asymptotics of the titchmarsh weyl m coefficient for integrable potentials ii this article is concerned with the titchmarsh weyl m coefficient for the differential equation y q x y 0 on 0 whee the potential q is integrable near 0 the m coefficient which is a function of the complex variable c depends on an initialization parameter a uniform as well as various non uniform asymptotic expansions of m are derived which are valid as in any sector of the complex plane that does not contain the real axis the results reported here extend those of kaper and kwong proc roy soc edinburgh sect a to appear

w symmetries of ito stochastic differential equations we discuss w symmetries of ito stochastic differential equations introduced in a recent paper by gaeta and spadaro j math phys 58 053503 2017 in particular we discuss the general form of acceptable generators for continuous lie point w symmetry arguing that they are related to the linear conformal group and how w symmetries can be used in the integration of ito stochastic equations along kozlov theory for standard deterministic or random symmetries it turns out that this requires in general considering more general classes of stochastic equations than just ito ones we discuss w symmetries of ito stochastic differential equations introduced in a recent paper by gaeta and spadaro j math phys 58 053503 2017 in particular we discuss the general form of acceptable generators for continuous lie point w symmetry arguing that they are related to the linear conformal group and how w symmetries can be used in the integration of ito stochastic equations along kozlov theory for standard deterministic or random symmetries it turns out that this requires in general considering more general classes of stochastic equations than just ito ones

a comparative study of the dynamic critical behavior of the four state potts like models abstract we investigate the short time critical dynamics of the baxter wu bw and n 3 turban 3tu models to estimate their global persistence exponent g we conclude that this new dynamical exponent can be useful in detecting differences between the critical behavior of these models which are very difficult to obtain in usual simulations in addition we estimate again the dynamical exponents of the four state potts fsp model in order to compare them with results previously obtained for the bw and 3tu models and to decide between two sets of estimates presented in the current literature we also revisit the short time dynamics of the 3tu model in order to check if as already found for the fsp model the anomalous dimension of the initial magnetization x 0 could be equal to zero

random lie point symmetries of stochastic differential equations we study the invariance of stochastic differential equations under random diffeomorphisms and establish the determining equations for random lie point symmetries of stochastic differential equations both in ito and in stratonovich forms we also discuss relations with previous results in the literature

approximate symmetries and conservation laws for ito and stratonovich dynamical systems a new definition for the approximate symmetries of ito dynamical system is given determining systems of approximate symmetries for ito and stratonovich dynamical systems have been obtained it has been shown that approximate conservation laws can be found from the approximate symmetries of stochastic dynamical systems which do not arise from a hamiltonian the results have been applied to an example

the group classification of a scalar stochastic differential equation lie point group classification of a scalar stochastic differential equation sde with one dimensional brownian motion is presented the admitted symmetry group can be zero one two or three dimensional if an equation admits a three dimensional symmetry group it can be transformed into the equation of brownian motion by a change of variables with non random time transformation such equations can be integrated by quadratures there are described drift coefficients for which sdes with constant diffusion coefficients admit two and three dimensional symmetry groups

lie point symmetries and stochastic differential equations we discuss lie point symmetries of stochastic ordinary differential equations and the interrelations between these and analogous symmetries of the associated fokker planck equation for the probability measure

non equilibrium critical dynamics of the two dimensional xy model with hamiltonian equations of motion non equilibrium critical dynamics of the two dimensional xy model is investigated with hamiltonian equations of motion critical relaxation starting from both ordered and random states is carefully analyzed and the short time dynamic scaling behavior is revealed logarithmic corrections to scaling are detected for relaxation with a random initial state while power law corrections to scaling are observed for relaxation with an ordered initial state the static exponent and dynamic exponent z are determined around and below the kosterlitz thouless phase transition temperature our results show that the deterministic dynamics described by hamiltonian equations is in the same universality class as the stochastic dynamics described by monte carlo algorithms and langevin equations

reduction and reconstruction of stochastic differential equations via symmetries an algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented and a natural definition of reconstruction inspired by the classical reconstruction by quadratures is proposed as a side result the well known solution formula for linear one dimensional stochastic differential equations is obtained within this symmetry approach the complete procedure is applied to several examples with both theoretical and applied relevance

equivalence and symmetries for variable coefficient linear heat type equations i a systematic and unified approach to transformations and symmetries of a class of variable coefficient heat type linear partial differential equations pdes is presented the complete symmetry group classification is re performed in the current context a useful criterion which is necessary and sufficient for being reducible to the standard heat equation by point transformations is established a similar criterion is also valid for the equations to have a four or six dimensional symmetry group nontrivial symmetry groups in this situation the basis elements are listed in terms of coefficients a number of illustrative examples are given in particular some applications from the recent literature are re examined in our new approach multidimensional parabolic pdes of heat and schrodinger type are also considered

