random walks on combs we develop techniques to obtain rigorous bounds on the behaviour of random walks on combs using these bounds we calculate exactly the spectral dimension of random combs with infinite teeth at random positions or teeth with random but finite length we also calculate exactly the spectral dimension of some fixed non translationally invariant combs we relate the spectral dimension to the critical exponent of the mass of the two point function for random walks on random combs and compute mean displacements as a function of walk duration we prove that the mean first passage time is generally infinite for combs with anomalous spectral dimension

anomalous diffusion of random walk on random planar maps we prove that the simple random walk on the uniform infinite planar triangulation uipt typically travels graph distance at most n 1 4 o n 1 in n units of time together with the complementary lower bound proven by gwynne and miller 2017 this shows that the typical graph distance displacement of the walk after n steps is n 1 4 o n 1 as conjectured by benjamini and curien geom funct anal 2 2 501 531 2013 arxiv 1202 5454 more generally we show that the simple random walks on a certain family of random planar maps in the gamma liouville quantum gravity lqg universality class for gamma in 0 2 including spanning tree weighted maps bipolar oriented maps and mated crt maps typically travels graph distance n 1 d gamma o n 1 in n units of time where d gamma is the growth exponent for the volume of a metric ball on the map which was shown to exist and depend only on gamma by ding and gwynne commun math phys 374 1877 1934 2018 arxiv 1807 01072 since d gamma 2 this shows that the simple random walk on each of these maps is subdiffusive our proofs are based on an embedding of the random planar maps under consideration into mathbb c wherein graph distance balls can be compared to euclidean balls modulo subpolynomial errors this embedding arises from a coupling of the given random planar map with a mated crt map together with the relationship of the latter map to sle decorated lqg

superintegrability and higher order integrals for quantum systems we refine a method for finding a canonical form of symmetry operators of arbitrary order for the schrodinger eigenvalue equation h 2 v e on any 2d riemannian manifold real or complex that admits a separation of variables in some orthogonal coordinate system the flat space equations with potentials v x iy k 1 x iy k 1 in cartesian coordinates and v r2 r2cos 2k r2sin 2k the tremblay turbiner and winternitz system in polar coordinates have each been shown to be classically superintegrable for all rational numbers k we apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k we develop the classical analog of the quantum canonical form for a symmetry it is clear that our methods will generalize to other hamiltonian systems

fast algorithms for computing isogenies between elliptic curves we survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime we introduce a new algorithm that computes an isogeny of degree l l different from the characteristic in time quasi linear with respect to l this is based in particular on fast algorithms for power series expansion of the weierstrass function and related functions

superintegrable three body systems on the line we consider classical three body interactions on a euclidean line depending on the reciprocal distance of the particles and admitting four functionally independent quadratic in the momentum first integrals these systems are multiseparable superintegrable and equivalent up to rescalings to a one particle system in the three dimensional euclidean space common features of the dynamics are discussed we show how to determine quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics the conformal multiseparability is discussed and examples of conformal first integrals are given the systems considered here in generality include the calogero wolfes and other three body interactions widely studied in mathematical physics

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

spectral dimension of trees with a unique infinite spine using generating functions techniques we develop a relation between the hausdorff and spectral dimension of trees with a unique infinite spine furthermore it is shown that if the outgrowths along the spine are independent and identically distributed then both the hausdorff and spectral dimension can easily be determined from the probability generating function of the random variable describing the size of the outgrowths at a given vertex provided that the probability of the height of the outgrowths exceeding n falls off as the inverse of n we apply this new method to both critical non generic trees and the attachment and grafting model which is a special case of the vertex splitting model resulting in a simplified proof for the values of the hausdorff and spectral dimension for the former and novel results for the latter

a new superintegrable hamiltonian we identify a new superintegrable hamiltonian in three degrees of freedom obtained as a reduction of pure keplerian motion in six dimensions the new hamiltonian is a generalization of the keplerian one and has the familiar 1 r potential with three barrier terms preventing the particle crossing the principal planes in three degrees of freedom there are five functionally independent integrals of motion and all bound classical trajectories are closed and strictly periodic the generalization of the laplace runge lenz vector is identified and shown to provide functionally independent isolating integrals they are quartic in the momenta and do not arise from separability of the hamilton jacobi equation a formulation of the system in action angle variables is presented

