unique normal forms the nilpotent hamiltonian case the problem we consider in this paper namely the characterization of unique normal forms derives from bifurcation theory in a typical situation one has an unperturbed vector field near a degenerate equilibrium in order to study the effect of generic perturbations suitable coordinates are introduced and the simplified system is said to be in normal form a well known case is the hopf bifurcation where a pair of conjugate eigenvalues cross the imaginary axis using averaging one can in this case compute the periodic orbit that splits off from the equilib um a second familiar example is the takens bogdanov bifurcation where two eigenvalues are zero in all these situations it is necessary to make certain choices both in the definition of what constitutes a normal form and in the computation of the transformation the idea here is to use this second kind of choice to sharpen the very definition of normal form while in standard treatments only the linear part of the field plays a role we show that one can effectively use its non linear part to remove additional terms and thus simplify the problem further the underlying philosophy in this type of game is that

characterization of a monodromic singular point of a planar vector field the newton diagram and the lowest degree quasi homogeneous terms of an analytic planar vector field allow us to determine whether an isolated singular point of the vector field is monodromic or has a characteristic trajectory

normal forms for hopf zero singularities with nonconservative nonlinear part abstract in this paper we are concerned with the simplest normal form computation of the systems 0 1 x 2 x f x y 2 z 2 y z y f x y 2 z 2 z y z f x y 2 z 2 where f is a formal function with real coefficients and without any constant term these are the classical normal forms of a larger family of systems with hopf zero singularity indeed these are defined such that this family would be a lie subalgebra for the space of all classical normal form vector fields with hopf zero singularity the simplest normal forms and simplest orbital normal forms of this family with nonzero quadratic part are computed we also obtain the simplest parametric normal form of any non degenerate perturbation of this family within the lie subalgebra the symmetry group of the simplest normal forms is also discussed this is a part of our results in decomposing the normal forms of hopf zero singular systems into systems with a first integral and nonconservative systems

an improved theory of asymptotic unfoldings abstract an asymptotic unfolding of a dynamical system near a rest point is a system with additional parameters such that every one parameter deformation of the original system can be embedded in the unfolding preserving all properties that can be detected by asymptotic methods asymptotic unfoldings are computed using normal and hypernormal form methods we present a simplified and improved method of computing such unfoldings that can be used in any normal form style

sub gaussian estimates of heat kernels on infinite graphs we prove that a two sided sub gaussian estimate of the heat kernel on an infinite weighted graph takes place if and only if the volume growth of the graph is uniformly polynomial and the green kernel admits a uniform polynomial decay

semi hyperbolic patches of solutions to the two dimensional euler equations we construct semi hyperbolic patches of solutions in which one family out of two nonlinear families of characteristics starts on sonic curves and ends on transonic shock waves to the two dimensional euler equations this type of solution appears in the transonic flow over an airfoil and guderley reflection and is common in the numerical solutions of riemann problems

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

unique normal forms for planar vector fields soit x un germe d un champ vectoriel analytique ou formel en equilibre a o r 2 avec des valeurs propres purement imaginaires a 0 i a 0 0 alors x est formellement equivalent a precisement un des champs vectoriels polynomiaux suivant ou x 1 2 cos y 1 2 sin pour des coordonnees appropriees x y 7b a 0 a 0 7b a k a 0 k 7b b j a 0 a 1 a j j j 1 b j j 0

harnack inequalities and sub gaussian estimates for random walks we show that the beta parabolic harnack inequality for random walks on graphs is equivalent on one hand to the sub gaussian estimate for the transition probability and on the other hand to the conjunction of the elliptic harnack inequality the doubling volume property and the fact that the mean exit time in any ball of radius r is of the order r beta the latter condition can be replaced by a certain estimate of the resistance of annuli

unique normal forms for vector fields and hamiltonians on construit des formes canoniques pour tous les champs vectoriels resp hamiltoniens en equilibre a 0 r n relativement au groupe des diffeomorphismes formels resp des transformations canoniques tangentiels a l identite