integration of stochastic ordinary differential equations from a symmetry standpoint we derive an algorithm for calculating lie point symmetries of systems of stochastic ordinary differential equations sodes of any order from this algorithm the following facts emerge symmetries of a sode do not in general form a lie algebra the determining equations for ito s equation are in general stochastic linear partial differential equations whereas for sodes of order n 2 the determining equations are linear deterministic partial differential equations that form an overdetemined system which is solvable by classical methods for scalar second order sodes we provide a complete classification of equations admitting finite dimensional symmetry lie algebras this classification is applied to the integration of scalar second order sodes in general a sode admitting a two dimensional symmetry algebra is not integrable by quadratures although it is reducible to a homogeneous ito equation in particular a scalar second order sode admitting a two dimensional symmetry algebra with connected operators is linearizable we also characterize integrable scalar second order sodes admitting three dimensional symmetry algebras finally we show that a sode can admit maximally a zero one two three or four dimensional lie algebra

help inequalities for limit circle and regular problems everitt s criterion for the validity of the generalized hardy littlewood inequality presupposes that the associated differential equation is singular at one end point of the interval of definition and is in the strong limit point case at the end point in this paper we investigate the cases when the differential equation is in the limit circle case and non oscillatory at the singular end point and when both end points of the interval are regular

the critical behaviour of a quantum spin problem with thee spin coupling the critical behaviour of an ising like model with three spin coupling is studied in a transverse field a self dual renormalisation group transformation can be applied which gives well converging results for the critical exponents when large enough cells are taken finite size scaling on the same model give comparable results

on the location of the weyl circles the paper deals with explicit estimates concerning certain circles in the complex plane which were associated with sturm liouville problems by h weyl by the use of riccati equations instead of linear integral equations improvements are obtained for results of everitt and halvorsen concerning the behaviour of the titchmarsh weyl m coefficient

symmetries of the fokker planck equation with a constant diffusion matrix in 2 1 dimensions we completely classify the symmetries of the fokker planck equation in two spatial dimensions with a constant positive definite diffusion matrix we apply these results to construct group invariant solutions for a physically interesting family of fokker planck equations

basic representations of a2l 2 and dl 1 2 and the polynomial solutions to the reduced bkp hierarchies basic representations of a2l 2 and dl 1 2 are studied the weight vectors are represented in terms of schur s q functions the method to get the polynomial solutions to the reduced bkp hierarchies is shown to be equivalent to a certain rule in the maya game

new conserved quantities derived from symmetry for stochastic dynamical systems recently the author has proposed an elementary theory of conserved quantities and symmetry for stochastic dynamical systems described by stochastic differential equations of stratonovich type within the framework a new method for deriving conserved quantities from symmetry is developed in a similar manner to that of hojman s work 1992 which gives a new conservation law constructed without using either lagrangians or hamiltonians for deterministic dynamical systems some examples of the conserved quantities obtained through the new method are given

normal forms for stochastic differential equations we address the following problem from the intersection of dynamical systems and stochastic analysis two sde dx t j 0 m f j x t dw t j and dx t j 0 m g j x t dw t j in d with smooth coefficients satisfying f j 0 g j 0 0 are said to be smoothly equivalent if there is a smooth random diffeomorphism coordinate transformation h with h 0 0 and dh 0 id which conjugates the corresponding local flows

computation of the m matrix for fourth order problems this paper is concerned with the numerical computation of the titchmarsh weyl m matrix we show how an algorithm may be developed which relates solutions of an initial value problem to an approximation of the m matrix in some special cases we compare the results from our algorithm with results obtained using special functions

on a series inequality for limit circle and finite problems brown and evans proposed a criterion for the validity of their series analogue of everitt s extension of the hardy littlewood inequality presupposes that the associated difference equation is in the strong limit point at infinity in this paper we investigate the case when the difference equation involved is in the limit circle case and non oscillatory and also when the interval is finite

a return to the hardy littlewood integral inequality this paper is concerned with the extension and generalization of a classic integral inequality first considered by hardy and littlewood in 1932 the integral inequality is big int b a p f 2 q f 2 big 2 leqslant k int b a w f 2 int b a w w 1 pf qf 2 f in delta where k is a positive number the coefficients p q and w are real valued and w 0 on the interval a b infty a b leqslant infty and p 1 q w in l loc a b delta is the maximal linear manifold of functions f for which the two integrals on the right hand side are both finite the hardy littlewood inequality is the special case a 0 b infty p w 1 and q 0 on 0 infty here k 4 and all the cases of equality can be identified in this paper the results of everitt in 1971 connecting the above inequality with the titchmarsh weyl m coefficient of the associated differential equation py qy lambda wy on a b are restated in an improved form and are shown to hold under the above minimal conditions on the coefficients p q and w it is proved that the best possible value for the number k satisfies 1 k leqslant infty and that within these bounds all values of k can be realized in the course of establishing these results new information of independent interest is obtained about the behaviour of the m coefficient near the real axis examples are given to exhibit the wide range of possibilities for the integral inequality recent results on the asymptotic form of the m coefficient and on the transformation theory of the associated differential equation are shown to have applications for the general properties of the inequality all recent contributions to this general integral inequality are listed in the references and their content is surveyed during the course of the paper