invariance principles for random bipartite planar maps it is conjectured in the physics literature that properly rescaled random planar maps when conditioned to have a large number of faces should converge to a limiting surface whose law does not depend up to scaling factors on details of the class of maps that are sampled previous works on the topic starting with chassaing schaeffer have shown that the radius of a random quadrangulation with n faces converges in distribution once rescaled by n 1 4 to the diameter of the brownian snake up to a scaling constant using a bijection due to bouttier di francesco guitter between bipartite planar maps and a family of labeled trees we show the corresponding invariance principle for a class of random maps that follow a boltzmann distribution the radius of such maps conditioned to have n faces or n vertices and under a criticality assumption converges in distribution once rescaled by n 1 4 to a scaled version of the diameter of the brownian snake convergence results for the so called profile of maps are also provided as well as convergence of rescaled bipartite maps to the brownian map as introduced by marckert mokkadem the proofs of these results rely on a new invariance principle for two type spatial galton watson trees

master symmetries non hamiltonian symmetries and superintegrability of the generalized smorodinsky winternitz system the first part of the paper is devoted to the theory of master symmetries using the geometric formalism as an approach it is shown that certain superintegrable systems are endowed with this property as a consequence of the existence of a family of master symmetries in the second part the properties of dynamical but non hamiltonian symmetries are studied it is proved that the higher order superintegrability of the generalized smorodinsky winternitz system is a consequence of the existence of symplectic symmetries not preserving the hamiltonian function

a mating of trees approach for graph distances in random planar maps we introduce a general technique for proving estimates for certain random planar maps which belong to the gamma liouville quantum gravity lqg universality class for gamma in 0 2 the family of random planar maps we consider are those which can be encoded by a two dimensional random walk with i i d increments via a mating of trees bijection and includes the uniform infinite planar triangulation uipt gamma sqrt 8 3 and planar maps weighted by the number of different spanning trees gamma sqrt 2 bipolar orientations gamma sqrt 4 3 or schnyder woods gamma 1 that can be put on the map using our technique we prove estimates for graph distances in the above family of random planar maps in particular we obtain non trivial upper and lower bounds for the cardinality of a graph distance ball consistent with the watabiki prog theor phys suppl 114 1 17 1993 prediction for the hausdorff dimension of gamma lqg and we establish the existence of an exponent for certain distances in the map the basic idea of our approach is to compare a given random planar map m to a mated crt map a random planar map constructed from a correlated two dimensional brownian motion using a strong coupling zaitsev in esaim probab stat 2 41 108 1998 of the encoding walk for m and the brownian motion used to construct the mated crt map this allows us to deduce estimates for graph distances in m from the estimates for graph distances in the mated crt map which we proved using continuum theory in a previous work in the special case when gamma sqrt 8 3 we instead deduce estimates for the sqrt 8 3 mated crt map from known results for the uipt the arguments of this paper do not directly use sle lqg and can be read without any knowledge of these objects

group theory of the smorodinsky winternitz system the three degrees of freedom smorodinsky winternitz system is a degenerate or super integrable hamiltonian that possesses five functionally independent globally defined and single valued integrals of the motion in both classical and quantum mechanics this is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the hamiltonian under a group of symmetry transformations isomorphic to the three dimensional unitary unimodular group su 3 in turn this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other the so called dynamical group all the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group so in common with the hydrogen atom and the harmonic oscillator the quantum mechanics of the smorodinsky winternitz system may be completely solved within the framework of group theory alone

uniform infinite planar triangulations the existence of the weak limit as n of the uniform measure on rooted triangulations of the sphere with n vertices is proved some properties of the limit are studied in particular the limit is a probability measure on random triangulations of the plane

on the master symmetries and bi hamiltonian structure of the toda lattice starting from a conformal symmetry higher order poisson tensors deformation relations and master symmetries for the toda lattice are obtained a hierarchy of time dependent symmetries is also constructed using reduction deformation relations previously known to hold up to a certain equivalence relation are shown to be exact

