operators preserving orthogonality are isometries we prove that every operator preserving orthogonality in a real banach space is an isometry multiplied by a constant

ergodic properties of levy flights coexisting with subdiffusion and related models abstract we analyze ergodic properties of two different stationary processes resulting from a combination of levy flights and subdiffusion the first one comes from the lamperti transformation of subordinated stable process the second one is a sequence of increments of subordinated levy process we prove that both processes are mixing and ergodic we also study increments and the lamperti transformation of time changed fractional brownian motion it appears that these models of anomalous diffusion are also ergodic and mixing we believe that the derived results will help to verify ergodicity in single particle tracking experiments

monodromy of a class of analytic generalized nilpotent systems through their newton diagram newton diagram of a planar vector field allows to determine whether a singular point of an analytic system is a monodromic singular point we solve the monodromy problem for the nilpotent systems and we apply our method to a wide family of systems with a degenerate singular point so called generalized nilpotent cubic systems

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

polynomial lienard equations near infinity abstract we study the behaviour near infinity of the generalized lienard equations x y y m k 0 a k x k y n k 0 b k x k providing a complete classification using poincare compactification as well as poincare lyapunov compactification we show that all the necessary information is contained in a m and b n except for the so called center focus problem occuring in case m 2 n 1 with m n odd and where the behaviour also depends on the value of the other coefficients a i and b j we also indicate how to take care about obtaining uniform information near infinity

non existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities abstract in this paper we study the limit cycles of the planar polynomial differential systems x a x y p n x y y x a y q n x y where p n and q n are homogeneous polynomials of degree n 2 and a r consider the functions p n cos sin cos q n cos sin sin q n cos sin cos p n cos sin sin 1 a 2 n 1 2 a first we prove that these differential systems have at most 1 limit cycle if there exists a linear combination of 1 and 2 with definite sign this result improves previous known results furthermore if 1 1 a 2 0 for some 1 2 0 we provide necessary and sufficient conditions for the non existence and the existence and uniqueness of the limit cycles of these differential systems when one of these mentioned limit cycles exists it is hyperbolic and surrounds the origin

rotationally symmetric p harmonic maps from d2 to s2 we consider rotationally symmetric p harmonic maps from the unit disk d2 r2 to the unit sphere s2 r3 subject to dirichlet boundary conditions and with 1 p we show that the associated energy functional admits a unique minimizer which is of class c in the interior and c1 up to the boundary we also show that there exist infinitely many global solutions to the associated euler lagrange equation and we completely characterize them

mappings approximately preserving orthogonality in normed spaces abstract we answer many open questions regarding approximately orthogonality preserving mappings in birkhoff james sense in normed spaces in particular we show that every approximately orthogonality preserving linear mapping in chmielinski sense is necessarily a scalar multiple of an e isometry thus whenever e isometries are close to isometries we obtain stability an example is given showing that approximately orthogonality preserving mappings are in general far from scalar multiples of isometries that is stability does not hold

integrability of centers perturbed by quasi homogeneous polynomials abstract the objective of this work is to give sufficient and necessary conditions for integrability of differential equation systems in the plane of the form y2p 1 x2 x y x2q 1 y2 x y wherex2 x y andy2 x y are p q quasi homogeneous polynomials of degree 2pq q 1 and 2pq p 1 respectively

topological invariants and equidesingularization for holomorphic vector fields on etudie l invariance du nombre de milnor et ses applications on demontre un theoreme d equidesingularisation

growth at infinity and index of polynomial maps abstract let f k n k n be a polynomial map such that f 1 0 is compact where k r or c then we give a condition implying that there is a uniform bound for the ojasiewicz exponent at infinity in certain deformations of f this fact gives a result about the invariance of the global index of f

canards in r3 abstract we give a geometric analysis of canard solutions in three dimensional singularly perturbed systems with a folded two dimensional critical manifold by analysing the reduced flow we obtain singular canard solutions passing through a singularity on the fold curve we classify these singularities called canard points as folded saddles folded nodes and folded saddle nodes we prove the existence of canard solutions in the case of the folded saddle we show the existence of canards in the folded node case provided a generic non resonance condition is satisfied and in a subcase of the folded saddle node the proof is based on the blow up method