convergence versus integrability in birkhoff normal form we show that any analytically integrable hamiltonian system near an equilibrium point admits a convergent birkhoff normalization the proof is based on a new geometric approach to the topic

versal normal form for nonsemisimple singularities abstract the theory of versal normal form has been playing a role in normal form since the introduction of the concept by v i arnol d in 1 2 but there has been no systematic use of it that is in line with the semidirect character of the group of formal transformations on formal vector fields that is the linear part should be done completely first before one computes the nonlinear terms in this paper we address this issue by giving a complete description of a first order calculation in the case of the two and three dimensional irreducible nilpotent cases which is then followed up by an explicit almost symplectic calculation to find the transformation to versal normal form in a particular fluid dynamics problem and in the celestial mechanics l 4 problem

on the two dimensional gas expansion for compressible euler equations we investigate the problem of two dimensional unsteady expansion of an inviscid polytropic gas which can be interpreted as the collapse of a wedge shaped dam containing water initially with a uniform velocity we model this problem by isentropic euler equations the flow is quasi stationary and using hodograph transform we describe it by a partial differential equation of second order in the state space if it is irrotational initially furthermore this equation is reduced to a linearly degenerate system of three partial differential equations with inhomogeneous source terms these properties are used to prove that the flow is globally smooth when a wedge of gas expands into a vacuum and to analyze that shocks may appear in the interaction of four planar rarefaction waves

uniformization of two dimensional metric surfaces we establish uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite hausdorff 2 measure applying the geometric definition of quasiconformality we give a necessary and sufficient condition for such spaces to be qc equivalent to the euclidean plane disk or sphere moreover we show that if such a qc parametrization exists then the dilatation can be bounded by 2 as an application we show that the euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2 qc parametrization this result gives a new approach to the bonk kleiner theorem on parametrizations of ahlfors 2 regular spheres by quasisymmetric maps

unique normal form of a class of 3 dimensional vector fields with symmetries abstract we present the unique normal form of a class of 3 dimensional vector fields bt zero singularity with symmetries the main technique applied to the computation is the combination of a linear grading function and the method of multiple lie brackets we introduce new notations for block matrices to simplify the expression of block matrices the new notations help to prove the non degeneracy of huge size matrices by an induction technique

the boundary of a square tiling of a graph coincides with the poisson boundary answering a question of benjamini and schramm ann probab 24 3 1219 1238 1996 we show that the poisson boundary of any planar uniquely absorbing e g one ended and transient graph with bounded degrees can be realised geometrically as a circle which arises from a discrete version of riemann s mapping theorem this implies a conjecture of northshield potential anal 2 4 299 314 1993 some of our technique apply to the non planar case and might have further applications when the graph is also hyperbolic then under mild conditions we prove the equivalence of several boundary constructions

orbital formal normal forms for general bogdanov takens singularity abstract we solve completely the problem of classification of germs of complex planar vector fields with nilpotent singularity with respect to formal orbital equivalence

analytic normalization of analytically integrable differential systems near a periodic orbit abstract for an analytic differential system in r n with a periodic orbit we will prove that if the system is analytically integrable around the periodic orbit i e it has n 1 functionally independent analytic first integrals defined in a neighborhood of the periodic orbit then the system is analytically equivalent to its poincare dulac type normal form this result is an extension of analytically integrable differential systems around a singularity to the ones around a periodic orbit