the complexity of class polynomial computation via floating point approximations we analyse the complexity of computing class polynomials that are an important ingredient for cm constructions of elliptic curves via complex floating point approximations of their roots the heart of the algorithm is the evaluation of modular functions in several arguments the fastest one of the presented approaches uses a technique devised by dupont to evaluate modular functions by newton iterations on an expression involving the arithmetic geometric mean it runs in time o d log 5 d log log d o d 1 epsilon o h 2 epsilon for any epsilon 0 where d is the cm discriminant and h is the degree of the class polynomial another fast algorithm uses multipoint evaluation techniques known from symbolic computation its asymptotic complexity is worse by a factor of log d up to logarithmic factors this running time matches the size of the constructed polynomials the estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary quadratic order and on a rigorously proven upper bound for the height of class polynomials

modular exponentiation via the explicit chinese remainder theorem fix pairwise coprime positive integers p 1 p 2 p a we propose representing integers u modulo m where m is any positive integer up to roughly p 1 p 2 p s as vectors u mod p 1 u mod p 2 u mod p s we use this representation to obtain a new result on the parallel complexity of modular exponentiation there is an algorithm for the common crcw pram that given positive integers x e and m in binary of total bit length n computes x e mod m in time o n lglgn using n o 1 processors for comparison a parallelization of the standard binary algorithm takes superlinear time adleman and kompella gave an o lg n 3 expected time algorithm using exp o n lg n processors von zur gathen gave an nc algorithm for the highly special case that m is polynomially smooth

master symmetries for volterra equation belov chaltikian and blaszak marciniak lattice equations it is shown how to derive master symmetries for nonlinear lattice equations systematically using the basic principles but without using either their zero curvature equations or the bi hamiltonian structure this has been illustrated for volterra equation two coupled belov chaltikian bc and three coupled blaszak marciniak bm lattice equations the existence of a sequence of master symmetries is one of the characteristics of completely integrable nonlinear partial differential and differential difference equations admitting hamiltonian structure

a p adic algorithm to compute the hilbert class polynomial classically the hilbert class polynomial p z x of an imaginary quadratic discriminant a is computed using complex analytic techniques in 2002 couveignes and henocq suggested a p adic algorithm to compute p unlike the complex analytic method it does not suffer from problems caused by rounding errors in this paper we give a detailed description of the algorithm in the paper by couveignes and henocq and our careful study of the complexity shows that if the generalized riemann hypothesis holds true the expected runtime of the p adic algorithm is o log 8 e instead of o 1 e we illustrate the algorithm by computing the polynomial p 639 using a 643 adic algorithm

a new approach to the higher order superintegrability of the tremblay turbiner winternitz system the higher order superintegrability of systems separable in polar coordinates is studied using an approach that was previously applied for the study of the superintegrability of a generalized smorodinsky winternitz system the idea is that the additional constant of motion can be factorized as the product of powers of two particular rather simple complex functions here denoted by m and n this technique leads to a proof of the superintegrability of the tremblay turbiner winternitz system and to the explicit expression of the constants of motion a second family related with the first one of superintegrable systems is also studied

recurrence of planar graph limits we prove that any distributional limit of nite planar graphs in which the degree of the root has an exponential tail is almost surely recurrent as a corollary we obtain that the uniform innite planar triangulation and quadrangulation uipt and uipq are almost surely recurrent resolving a conjecture of angel benjamini and schramm we also settle another related problem of benjamini and schramm we show that in any bounded degree nite planar graph the probability that the simple random walk started at a uniform random vertex avoids its initial location for t steps is at most c logt

computing modular polynomials in quasi linear time we analyse and compare the complexity of several algorithms for computing modular polynomials we show that an algorithm relying on floating point evaluation of modular functions and on interpolation which has received little attention in the literature has a complexity that is essentially up to logarithmic factors linear in the size of the computed polynomials in particular it obtains the classical modular polynomials phi ell of prime level ell in time o ell 3 log 4 ell log log ell besides treating modular polynomials for gamma 0 ell which are an important ingredient in many algorithms dealing with isogenies of elliptic curves the algorithm is easily adapted to more general situations composite levels are handled just as easily as prime levels as well as polynomials between a modular function and its transform of prime level such as the schlafli polynomials and their generalisations our distributed implementation of the algorithm confirms the theoretical analysis by computing modular equations of record level around 10000 in less than two weeks on ten processors