an identity for δ e approximately orthogonality preserving mappings abstract suppose t is a hilbert space operator given 0 1 we define e t to be the smallest e for which t is e approximately orthogonality preserving and then obtain an exact formula for e t in terms of t and the minimum modulus m t of t for two nonzero operators t s it follows from the formula that t is e s e aop if and only if s is e t e aop where e t e 0 t finally we show that an operator t is e aop if and only if there exists a special aop operator s such that ts is e aop theorem 3 8

structural stability of planar homogeneous polynomial vector fields applications to critical points and to infinity lethmbe the space of planar homogeneous polynomial vector fields of degreemendowed with the coefficient topology we characterize the set mof the vector fields ofhmthat are structurally stable with respect to perturbations inhmand we determine the exact number of the topological equivalence classes in m the study of structurally stable homogeneous polynomial vector fields is very useful for understanding some interesting features of inhomogeneous vector fields thus by using this characterization we can do first an extension of the hartman grobman theorem which allows us to study the critical points of planar analytical vector fields whosek jets are zero for allk munder generic assumptions and second the study of the flows of the planar polynomial vector fields in a neighborhood of the infinity also under generic assumptions

the index of singularities of vector fields and finite jets abstract we describe when the index of a singularity of a smooth vector field is determined by a finite jet at the singularity we also give some criteria to determine some terms from the formal series expansion of the vector field at the singularity which determine the index

an upper bound of the index of an equilibrium point in the plane abstract we give an upper bound of the index of an isolated equilibrium point of a c 1 vector field in the plane the vector field is decomposed in gradient and hamiltonian components this decomposition is related with the loewner vector field associated to this decomposition we consider the set where the gradient and hamiltonian components are linearly dependent the number of branches of starting at the equilibrium point determines the upper bound of the index

global phase portraits and bifurcation diagrams for hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the y axis abstract this paper studies the global dynamics for hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the y axis by linear changes of variables it can be written as four systems one of them has been characterized in llibre 2018 in this paper the remaining three systems are investigated firstly the normal forms of these systems are given by an algebraic classification of their infinite singular points then we classify the global phase portraits of these systems having canonical forms on the poincare disk moreover we obtain the bifurcation diagrams for the corresponding global phase portraits combining this work with llibre 2018 we provide the complete classification of global phase portraits for hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the y axis

on mappings approximately preserving orthogonality abstract we answer a question posed by chmielinski whether a linear map which approximately preserves orthogonality must be close to an orthogonality preserving one furthermore we give a short proof of the stability of the orthogonality equation on finite dimensional hilbert spaces

isosceles orthogonality preserving property and its stability for real normed spaces we consider the class of linear operators preserving approximately the relation of isosceles orthogonality we show some general properties of such mappings next we examine whether an approximately orthogonality preserving mapping admits an approximation by an orthogonality preserving one in regard to this we generalize some results obtained earlier for inner product spaces with standard orthogonality relation

geometric treatments and a common mechanism in finite time singularities for autonomous odes abstract geometric treatments of blow up solutions for autonomous ordinary differential equations and their blow up rates are concerned our approach focuses on the type of invariant sets at infinity via compactifications of phase spaces and dynamics on their center stable manifolds in particular we show that dynamics on center stable manifolds of invariant sets at infinity with appropriate time scale desingularizations as well as blowing up of singularities characterize dynamics of blow up solutions as well as their rigorous blow up rates similarities for characterizing finite time extinction and asymptotic behavior of compacton traveling waves to blow up solutions are also shown

limit cycles for planar semi quasi homogeneous polynomial vector fields abstract this paper is concerned with the limit cycles for planar semi quasi homogeneous polynomial systems we give some explicit criteria for the nonexistence and existence of periodic orbits a lower bound is given for the maximum number of limit cycles of such a system the cyclicity and center problems are studied for some subfamilies of semi quasi homogeneous polynomial systems our results generalize those obtained for polynomial semi homogeneous systems

stable non gaussian random processes stochastic models with infinite variance stable random variables on the real line multivariate stable distributions stable stochastic integrals dependence structures of multivariate stable distributions non linear regression complex stable stochastic integrals and harmonizable processes self similar processes chentsov random fields introduction to sample path properties boundedness continuity and oscillations measurability integrability and absolute continuity boundedness and continuity via metric entropy integral representation historical notes and extensions