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

the analytic and formal normal form for the nilpotent singularity abstract we study orbital normal forms for analytic planar vector fields with nilpotent singularity we show that the takens normal form is analytic in the case of generalized cusp we present the complete formal orbital normal form it contains functional moduli we interprete the coefficients of these moduli in terms of the hidden holonomy group

global solution of an initial value problem for two dimensional compressible euler equations abstract this paper is concerned with the existence of global continuous solutions of the expansion of a wedge of gas into a vacuum for compressible euler equations by hodograph transformation we first prove that the flow is governed by a partial differential equation of second order which is further reduced to a system of two nonhomogeneous linearly degenerate equations in the phase space under an irrotationality condition then this conclusion is applied to solving the problem that a wedge of gas expands into a vacuum which is actually a goursat type problem for these two equations in the supersonic domain

brownian motion on the sierpinski gasket we construct a brownian motion taking values in the sierpinski gasket a fractal subset of 2 and study its properties this is a diffusion process characterized by local isotropy and homogeneity properties we show for example that the process has a continuous symmetric transition density p t x y with respect to an appropriate hausdorff measure and obtain estimates on p t x y

normal form for the 2 n nilpotent vector field using unvariant theory abstract here we give an algorithm for finding the normal form of formal power series vector fields on r 2n whose linear terms are the nilpotent matrix made up of 2 2 jordan blocks the problem is reduced to finding the stanley decomposition of a bracket algebra which is then determined using a straightening algorithm the 4 and 6 dimensional cases are explicitly worked out

stability of parabolic harnack inequalities let g e be a graph with weights a xy for which a parabolic harnack inequality holds with space time scaling exponent 2 suppose xy is another set of weights that are comparable to a xy we prove that this parabolic harnack inequality also holds for g e with the weights a xy we also give stable necessary and sufficient conditions for this parabolic harnack inequality to hold

axisymmetric solutions of the euler equations for polytropic gases we construct rigorously a three parameter family of self similar globally bounded and continuous weak solutions in two space dimensions for all positive time to the euler equations with axisymmetry for polytropic gases with a quadratic pressure density law we use the axisymmetry and self similarity assumptions to reduce the equations to a system of three ordinary differential equations from which we obtain detailed structures of solutions besides their existence these solutions exhibit familiar structures seen in hurricanes and tornadoes they all have finite local energy and vorticity with well defined initial and boundary values these solutions include the one parameter family of explicit solutions reported in a recent article of ours

reversibility and quasi homogeneous normal forms of vector fields abstract this paper uses tools in quasi homogeneous normal form theory to discuss certain aspects of reversible vector fields around an equilibrium point our main result provides an algorithm via lie triangle that detects the non reversibility of vector fields that is it is possible to decide whether a planar center is not reversible some of the theory developed is also applied to get further results on nilpotent and degenerate polynomial vector fields we find several families of nilpotent centers which are non reversible

the center problem a view from the normal form theory abstract in this work we analyze some aspects of the center problem from the perspective of the normal form theory we provide alternative proofs of some well known results in the case of non vanishing linear part nondegenerate and nilpotent centers moreover some new results are also derived in particular we show that the unique characterization for centers in this situation is the orbital reversibility also we include some examples illustrating that this characterization is no longer valid for the case of null linear part

further reductions of normal forms for dynamical systems abstract we propose in this paper a method for obtaining a significant refinement of normal forms for dynamical systems or vector fields with concrete and interesting applications we use lower order nonlinear terms in the normal form for the simplifications of higher order terms our approach is applicable for both the non nilpotent and the nilpotent cases for dynamical systems of dimensions 2 and 3 we give an algorithm that leads to interesting finite order normal forms which are optimal or unique with respect to equivalence by formal near identity transformations we can compute at the same time a formal diffeormorphism that realizes the normalization comparisons with other methods are given for several examples

spectral sequences and parametric normal forms abstract this paper extends recent developments on spectral sequence approach on the simplest normal form theory to parametric cases in this paper the method is first applied to two simple examples parametric single zero and parametric resonant saddle singularities and then to parametric generalized hopf singularity we provide the simplest parametric normal form for these cases

analytic integrability around a nilpotent singularity abstract in this work it is characterized the analytic integrability problem around a nilpotent singularity of a differential system in the plane under generic conditions

linear grading function and further reduction of normal forms abstract in this note an idea of quasi homogeneous normal form theory using new grading functions is introduced the definition ofnth order normal form is given and some sufficient conditions for the uniqueness of normal forms are derived a special case of the unsolved problem in a paper of baider and sanders for the unique normal form of bogdanov takens singularities is solved

