation kim et al proposed a general construction method of unit quaternion curves which can transform the closed form equation for kth order b spline basis functions in r3 into its unit quaternion analogue in so 3 while preserving the c k 2 continuity juhasz and roth generalized the classical b spline functions by means of monotone increasing continuously differentiable core functions based on the recurrence formula of b spline functions in order to extend the applications of the generalized b spline functions in computer animation the definition and construction scheme of generalized b spline quaternion curves in s3 are put forward in this paper the introduced nonlinear core functions are not only theoretically interesting but also offer a large variety of shapes some properties of this class of unit quaternion curves such as continuity and local controllability are also discussed experimental results show the effectiveness and usefulness of our construction methods of generalized b spline quaternion curves

universality of the excess number of clusters and the crossing probability function in three dimensional percolation extensive monte carlo simulations were performed to evaluate the excess number of clusters and the crossing probability function for three dimensional percolation on the simple cubic s c face centred cubic f c c and body centred cubic b c c lattices systems l l l0 with l0 l were studied for both bond s c f c c b c c and site f c c percolation the excess number of clusters q b per unit length was confirmed to be a universal quantity with a value q b 0 412 likewise the critical crossing probability in the l0 direction with periodic boundary conditions in thell plane was found to follow a universal exponential decay as a function of r d l 0 l for large r simulations were also carried out to find new precise values of the critical thresholds for site percolation on the f c c and b c c lattices yielding pc f c c d 0 199 2365 0 000 0010 pc b c c d 0 245 9615 0 000 0010 we also report the value pc s c d 0 311 6080 0 000 0004 for site percolation

proof of crossing formula for 2d percolation the author s recently conjectured expression ibid vol 28 p 1249 1995 for cardy s 1992 crossing formula in 2d percolation is rewritten in terms of theta and elliptic functions and verified explicitly exact results for the aspect ratio r equal to integral powers of two are also given

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems

conformal invariance of a model of percolation on random lattices abstract a new model of percolation on random lattices is introduced it allows for simulations on geometries like the sphere for which there do not exist periodic lattices of arbitrary finess horizontal crossing probabilities h are measured on rectangles of various aspect ratios r these measurements agree with cardy s prediction though there are small discrepancies for rectangles with large aspect ratio further crossing probabilities are measured on a cylinder these probe the hypothesis of conformal invariance stated in bull ams 30 1994 1

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

the number of incipient spanning clusters in two dimensional percolation using methods of conformal field theory we conjecture an exact form for the probability that n distinct clusters span a large rectangle or open cylinder of aspect ratio k i n the limit when k is large the study of the structure of large clusters at the percolation threshold continues to pose interesting problems whose solution sheds light on the nature of the critical state in general recently some attention has been paid to incipient spanning clusters iscs these are clusters which connect two disjoint segments of the boundary of a macroscopically large region langlands et al 1 conjectured that the probability that at least one such cluster exists that is that the segments are connected is invariant under conformal transformations this statement was placed in the context of conformal field theory in 2 where an explicit formula was given for this crossing probability for a review of the status of conformal invariance in critical percolation see 3 more recently aizenman 4 has considered among other things the probability that there exist n distinct iscs connecting the two segmentsy in the case of a rectangular region 0 kl 0 l he has proved that the probability p n k l that the strip is traversed in the direction of length kl b y nindependent clusters satisfies the bounds a e n 2 k

crossing formulae for critical percolation in an annulus an exact formula is given for the probability that there exists a spanning cluster between opposite boundaries of an annulus in the scaling limit of critical percolation the entire distribution function for the number of distinct spanning clusters is also derived these results are found using coulomb gas methods their forms are compared with the expectations of conformal field theory

rigorous confidence intervals for critical probabilities we use the method of balister bollobas and walters random struct algorithms 26 392 2005 to give rigorous 99 9999 confidence intervals for the critical probabilities for site and bond percolation on the 11 archimedean lattices in our computer calculations the emphasis is on simplicity and ease of verification rather than obtaining the best possible results nevertheless we obtain intervals of width at most 0 0005 in all cases

crossing probabilities in critical 2 d percolation and modular forms abstract crossing probabilities for critical 2 d percolation on large but finite lattices have been derived via boundary conformal field theory these predictions agree very well with numerical results however their derivation is heuristic and there is evidence of additional symmetries in the problem this contribution gives a preliminary examination some unusual modular behavior of these quantities in particular the derivatives of the horizontal and horizontal vertical crossing probabilities transform as a vector modular form one component of which is an ordinary modular form and the other the product of a modular form with the integral of a modular form we include consideration of the interplay between conformal and modular invariance that arises