critical exponents on fortuin kasteleyn weighted planar maps in this paper we consider random planar maps weighted by the self dual fortuin kastelyn model with parameter q2 0 4 using a bijection due to sheeld and a connection to planar brownian motion in a cone we obtain rigorously the value of critical exponents associated with the length of cluster interfaces which is shown to be 4 arccos p 2 p q 2

reduction of superintegrable systems the anisotropic harmonic oscillator we introduce a 2n parametric family of maximally superintegrable systems in n dimensions obtained as a reduction of an anisotropic harmonic oscillator in a 2n dimensional configuration space these systems possess closed bounded orbits and integrals of motion which are polynomial in the momenta they generalize known examples of superintegrable models in the euclidean plane

generic ising trees the ising model on a class of infinite random trees is defined as a thermodynamic limit of finite systems a detailed description of the corresponding distribution of infinite spin configurations is given as an application we study the magnetization properties of such systems and prove that they exhibit no spontaneous magnetization furthermore the values of the hausdorff and spectral dimensions of the underlying trees are calculated and found to be respectively and

a rigorous time bound for factoring integers in this paper a probabilistic algorithm is exhibited that factors any positive integer n into prime factors in expected time at most ln 2 1 o for n oo where l a b exp b logx a loglogx l a many practical factoring algorithms including the quadratic sieve and the elliptic curve method are conjectured to have an expected running time that satisfies the same bound but this is the first algorithm for which the bound can be rigorously proved nevertheless this does not close the gap between rigorously established time bounds and merely conjectural ones for factoring algorithms this is due to the advent of a new factoring algorithm the number field sieve which is conjectured to factor any positive integer n in time ln i 0 1 the algorithm analyzed in this paper is a variant of the class group relations method which makes use of class groups of binary quadratic forms of negative discriminant this algorithm was first suggested by seysen and later improved by a k lenstra who showed that the algorithm runs in expected time at most ln 12 1 o 1 if one assumes the generalized riemann hypothesis the main device for removing the use of the generalized riemann hypothesis from the proof is the use of multipliers in addition a character sum estimate for algebraic number fields is used with an explicit dependence on possible exceptional zeros of the corresponding l functions another factoring algorithm using class groups that has been proposed is the random class groups method it is shown that there is a fairly large set of numbers that this algorithm cannot be expected to factor as efficiently as had previously been thought department of mathematics university of california berkeley california 94720 e mail address hwl math berkeley edu department of mathematics university of georgia athens georgia 30602 e mail address carl ada math uga edu this content downloaded from 157 55 39 186 on tue 12 apr 2016 08 54 17 utc all use subject to http about jstor org terms

superintegrability of the calogero moser system constants of motion master symmetries and time dependent symmetries the classical n dimensional calogero moser system is a maximally superintegrable system endowed with a rich variety of symmetries and constants of motion in the first part of the article some properties related with the existence of several families of constants of motion are analyzed in the second part the master symmetries and the time dependent symmetries of this system are studied

local convergence of large random triangulations coupled with an ising model we prove the existence of the local weak limit of the measure obtained by sampling random triangulations of size n decorated by an ising configuration with a weight proportional to the energy of this configuration to do so we establish the algebraicity and the asymptotic behaviour of the partition functions of triangulations with spins for any boundary condition in particular we show that these partition functions all have the same phase transition at the same critical temperature some properties of the limiting object called the infinite ising planar triangulation are derived including the recurrence of the simple random walk at the critical temperature

tangent bundle geometry lagrangian dynamics various aspects of the differential geometry of the tangent bundle of a differentiable manifold are examined and the results applied to time independent lagrangian dynamics it is shown that a certain type 1 1 tensor field which is part of the intrinsic geometry of a tangent bundle being a tensorial equivalent of the projection map of tangent vectors plays a role in lagrangian theory scarcely less important than that of the canonical one form on a cotangent bundle in hamiltonian theory recent results in lagrangian theory are interpreted from this new viewpoint