harmonic functions on planar and almost planar graphs and manifolds via circle packings the circle packing theorem is used to show that on any bounded valence transient planar graph there exists a non constant harmonic bounded dirichlet function if p is a bounded circle packing in bbb r 2 whose contacts graph is a bounded valence triangulation of a disk then with probability 1 the simple random walk on p converges to a limit point moreover in this situation any continuous function on the limit set of p extends to a continuous harmonic function on the closure of the contacts graph of p that is this dirichlet problem is solvable we define the notions of almost planar graphs and manifolds and show that under the assumptions of transience and bounded local geometry these possess non constant harmonic bounded dirichlet functions let us stress that an almost planar graph is not necessarily roughly isometric to a planar graph

hypernormal form theory foundations and algorithms abstract hypernormal forms unique normal forms simplest normal forms are investigated both from the standpoint of foundational theory and algorithms suitable for use with computer algebra the baider theory of the campbell hausdorff group is refined by a study of its subgroups to determine the smallest substages into which the hypernormalization process can be divided this leads to a linear algebra algorithm to compute the generators needed for each substage with the least amount of work a concrete interpretation of jan sanders spectral sequence for hypernormal forms is presented examples are given and a proof is given for a little known theorem of belitskii expressing the hypernormal form space in the inner product style as the kernel of a higher order differential operator

box products in nilpotent normal form theory the factoring method abstract let n be a nilpotent matrix and consider vector fields x n x v x in normal form then v is equivariant under the flow e n t for the inner product normal form or e m t for the sl 2 normal form these vector equivariants can be found by finding the scalar invariants for the jordan blocks in n or m taking the box product of these to obtain the invariants for n or m itself and then boosting the invariants to equivariants by another box product these methods developed by murdock and sanders in 2007 are here given a self contained exposition with new foundations and new algorithms yielding improved simpler stanley decompositions for the invariants and equivariants ideas used include transvectants from classical invariant theory stanley decompositions from commutative algebra and integer cones from integer programming this approach can be extended to covariants of sl 2 k for k 1 known as slocc in quantum computing

bifurcations from a center at infinity in 3d piecewise linear systems with two zones abstract we consider continuous piecewise linear systems in r 3 with two zones under the assumption of having a linear center in the invariant manifold of the point at infinity a specific conic projection is introduced so that it is possible to analyze in a convenient way the dynamics near such a center at infinity in two qualitative different situations in the semi homogeneous case such a center is associated to the existence of a continuum of invariant semi cones sharing the vertex at the origin perturbing the configuration it is possible to detect the bifurcation of a limit cycle at infinity leading to the bifurcation of isolated invariant semi cones for the non homogeneous case the cycles at infinity does not imply invariant semi cones however the non generic case when the center at infinity is associated to the existence of invariant cylinders for one of the involved vector fields becomes rather interesting it is possible then by perturbing the other vector field to get the bifurcation of a big limit cycle from infinity without destroying the center

vector potential normal form classification for completely integrable solenoidal nilpotent singularities abstract we introduce an sl 2 invariant family of polynomial vector fields with an irreducible nilpotent singularity in this paper we are concerned with characterization and normal form classification of these vector fields we show that the family is a lie subalgebra and each vector field from this family is volume preserving completely integrable and rotational all such vector fields share a common quadratic invariant we provide a poisson structure for the lie subalgebra from which the second invariant for each vector field can be readily derived we show that each vector field from this family can be uniquely characterized by two alternative representations which can be found in applications one uses a vector potential while the other uses two functionally independent clebsch potentials our normal form results are designed to preserve these structures and representations