a posteriori error bound methods for the inclusion of polynomial zeros using carstensen s results from 1991 we state a theorem concerning the localization of polynomial zeros and derive two a posteriori error bound methods with the convergence order 3 and 4 these methods possess useful property of inclusion methods to produce disks containing all simple zeros of a polynomial we establish computationally verifiable initial conditions that guarantee the convergence of these methods some computational aspects and the possibility of implementation on parallel computers are considered including two numerical examples a comparison of a posteriori error bound methods with the corresponding circular interval methods regarding the computational costs and sizes of produced inclusion disks were given

critical ising on the square lattice mixes in polynomial time the ising model is widely regarded as the most studied model of spin systems in statistical physics the focus of this paper is its dynamic stochastic version the glauber dynamics introduced in 1963 and by now the most popular means of sampling the ising measure intensive study throughout the last three decades has yielded a rigorous understanding of the spectral gap of the dynamics on mathbb z 2 everywhere except at criticality while the critical behavior of the ising model has long been the focus for physicists mathematicians have only recently developed an understanding of its critical geometry with the advent of sle cle and new tools to study conformally invariant systems

laguerre like inclusion method for polynomial zeros a new method of the fourth order for the simultaneous inclusion of simple polynomial zeros is proposed the presented method is based on the fixed point relation of laguerre s type and realized in circular complex interval arithmetic the established initial conditions guarantee the convergence of the considered method and the fourth order of convergence these conditions are computationally verifiable which is of practical importance the presented iteration formula is suitable for the construction of modified interval methods with improved convergence rate with negligible additional operations convergence analysis and numerical results are given

the root and bell s disk iteration methods are of the same error propagation characteristics in the simultaneous determination of the zeros of a polynomial part i correction methods in this paper we consider the error propagation of the root and bell s disk iteration methods enhanced by incorporating a correction term and a choice of a disk inversion formula in the methods for the simultaneous computation of the zeros of a polynomial the asymptotic error propagation is proved to be the same in both methods this result is important considering the fact that these methods are in popular usage in the simultaneous computation of the zeros of a polynomial the proof of the results herein follows the ideas of m s petkovic c carstensen some improved inclusion methods for polynomial roots with weierstrass corrections comput math appl 25 3 1993 59 67 when the refinement process of correction is efficient it is this mode of correction we have desired to propose

algebraic tools for the study of quaternionic behavioral systems in this paper we study behavioral systems whose trajectories are given as solutions of quaternionic difference equations as happens in the commutative case it turns out that quaternionic polynomial matrices play an important role in this context therefore we pay special attention to such matrices and derive new results concerning their smith form based on these results we obtain a characterization of system theoretic properties such as controllability and stability of a quaternionic behavior

higher order iterative methods for approximating zeros of analytic functions abstract iterative methods with extremely rapid convergence in floating point arithmetic and circular arithmetic for simultaneously approximating simple zeros of analytic functions inside a simple smooth closed contour in the complex plane are presented the r order of convergence of the basic total step and single step methods as well as their improvements which use newton s and halley s corrections is given some hybrid algorithms that combine the efficiency of ordinary floating point iterative methods with the accuracy control provided by interval arithmetic are also considered

a new higher order family of inclusion zero finding methods starting from a suitable fixed point relation a new one parameter family of iterative methods for the simultaneous inclusion of complex zeros in circular complex arithmetic is constructed it is proved that the order of convergence of this family is four the convergence analysis is performed under computationally verifiable initial conditions an approach for the construction of accelerated methods with negligible number of additional operations is discussed to demonstrate convergence properties of the proposed family of methods two numerical examples results are given

on the scaling limits of planar percolation we prove tsirelson s conjecture that any scaling limit of the critical planar percolation is a black noise our theorems apply to a number of percolation models including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid we also suggest a natural construction for the scaling limit of planar percolation and more generally of any discrete planar model describing connectivity properties

on the efficiency of some combined methods for polynomial complex zeros abstract interval methods for the simultaneous inclusion of polynomial zeros produce the approximations that contain the exact zeros providing not only error bounds automatically but also take into account rounding errors without altering the fundamental structure of the interval formula however at present the computational costs of most interval methods are still great in general in this paper several effective algorithms which preserve the inclusion property concerning the complex zeros and which have a high computational efficiency are constructed these algorithms combine the efficiency of ordinary floating point iterations with the accuracy control that may be obtained by the iterations in interval arithmetic several examples are included to illustrate the efficiency and some advantages of the proposed combined methods

characters of irreducible representations of the virasoro algebra on obtient les caracteres des modules irreductibles non integrables de plus haut poids l h 0 on determine les caracteres des familles l t 25 et l t 26 duales de l h 1 et de l h 0