random walks polymers percolation and quantum gravity in two dimensions we consider l planar random walks or brownian motions of large length t starting at neighboring points and the probability pl t t l that their paths do not intersect by a 2d quantum gravity method i e a non linear map onto a random riemann surface the former conjecture that l 124 4l2 1 is established this also applies to the half plane where l l3 1 2l as well as to non intersection exponents of unions of paths mandelbrot s conjecture for the hausdorff dimension dh 43 of the frontier of a brownian path follows from dh 2 32 we then consider in two dimensions the most general star shaped copolymer mixing random rw or self avoiding walks saw with specific mutual avoidance interactions thereof its exact conformal scaling dimensions in the plane are derived the harmonic measures or electrostatic potential or diffusion field near a rw or a saw or near a critical percolation cluster are also considered their moments exhibit a multifractal spectrum the generalized dimensions d n as well as the function are derived and are shown to be all identical these are examples of exact conformal multifractality

critical ising model on random triangulations of the disk enumeration and local limits we consider boltzmann random triangulations coupled to the ising model on their faces under dobrushin boundary conditions and at the critical point of the model the first part of this paper computes explicitly the partition function of this model by solving its tutte s equation extending a previous result by bernardi and bousquet melou j combin theory ser b 101 5 315 377 2011 to the model with dobrushin boundary conditions we show that the perimeter exponent of the model is 7 3 in contrast to the exponent 5 2 for uniform triangulations in the second part we show that the model has a local limit in distribution when the two components of the dobrushin boundary tend to infinity one after the other the local limit is constructed explicitly using the peeling process along an ising interface moreover we show that the main interface in the local limit touches the infinite boundary almost surely only finitely many times a behavior opposite to that of the bernoulli percolation on uniform maps some scaling limits closely related to the perimeters of finite clusters are also obtained

higher order superintegrability of separable potentials with a new approach to the post winternitz system the higher order superintegrability of separable potentials is studied it is proved that these potentials possess in addition to the two quadratic integrals a third integral of higher order in the momenta that can be obtained as the product of powers of two particular rather simple complex functions some systems related to the harmonic oscillator such as the generalized sw system and the ttw system were studied in previous papers now a similar analysis is presented for superintegrable systems related to the kepler problem in this way a new proof of the superintegrability of the post winternitz system is presented and the explicit expression for the integral is obtained finally the relations between the superintegrable systems with quadratic constants of motion separable in several different coordinate systems and the superintegrable systems with higher order constants of motion are analyzed

explicit bounds for primality testing and related problems many number theoretic algorithms rely on a result of ankeny which states that if the extended riemann hypothesis erh is true any nontrivial multiplicative subgroup of the integers modulo m omits a number that is o log2 m this has been generalized by lagarias montgomery and odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields this paper gives a different proof of this theorem in which explicit constants are supplied the bounds imply that if the erh holds a composite number m has a witness for its compositeness in the sense of miller or solovay strassen that is at most 2 log2m

growth of uniform infinite causal triangulations we introduce a growth process which samples sections of uniform infinite causal triangulations by elementary moves in which a single triangle is added a relation to a random walk on the integer half line is shown this relation is used to estimate the geodesic distance of a given triangle to the rooted boundary in terms of the time of the growth process and to determine from this the fractal dimension furthermore convergence of the boundary process to a diffusion process is shown leading to an interesting duality relation between the growth process and a corresponding branching process

the topological structure of scaling limits of large planar maps we discuss scaling limits of large bipartite planar maps if p 2 is a fixed integer we consider for every integer n 2 a random planar map mn which is uniformly distributed over the set of all rooted 2p angulations with n faces then at least along a suitable subsequence the metric space consisting of the set of vertices of mn equipped with the graph distance rescaled by the factor n 1 4 converges in distribution as n towards a limiting random compact metric space in the sense of the gromov hausdorff distance we prove that the topology of the limiting space is uniquely determined independently of p and of the subsequence and that this space can be obtained as the quotient of the continuum random tree for an equivalence relation which is defined from brownian labels attached to the vertices we also verify that the hausdorff dimension of the limit is almost surely equal to 4

limit of normalized quadrangulations the brownian map consider qn a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces in this paper we show that when n goes to qn suitably normalized converges weakly in a certain sense to a random limit object which is continuous and compact and that we name the brownian map the same result is shown for a model of rooted quadrangulations and for some models of rooted quadrangulations with random edge lengths a metric space of rooted resp pointed abstract maps that contains the model of discrete rooted resp pointed quadrangulations and the model of the brownian map is defined the weak convergences hold in these metric spaces