computation of the simplest normal forms with perturbation parameters based on lie transform and rescaling normal form theory is one of the most power tools for the study of nonlinear differential equations in particular for stability and bifurcation analysis recently many researchers have paid attention to further reduction of conventional normal forms cnf to so called the simplest normal form snf however the computation of normal forms has been restricted to systems which do not contain perturbation parameters unfolding the computation of the snf is more involved than that of cnfs and the computation of the snf with unfolding is even more complicated than the snf without unfolding although some author mentioned further reduction of the snf no results have been reported on the exact computation of the snf of systems with perturbation parameters this paper presents an efficient method for computing the snf of differential equations with perturbation parameters unlike cnf theory which uses an independent nonlinear transformation at each order this approach uses a consistent nonlinear transformation through all order computations the particular advantage of the method is able to provide an efficient recursive formula which can be used to obtain the nth order equations containing the nth order terms only this greatly saves computational time and computer memory the recursive formulations have been implemented on computer systems using maple as an illustrative example the snf for single zero singularity is considered using the new approach

quasi homogeneous normal forms in this paper we extend the concepts of the normal form theory for vector fields that are expanded in quasi homogeneous components of a fixed type these expansions have been used by some authors in the analysis of the determinacy of a given singularity also the use of reparametrizations in the time are considered namely beyond the use of c conjugation to determine normal forms we present a method useful to determine how much a vector field can be simplified by using c equivalence the results obtained are applied in the case of the bogdanov takens singularity firstly using c conjugation and later showing the improvements provided by the c equivalence

topological equivalence of a plane vector field with its principal part defined through newton polyhedra abstract we introduce a blowing up of singularities of vector fields associated with newton polyhedra in the space of the exponents by means of which we prove a generalization of the hartman grobman theorem namely a plane vector field possessing characteristic orbits is locally topologically equivalent with its principal part under suitable non degeneracy hypotheses under stronger hypotheses a similar equivalence result is proven between a plane vector field and a single quasihomogeneous component of its principal part the case of second order equations is also studied

characterization of sub gaussian heat kernel estimates on strongly recurrent graphs sub gaussian estimates for random walks are typical of fractal graphs we characterize them in the strongly recurrent case in terms of resistance estimates only without assuming elliptic harnack inequalities

axisymmetric solutions of the pressure gradient system a family of self similar and global bounded weak solutions are constructed rigorously for all positive time to the two dimensional pressure gradient system in this paper by using the axisymmetric and self similar assumptions the equations are reduced to a system of ordinary differential equations from which we obtain detailed structures of solutions as well as their existence on this basis we explore the behavior of characteristics near the sonic boundary

interaction of four rarefaction waves in the bi symmetric class of the two dimensional euler equations the global existence and structures of solutions to multi dimensional unsteady compressible euler equations are interesting and important open problems in this paper we construct global classical solutions to the interaction of four orthogonal planar rarefaction waves with two axes of symmetry for the euler equations in two space dimensions in the case where the initial rarefaction waves are large the bi symmetric initial data is a basic type of four wave two dimensional riemann problems the solutions in this case are continuous bounded and self similar and we characterize how large the rarefaction waves must be we use the methods of hodograph transformation characteristic decomposition and phase space analysis we resolve binary interactions of simple waves in the process

analytic integrability for some degenerate planar vector fields abstract in this paper we study the analytic integrability of degenerate vector fields of the form y 3 2 a x 3 y x 5 3 a x 2 y 2 around the origin for these vector fields it is proved that integrability does not imply formal orbital equivalence to the hamiltonian leading part moreover it is shown the existence of a system in this class which has a center but is neither analytically integrable nor formal orbital reversible

normal form theory and spectral sequences abstract the concept of unique normal form is formulated in terms of a spectral sequence as an illustration of this technique some results of baider and churchill concerning the normal form of the anharmonic oscillator are reproduced the aim of this paper is to show that spectral sequences give us a natural framework in which to formulate normal form theory

global solutions to the compressible euler equations with geometrical structure we prove the existence of global solutions to the euler equations of compressible isentropic gas dynamics with geometrical structure including transonic nozzle flow and spherically symmetric flow due to the presence of the geometrical source terms the existence results themselves are new especially as they pertain to radial flow in an unbounded region vec x geqq 1 and to transonic nozzle flow arbitrary data withl bounds are allowed in these results a shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions the convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