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

planar quasi homogeneous polynomial differential systems and their integrability abstract in this paper we study the quasi homogeneous polynomial differential systems and provide an algorithm for obtaining all these systems with a given degree using this algorithm we obtain all quasi homogeneous vector fields of degree 2 and 3 the quasi homogeneous polynomial differential systems are liouvillian integrable in particular we characterize all the quasi homogeneous vector fields of degree 2 and 3 having a polynomial rational or global analytical first integral

generalization of the khinchin theorem to levy flights one of the most fundamental theorems in statistical mechanics is the khinchin ergodic theorem which links the ergodicity of a physical system with the irreversibility of the corresponding autocorrelation function however the khinchin theorem cannot be successfully applied to processes with infinite second moment in particular to the relevant class of levy flights here we solve this challenging problem namely using the recently developed measure of dependence called levy correlation cascade we derive a version of the khinchin theorem for levy flights this result allows us to verify the boltzmann hypothesis for systems displaying levy flight type dynamics

distributed order diffusion equations and multifractality models and solutions we study distributed order time fractional diffusion equations characterized by multifractal memory kernels in contrast to the simple power law kernel of common time fractional diffusion equations based on the physical approach to anomalous diffusion provided by the seminal scher montroll weiss continuous time random walk we analyze both natural and modified form distributed order time fractional diffusion equations and compare the two approaches the mean squared displacement is obtained and its limiting behavior analyzed we derive the connection between the wiener process described by the conventional langevin equation and the dynamics encoded by the distributed order time fractional diffusion equation in terms of a generalized subordination of time a detailed analysis of the multifractal properties of distributed order diffusion equations is provided

approximate orthogonality in normed spaces and its applications abstract in a normed space we consider an approximate orthogonality relation related to the birkhoff orthogonality we give some properties of this relation as well as applications in particular we characterize the approximate orthogonality in the class of linear bounded operators on a hilbert space

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

aop mappings and the distance to the scalar multiples of isometries in this paper linear e orthogonality preserving mappings are studied we define e t as the smallest e for which t is e orthogonality preserving and then derive an exact formula for e t in terms of t and the minimum modulus m t of t we see that e orthogonality preserving mappings for some e 1 are exactly the operators that are bounded from below we improve an upper bound in the stability equation given in 7 theorem 2 3 which was thought to be sharp

the projective vector field of a kind of three dimensional quasi homogeneous system on s2 abstract in this paper we study the projective vector field q t of a three dimensional quasi homogeneous system with weight 1 1 3 and degree 2 3 2 projective vector fields q t of this kind are classified into two types for one type q t has no closed orbit and at most eight singularities which lead to a global topology of the three dimensional system for the other type q t has at most ten singularities in addition we show a relationship between q t and a lienard system of this type for both of them we obtain some conditions for the existence of limit cycles

superaging correlation function and ergodicity breaking for brownian motion in logarithmic potentials we consider an overdamped brownian particle moving in a confining asymptotically logarithmic potential which supports a normalized boltzmann equilibrium density we derive analytical expressions for the two time correlation function and the fluctuations of the time averaged position of the particle for large but finite times we characterize the occurrence of aging and nonergodic behavior as a function of the depth of the potential and we support our predictions with extensive langevin simulations while the boltzmann measure is used to obtain stationary correlation functions we show how the non normalizable infinite covariant density is related to the superaging behavior

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

an approach to mel nikov theory in celestial mechanics using a completely analytic procedure based on a suitable extension of a classical method we discuss an approach to the poincare mel nikov theory which can be conveniently applied also to the case of nonhyperbolic critical points and even if the critical point is located at the infinity in this paper we concentrate our attention on the latter case and precisely on problems described by kepler type potentials in one or two degrees of freedom in the presence of general time dependent perturbations we show that the appearance of chaos possibly including arnol d diffusion can be proved quite easily and in a direct way without resorting to singular coordinate transformations such as the mcgehee or blowing up transformations natural examples are provided by the classical gylden problem originally proposed in celestial mechanics but also of interest in different fields and by the general three body problem in classical mechanics