polynomial constants of motion for calogero type systems in three dimensions we give an explicit and concise formula for higher degree polynomial first integrals of a family of calogero type hamiltonian systems in dimension three these first integrals together with the already known ones prove the maximal superintegrability of the systems

a glimpse of the conformal structure of random planar maps we present a way to study the conformal structure of random planar maps the main idea is to explore the map along an sle schramm loewner evolution process of parameter kappa 6 and to combine the locality property of the sle6 together with the spatial markov property of the underlying lattice in order to get a non trivial geometric information we follow this path in the case of the conformal structure of random triangulations with a boundary

n 2 supersymmetric extension of the tremblay turbiner winternitz hamiltonians on a plane the family of tremblay turbiner winternitz hamiltonians hk on a plane corresponding to any positive real value of k is shown to admit an supersymmetric extension of the same kind as that introduced by freedman and mende for the calogero problem and based on an superalgebra the irreducible representations of the latter are characterized by the quantum number specifying the eigenvalues of the first integral of motion xk of hk bases for them are explicitly constructed the ground state of each supersymmetrized hamiltonian is shown to belong to an atypical lowest weight state irreducible representation

modular polynomials via isogeny volcanoes we present a new algorithm to compute the classical modular polynomial l in the rings z x y and z mz x y for a prime l and any positive integer m our approach uses the graph of l isogenies to eciently compute l mod p for many primes p of a suitable form and then applies the chinese remainder theorem crt under the generalized riemann hypoth esis grh we achieve an expected running time of o l 3 logl 3 log logl and compute l mod m using o l 2 logl 2 l 2 logm space we have used the new algorithm to compute l with l over 5000 and l mod m with l over 20000 we also consider several modular functions g for which g is smaller than l allowing us to handle l over 60000

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

hamiltonians separable in cartesian coordinates and third order integrals of motion we present in this article all hamiltonian systems in e 2 that are separable in cartesian coordinates and that admit a third order integral both in quantum and in classical mechanics many of these superintegrable systems are new and it is seen that there exists a relation between quantum superintegrable potentials invariant solutions of the korteweg de vries equation and the painleve transcendents

appearance of vertices of infinite order in a model of random trees we study an equilibrium statistical mechanical model of tree graphs which are made up of a linear subgraph the spine to which leaves are attached we prove that the model has two phases a generic phase where the spine becomes infinitely long in the thermodynamic limit and all vertices have finite order and a condensed phase where the spine is finite with probability 1 and a single vertex of infinite order appears in the thermodynamic limit we calculate the spectral dimension of the graphs in both phases and prove the existence of a gibbs measure we discuss generalizations of this model and the relationship with models of nongeneric random trees

third order superintegrable systems separable in parabolic coordinates in this paper we investigate superintegrable systems which separate in parabolic coordinates and admit a third order integral of motion we give the corresponding determining equations and show that all such systems are multi separable and so admit two second order integrals the third order integral is their lie or poisson commutator we discuss how this situation is different from the cartesian and polar cases where new potentials were discovered which are not multi separable and which are expressed in terms of painleve transcendents or elliptic functions

computing l l isogenies in polynomial time on jacobians of genus 2 curves in this paper we compute l isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in l when l is an odd prime which is coprime to the characteristic we use level n symmetric theta structure where n 2 or n 4 in a second part of this paper we explain how to convert between mumford coordinates of jacobians of genus 2 hyperelliptic curves to theta coordinates of level 2 or 4 combined with the preceding algorithm this gives a method to compute l l isogenies in polynomial time on jacobians of genus 2 curves

a quantum exactly solvable nonlinear oscillator related to the isotonic oscillator a nonpolynomial one dimensional quantum potential representing an oscillator which can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator harmonic oscillator with a centripetal barrier is studied first the general case that depends on a parameter a is considered and then a particular case is studied with great detail it is proven that it is schr dinger solvable and then the wavefunctions n and the energies en of the bound states are explicitly obtained finally it is proven that the solutions determine a family of orthogonal polynomials related to the hermite polynomials and such that i every is a linear combination of three hermite polynomials and ii they are orthogonal with respect to a new measure obtained by modifying the classic hermite measure