unique normal form of bogdanov takens singularities abstract in this note the method introduced by kokubu oka and wang is improved and the unsolved problem in a paper of baider and sanders for the unique normal form of bogdanov takens singularities is solved under a very general condition

measurable riemannian geometry on the sierpinski gasket the kusuoka measure and the gaussian heat kernel estimate we study the standard dirichlet form and its energy measure called the kusuoka measure on the sierpinski gasket as aprototype of measurable riemannian geometry the shortest pathmetric on the harmonic sierpinski gasket is shown to be thegeodesic distance associated with the measurable riemannianstructure the kusuoka measure is shown to have the volumedoubling property with respect to the euclidean distance and alsoto the geodesic distance li yau type gaussian off diagonal heatkernel estimate is established for the heat kernel associated withthe kusuoka measure

a new transvectant algorithm for nilpotent normal forms we offer an algorithm to determine the form of the normal form for a vector field with a nilpotent linear part when the form of the normal form is known for each jordan block of the linear part taken separately the algorithm is based on the notion of transvectant from classical invariant theory

asymptotic unfoldings of dynamical systems by normalizing beyond the normal form abstract systematic methods are given for simplification of the normal form of a perturbed dynamical system near a rest point it then becomes easy to write down an unfolding of the dynamical system which can be rigorously justified as a universal unfolding with respect to asymptotic equivalence these unfoldings always have finite codimension they are both weaker and stronger than topological unfoldings they may fail to be universal unfoldings in the topological sense because of behavior which escapes detection by asymptotic methods and they also may distinguish asymptotic behaviors that are topologically equivalent the method is illustrated by computing an unfolding of codimension 14 as well as some standard examples of codimension 2 and 3

monodromy center focus and integrability problems for quasi homogeneous polynomial systems abstract this paper deals with planar quasi homogeneous polynomial vector fields and addresses three major questions the monodromy the center focus and the integrability problems we characterize the monodromic planar quasi homogeneous polynomial vector fields and we give a condition to distinguish between a center and a focus in this case also we provide conditions which characterize the integrability of quasi homogeneous polynomial systems under non resonance conditions the results obtained allow us to analyse two monodromic planar systems with degenerate linear part one of them with nilpotent linearization and another one with null linear part

on the representations and formula presented equivariant normal form for solenoidal hopf zero singularities abstract in this paper we deal with the solenoidal conservative lie algebra associated to the classical normal form of hopf zero singular system we concentrate on the study of some representations and z 2 equivariant normal form for such singular differential equations first we list some of the representations that this lie algebra admits the vector fields from this lie algebra could be expressed by the set of ordinary differential equations where the first two of them are in the canonical form of a one degree of freedom hamiltonian system and the third one depends upon the first two variables this representation is governed by the associated poisson algebra to one sub family of this lie algebra euler s form vector potential and clebsch representation are other representations of this lie algebra that we list here we also study the non potential property of vector fields with hopf zero singularity from this lie algebra finally we examine the unique normal form with non zero cubic terms of this family in the presence of the symmetry group z 2 the theoretical results of normal form theory are illustrated with the modified chua s oscillator

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

snowballs are quasiballs we introduce snowballs which are compact sets in r 3 homeomorphic to the unit ball they are 3 dimensional analogs of domains in the plane bounded by snowflake curves for each snowball b a quasiconformal map f colon r 3 to r 3 is constructed that maps b to the unit ball

the center problem for a family of systems of differential equations having a nilpotent singular point abstract we study the analytic system of differential equations in the plane x y t i 0 f q p 2 i s where p q n p q s n 1 p q 0 n n and f i p i q i t are quasi homogeneous vector fields of type t p q and degree i with f q p y 0 t and q q p 2 s 1 0 0 the origin of this system is a nilpotent and monodromic isolated singular point we prove for this system the existence of a lyapunov function and we solve theoretically the center problem for such system finally as an application of the theoretical procedure we characterize the centers of several subfamilies