approximate symmetry of birkhoff orthogonality abstract in a real normed space x we consider an approximate symmetry of the birkhoff orthogonality b and establish its connections with some properties of the space x moreover we introduce and study a new geometric constant for x connected with the considered property

non stabilisable jets of diffeomorphisms in r2 and of vector fields in r3 we essentially prove two results firstly that for any x germ of a coo vector field on r2 with a certain specific 9 jet there is a germ of a c diffeomorphism f with j f o j x o x denoting the flow of x such that f and x are not c0 conjugate theorem 1 and secondly that whenever x is a germ of a coo vector field on r3 whose 9 jet belongs to a certain class of 9 jets there can be found a germ of a coo vector field y on r3 with jo x 0 jo y 0 such that x and y are not c0 equivalent theorem 2 the second result implies the existence of jets of vector fields in r3 which are not stabilisable for the notion of c0 equivalence the question of the existence of such jets was raised by thom in 1970 and very soon takens found examples in rn with n 2 4 ta ta3 the first result which we use in proving the second also implies the existence of jets of diffeomorphisms in r2 which are not stabilisable for the notion of c conjugacy the diffeomorphisms having such a jet contain hyperbolic or saddle sectors with a different topological structure than the hyperbolic sectors occurring for vector fields in finite codimension in these sectors the singularity scheme of a nice desingularisation after blowing up contains certain successions of semi hyperbolic singularities these cases had not been treated in d r r and our example now shows that the problems we encountered in d r r in trying to add this case were not technical but essential our examples provide germs of coo diffeomorphisms in r2 which are infinitely tangent to the time l mapping of a vector field of finite codimension and not c embeddable in the flow of a vector field of finite codimension this is definitely not possible in r germs of orientable coo diffeomorphisms of finite codimension are coo embeddable in a flow ta4 even the germs of diffeomorphisms infinitely tangent to the identity and with isolated fixed point show up in a c flow but they do not necessarily show up in a c 2 flow se possibly as

on mappings approximately transferring relations in finite dimensional normed spaces abstract we discuss the problem of stability of relations preserving property for finite dimensional normed spaces moreover we obtain similar results for other orthogonality relations next we show a counterexample for infinite dimensional normed spaces

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

phase portraits of planar semi homogeneous vector fields ii we determine all the phase portraits of the semi homogeneous polynomial vector fields of the form ax by ax4 bx3y cx2y2 dxy3 ey4

topological equivalence of planar vector fields and their generalised principal part abstract let x r2 be the space of c planar vector fields we consider the space v x r2 of vector fields with an isolated singularity and a fixed newton diagram we define the generalised principal part x of the vector field x v and give the nondegeneracy condition on x using the newton diagram we prove that x v is locally topologically equivalent to its minimal generalised principal part x if x is nondegenerate and x is not a monodromic vector field in the proof we use the normal form method and the blowing up method

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

transition functions and moduli of stability for 3 dimensional homogeneous vector fields with a hyperbolic blowing up abstract in the first part of the paper we introduce the normal transition function for saddle connections of planar diffeomorphisms it is a positive multiple of the usual transition function but in its definition we do not need c 1 linearizing coordinates among other nice properties it is found to be analytic when the diffeomorphism is the second part of the paper deals with the existence of a modulus of stability for germs on r 3 of homogeneous vector fields with a hyperbolic blowing up we show that inside a specific class of examples the modulus occurs for a sufficiently high degree

structural stability of planar quasi homogeneous vector fields abstract in this work we study the structural stability of planar quasi homogeneous vector fields under quasi homogeneous perturbations and provide a complete classification this study which has been the subject of previous works is only complete in the homogeneous case the main tool in our analysis is a splitting of planar quasi homogeneous vector fields into conservative dissipative parts moreover we describe the topological equivalence classes in the set of the structurally stable planar quasi homogeneous vector fields finally we include some examples where the equivalence classes are determined

on certain basis connected with operator and its applications abstract we assume that h k are the complex hilbert spaces and a linear operator t b h k is given the object of this paper is to define and consider the certain basis of h connected with the operator t b h k this basis will play an important role in describing the approximately orthogonality preserving operators next we will give some application of our results we give new results and more general ones namely we answer a question whether a linear operator which approximately preserves orthogonality must be close to an orthogonality preserving one