logarithmic conformal field theory beyond an introduction this article aims to review a selection of central topics and examples in logarithmic conformal field theory it begins with the remarkable observation of cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory cardy s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators for this a short introduction to the fusion algorithm of nahm gaberdiel and kausch is provided while the percolation logarithmic conformal field theory is still not completely understood there are several examples for which the formalism familiar from rational conformal field theory including bulk partition functions correlation functions modular transformations fusion rules and the verlinde formula has been successfully generalized this is illustrated for three examples the singlet model related to the triplet model symplectic fermions and the fermionic bc ghost system the fractional level wess zumino witten model based on at related to the bosonic ghost system and the wess zumino witten model for the lie supergroup related to at and 1 the bershadsky polyakov algebra and the feigin semikhatov algebras these examples have been chosen because they represent the most accessible and most useful members of the three best understood families of logarithmic conformal field theories the logarithmic minimal models the fractional level wess zumino witten models and the wess zumino witten models on lie supergroups excluding in this review the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum which turns out to be continuous as well as a selection of natural reducible but indecomposable modules this is followed by a detailed description of how to obtain character formulae for each irreducible a derivation of the action of the modular group on the characters and an application of the verlinde formula to compute the grothendieck fusion rules in each case the genuine fusion rules are known so comparisons can be made and favourable conclusions drawn in addition each example admits an infinite set of simple currents hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed the spectrum of such an extended theory is typically discrete and this is how the triplet model arises for example moreover simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory finally each example is concluded by a brief description of the computation of some bulk correlators a discussion of the structure of the bulk state space and remarks concerning more advanced developments and generalizations the final part gives a very short account of the theory of staggered modules the simplest class of representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins these modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven then the important quantities known as logarithmic couplings are reviewed for virasoro staggered modules and their role as fundamentally important parameters akin to the three point constants of rational conformal field theory is discussed an appendix is also provided in order to introduce some of the necessary but perhaps unfamiliar language of homological algebra

the self validated method for polynomial zeros of high efficiency the improved iterative method of newton s type for the simultaneous inclusion of all simple complex zeros of a polynomial is proposed the presented convergence analysis which uses the concept of the r order of convergence of mutually dependent sequences shows that the convergence rate of the basic third order method is increased from 3 to 6 using ostrowski s corrections the new inclusion method with ostrowski s corrections is more efficient compared to all existing methods belonging to the same class to demonstrate the convergence properties of the proposed method two numerical examples are given

monte carlo method for critical systems in infinite volume the planar ising model in this paper we propose a monte carlo method for generating finite domain marginals of critical distributions of statistical models in infinite volume the algorithm corrects the problem of the long range effects of boundaries associated to generating critical distributions on finite lattices it uses the advantage of scale invariance combined with ideas of the renormalization group in order to construct a type of holographic boundary condition that encodes the presence of an infinite volume beyond it we check the quality of the distribution obtained in the case of the planar ising model by comparing various observables with their infinite plane prediction we accurately reproduce planar two three and four point of spin and energy operators we also define a lattice stress energy tensor and numerically obtain the associated conformal ward identities and the ising central charge

on an efficient inclusion method for finding polynomial zeros new efficient iterative method of halley s type for the simultaneous inclusion of all simple complex zeros of a polynomial is proposed the presented convergence analysis which uses the concept of the r order of convergence of mutually dependent sequences shows that the convergence rate of the basic fourth order method is increased from 4 to 9 using a two point correction the proposed inclusion method possesses high computational efficiency since the increase of convergence is attained with only one additional function evaluation per sought zero further acceleration of the proposed method is carried out using the gauss seidel procedure some computational aspects and three numerical examples are given in order to demonstrate high computational efficiency and the convergence properties of the proposed methods

on the convergence of wang zheng s method the choice of initial conditions ensuring safe convergence of the implemented iterative method is one of the most important problems in solving polynomial equations these conditions should depend only on the coefficients of a given polynomial p and initial approximations to the zeros of p in this paper we state initial conditions with the described properties for the wang zheng method for the simultaneous approximation of all zeros of p the safe convergence and the fourth order convergence of this method are proved

on halley like algorithms for simultaneous approximation of polynomial complex zeros let p z be a polynomial of degree n with n distinct zeros iterative methods for the simultaneous inclusion of k leqq n simple or multiple zeros are given the iterative formulas presented can be regarded as an interval version of halley s iterative formula for multiple zeros realized in circular arithmetic the convergence analysis is done in the presence of multiplicity the convergence order is three for k n and four if k n applying the gauss seidel approach to the basic iterative formula the single step method with accelerated convergence is stated and the lower bound of the r order of convergence of this method is obtained the dependence of convergence rate of the basic methods on the rounding errors is also studied some numerical examples illustrate the theoretical results

a crossing probability for critical percolation in two dimensions langlands et al considered two crossing probabilities and in their extensive numerical investigations of critical percolation in two dimensions cardy was able to find the exact form of by treating it as a correlation function of boundary operators in the limit of the q state potts model we extend his results to find an analogous formula for which compares very well with the numerical results