realizability of the normal form for the triple zero nilpotency in a class of delayed nonlinear oscillators abstract the effects of delayed feedback terms on nonlinear oscillators have been extensively studied and there are important applications in many areas of science and engineering we study a particular class of second order delay differential equations near a point of triple zero nilpotent bifurcation using center manifold and normal form reduction we show that the three dimensional nonlinear normal form for the triple zero bifurcation can be fully realized at any given order for appropriate choices of nonlinearities in the original delay differential equation

random walks and harmonic functions on infinite planar graphs using square tilings we study a wide class of transient planar graphs through a geometric model given by a square tiling of a cylinder for many graphs the geometric boundary of the tiling is a circle and is easy to describe in general the simple random walk on the graph converges with probability 1 to a point in the geometric boundary we obtain information on the harmonic measure and estimates on the rate of convergence this allows us to extend results we previously proved for triangulations of a disk

quasisymmetric parametrizations of two dimensional metric spheres according to the classical uniformization theorem every smooth riemannian surface z homeomorphic to the 2 sphere is conformally diffeomorphic to s2 the unit sphere inr3 equipped with the riemannian metric induced by the ambient euclidean metric the availability of a similar uniformization procedure for spheres with a generalized conformal structure is highly desirable in particular in connection with thurston s hyperbolization conjecture this was addressed by cannon in his combinatorial riemann mapping theorem 7 he considers topological surfaces equipped with a sequence of shinglings a combinatorial structure that leads to a notion of approximate conformal moduli of rings he then finds conditions that imply the existence of coordinate systems on the surface that relate these combinatorial moduli to classical analytic moduli in the plane in this paper we develop a uniformization theory for a different type of generalized conformal structure we start with a metric space z homeomorphic to s2 and ask for conditions under which z can be mapped onto s2 by a quasisymmetric homeomorphism the class of quasisymmetries is an appropriate analog of conformal1 mappings in a metric space context quasisymmetric homeomorphisms also arise in the theory of gromov hyperbolic metric spaces quasi isometries between gromov hyperbolic spaces induce quasisymmetric boundary homeomorphisms our setup has the advantage that we can exploit recent notions and methods from analysis

on the structure of nilpotent normal form modules abstract the set of vector fields that are in normal form with respect to a given linear part has the structure of a module and is best described by giving the stanley decomposition of that module an algorithm is presented that produces a stanley decomposition for the module of equivariants of the flow of a nilpotent linear vector field given a stanley decomposition for the ring of invariants this reduces the study of nilpotent normal forms to classical invariant theory plus this one additional algorithm both the inner product normal form elphick and the sl 2 normal form cushman sanders are covered within a single theory and simplified non equivariant versions of each normal form are presented

boundaries of planar graphs via circle packings we provide a geometric representation of the poisson and martin boundaries of a transient bounded degree triangulation of the plane in terms of its circle packing in the unit disc this packing is unique up to mobius transformations more precisely we show that any bounded harmonic function on the graph is the harmonic extension of some measurable function on the boundary of the disk and that the space of extremal positive harmonic functions that is the martin boundary is homeomorphic to the unit circle all our results hold more generally for any good embedding of planar graphs that is an embedding in the unit disc with straight lines such that angles are bounded away from 0 and uniformly and lengths of adjacent edges are comparable furthermore we show that in a good embedding of a planar graph the probability that a random walk exits a disc through a sufficiently wide arc is at least a constant and that brownian motion on such graphs takes time of order r2 to exit a disc of radius r these answer a question recently posed by chelkak 2014

a new algorithm for determining the monodromy of a planar differential system abstract we give a new algorithmic criterium that determines wether an isolated degenerate singular point of a system of differential equations on the plane is monodromic this criterium involves the conservative and dissipative parts associated to the edges and vertices of the newton diagram of the vector field