lagrangian formalism for nonlinear second order riccati systems one dimensional integrability and two dimensional superintegrability the existence of a lagrangian description for the second order riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized riccati equation the lagrangians are non natural and the forces are not derivable from a potential the constant value e of a preserved energy function can be used as an appropriate parameter for characterizing the behavior of the solutions of these two systems in the second part the existence of two dimensional versions endowed with superintegrability is proved the explicit expressions of the additional integrals are obtained in both cases finally it is proved that the orbits of the second system that represents a nonlinear oscillator can be considered as nonlinear lissajous figures

planar stochastic hyperbolic triangulations pursuing the approach of angel and ray ann probab 2015 we introduce and study a family of random infinite triangulations of the full plane that satisfy a natural spatial markov property these new random lattices naturally generalize angel and schramm s uniform infinite planar triangulation uipt and are hyperbolic in flavor we prove that they exhibit a sharp exponential volume growth are non liouville and that the simple random walk on them has positive speed almost surely we conjecture that these infinite triangulations are the local limits of uniform triangulations whose genus is proportional to the size an artistic representation of a random 3 connected triangulation of the plane with hyperbolic flavor

local convergence for permutations and local limits for uniform rho avoiding permutations with rho 3 we set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of consecutive pattern occurrences we also characterize random limiting objects for this new topology introducing a notion of shift invariant property corresponding to the notion of unimodularity for random graphs we then study two models in the framework of random pattern avoiding permutations we compute the local limits of uniform rho avoiding permutations for rho 3 when the size of the permutations tends to infinity the core part of the argument is the description of the asymptotics of the number of consecutive occurrences of any given pattern for this result we use bijections between rho avoiding permutations and rooted ordered trees local limit results for galton watson trees the second moment method and singularity analysis

a new potential with the spectrum of an isotonic oscillator with the method of factorisation a new potential with free parameters is generated this potential possesses exactly the same energy spectrum as the isotonic oscillator harmonic oscillator with centripetal barrier the wavefunctions and shift operators of the new system are also presented

the half plane uipt is recurrent we prove that the half plane version of the uniform infinite planar triangulation uipt is recurrent the key ingredients of the proof are a construction of a new full plane extension of the half plane uipt based on a natural decomposition of the half plane uipt into independent layers and an extension of previous methods for proving recurrence of weak local limits while still using circle packings

an infinite family of solvable and integrable quantum systems on a plane an infinite family of exactly solvable and integrable potentials on a plane is introduced it is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family the underlying algebraic structure of the new potentials is revealed as well as its hidden algebra we conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases a quasi exactly solvable and integrable generalization of the family is found

percolation on random triangulations and stable looptrees we study site percolation on angel and schramm s uniform infinite planar triangulation we compute several critical and near critical exponents and describe the scaling limit of the boundary of large percolation clusters in all regimes subcritical critical and supercritical we prove in particular that the scaling limit of the boundary of large critical percolation clusters is the random stable looptree of index 3 2 which was introduced in curien and kortchemski random stable looptrees arxiv 1304 1044 2014 we also give a conjecture linking looptrees of any index alpha in 1 2 with scaling limits of cluster boundaries in random triangulations decorated with o n models

third order superintegrable systems separating in polar coordinates a complete classification of quantum and classical superintegrable systems in e2 is presented that allow the separation of variables in polar coordinates and admit an additional integral of motion of order 3 in the momentum new quantum superintegrable systems are discovered for which the potential is expressed in terms of the sixth painleve transcendent or in terms of the weierstrass elliptic function

unimodular hyperbolic triangulations circle packing and random walk we show that the circle packing type of a unimodular random plane triangulation is parabolic if and only if the expected degree of the root is six if and only if the triangulation is amenable in the sense of aldous and lyons 1 as a part of this we obtain an alternative proof of the benjamini schramm recurrence theorem 19 secondly in the hyperbolic case we prove that the random walk almost surely converges to a point in the unit circle that the law of this limiting point has full support and no atoms and that the unit circle is a realisation of the poisson boundary finally we show that the simple random walk has positive speed in the hyperbolic metric

scaling limits of random planar maps with a unique large face we study random bipartite planar maps defined by assigning nonnegative weights to each face of a map we prove that for certain choices of weights a unique large face having degree proportional to the total number of edges in the maps appears when the maps are large it is furthermore shown that as the number of edges n of the planar maps goes to infinity the profile of distances to a marked vertex rescaled by n 1 2 is described by a brownian excursion the planar maps with the graph metric resealed by n 1 2 are then shown to converge in distribution toward aldous brownian tree in the gromov hausdorff topology in the proofs we rely on the bouttier di francesco guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large

on the spectral dimension of causal triangulations we introduce an ensemble of infinite causal triangulations called the uniform infinite causal triangulation and show that it is equivalent to an ensemble of infinite trees the uniform infinite planar tree it is proved that in both cases the hausdorff dimension almost surely equals 2 the infinite causal triangulations are shown to be almost surely recurrent or equivalently their spectral dimension is almost surely less than or equal to 2 we also establish that for certain reduced versions of the infinite causal triangulations the spectral dimension equals 2 both for the ensemble average and almost surely the triangulation ensemble we consider is equivalent to the causal dynamical triangulation model of two dimensional quantum gravity and therefore our results apply to that model

hecke stability and weight 1 modular forms the galois representations associated to weight 1 eigenforms over bar mathbb f p are remarkable in that they are unramified at p but the effective computation of these modular forms presents challenges one complication in this setting is that a weight 1 cusp form over bar mathbb f p need not arise from reducing a weight 1 cusp form over bar mathbb q in this article we propose a unified hecke stability method for computing spaces of weight 1 modular forms of a given level in all characteristics simultaneously our main theorems outline conditions under which a finite dimensional hecke module of ratios of modular forms must consist of genuine modular forms we conclude with some applications of the hecke stability method that are motivated by the refined inverse galois problem

martingales in self similar growth fragmentations and their connections with random planar maps the purpose of the present work is twofold first we develop the theory of general self similar growth fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for branching random walks in this framework in particular we establish many to one formulas for growth fragmentations and define the notion of intrinsic area of a growth fragmentation second we identify a distinguished family of growth fragmentations closely related to stable levy processes which are then shown to arise as the scaling limit of the perimeter process in markovian explorations of certain random planar maps with large degrees which are roughly speaking the dual maps of the stable maps of le gall and miermont in ann probab 39 1 69 2011 as a consequence of this result we are able to identify the law of the intrinsic area of these distinguished growth fragmentations this generalizes a geometric connection between large boltzmann triangulations and a certain growth fragmentation process which was established in bertoin et al ann probab accepted

local limit of labeled trees and expected volume growth in a random quadrangulation exploiting a bijective correspondence between planar quadrangulations and well labeled trees we define an ensemble of infinite surfaces as a limit of uniformly distributed ensembles of quadrangulations of fixed finite volume the limit random surface can be described in terms of a birth and death process and a sequence of multitype galton watson trees as a consequence we find that the expected volume of the ball of radius r around a marked point in the limit random surface is r4

periodic orbits for an infinite family of classical superintegrable systems we show that all bounded trajectories in the two dimensional classical system with the potential are closed for all integer and rational values of k the period is and does not depend on k this agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable

non symplectic symmetries and bi hamiltonian structures of the rational harmonic oscillator the existence of bi hamiltonian structures for the rational harmonic oscillator non central harmonic oscillator with rational ratio of frequencies is analysed by making use of the geometric theory of symmetries we prove that these additional structures are a consequence of the existence of dynamical symmetries of non symplectic non canonical type the associated recursion operators are also obtained

geodesics in large planar maps and in the brownian map we study geodesics in the random metric space called the brownian map which appears as the scaling limit of large planar maps in particular we completely describe geodesics starting from the distinguished point called the root and we characterize the set s of all points that are connected to the root by more than one geodesic the set s is dense in the brownian map and homeomorphic to a non compact real tree furthermore for every x in s the number of distinct geodesics from x to the root is equal to the number of connected components of s x in particular points of the brownian map can be connected to the root by at most three distinct geodesics our results have applications to the behavior of geodesics in large planar maps

computing hilbert class polynomials with the chinese remainder theorem we present a space efficient algorithm to compute the hilbert class polynomial h d x modulo a positive integer p based on an explicit form of the chinese remainder theorem under the generalized riemann hypothesis the algorithm uses o d 1 2 e log p space and has an expected running time of o d 1 e we describe practical optimizations that allow us to handle larger discriminants than other methods with d as large as 10 13 and h d up to 10 6 we apply these results to construct pairing friendly elliptic curves of prime order using the cm method