noether s theorem in symmetric stochastic calculus of variations quite recently a symmetric stochastic calculus of variations was proposed to formulate canonical stochastic dynamics which is an extension of nelson s stochastic mechanics in this article a noether s theorem is formulated within this calculus of variations conservation laws of momentum angular momentum and energy are proved which are related with the same laws in quantum mechanics

conformal invariance and the critical behaviour of a quantum spin hamiltonian with three spin coupling a quantum spin model with three spin coupling is studied by finite lattice methods the critical indices nu and alpha are obtained by standard finite size scaling analysis conformal invariance is used to estimate the exponents nu and eta the central charge or conformal anomaly c and the anomalous dimension of a spin 1 4 parafermion the authors conclude that the model belongs to the same universality class as the four state potts and baxter wu models although the convergence of the various finite lattice estimators is extremely slow

on the definition of an admitted lie group for stochastic differential equations with multi brownian motion the definition of an admitted lie group of transformations for stochastic differential equations has been already presented for equations with one dimensional brownian motion the transformation of the dependent variables involves time as well and it has been proven that brownian motion is transformed to brownian motion in this paper we will discuss this concept for stochastic differential equations involving multi dimensional brownian motion and present applications to a variety of stochastic differential equations

scaling relations for logarithmic corrections multiplicative logarithmic corrections to scaling are frequently encountered in the critical behavior of certain statistical mechanical systems here a lee yang zero approach is used to systematically analyze the exponents of such logarithms and to propose scaling relations between them these proposed relations are then confronted with a variety of results from the literature

lie point symmetries of stratonovich stochastic differential equations this paper considers lie point symmetries of stochastic differential equations sdes in stratonovich form first we derive the determining equations of the random symmetries which correspond to lie group transformations acting in the space of the independent variable time the dependent variables and the brownian motion then the determining equations of the random symmetries are used to obtain the determining equations of the deterministic symmetries which correspond to lie group transformations acting in the space of the independent variable and dependent variables in the case of the deterministic symmetries the transformation of the brownian motion is induced by the random time change it is proved that the corresponding ito and stratonovich sdes admit the same random symmetries and the same deterministic symmetries

persistence and dynamics in the annni chain we investigate both the local and global persistence behaviour in the annni axial next nearest neighbour ising model we find that when the ratio of the second neighbour interaction to the first neighbour interaction is less than 1 p t the probability of a spin to remain in its original state up to time t shows a stretched exponential decay for 1 p t has an algebraic decay but the exponent is different from that of the nearest neighbour ising model the global persistence behaviour shows similar features we also conduct some deeper investigations in the dynamics of the annni model and conclude that it has a different dynamical behaviour compared to the nearest neighbour ising model

non equilibrium relaxation at a tricritical point we study the purely relaxational tricritical dynamics of a non conserved order parameter model a in the upper critical dimension dc 3 and in 3 epsilon dimensions we are especially interested in the relaxation starting from a macroscopically prepared initial state with a small correlation length using the methods of renormalized field theory we obtain the scaling behaviour of the correlation and response functions and study the nonlinear relaxation of the order parameter m t in three dimensions m t displays a crossover from the purely logarithmic short time behaviour m t approximately in t t0 a to a t 14 power law with logarithmic corrections for dimensions d 3 we obtain the exponents which govern the tricritical relaxation at lowest non trivial order in epsilon 3 d the dynamic scaling exponent z is calculated at second order in epsilon

symmetries of systems of stochastic differential equations with diffusion matrices of full rank lie point symmetries of a system of stochastic differential equations sdes with diffusion matrices of full rank are considered it is proved that the maximal dimension of a symmetry group admitted by a system of n sdes is n 2 in addition such systems cannot admit symmetry operators whose coefficients are proportional to a nonconstant coefficient of proportionality these results are applied to compute the lie group classification of a system of two sdes the classification is obtained with the help of non equivalent realizations of real lie algebras by fiber preserving vector fields in 1 2 variables possibilities of using symmetries for integration of sdes by quadratures are discussed

repeated diagonalization and the numerical computation of the titchmarsh weyl m λ function the repeated diagonalization techniques used by eastham to obtain asymptotic results for solutions to linear differential systems are further developed to produce a numerical algorithm for estimating the titchmarsh weyl m coefficient in the second order case these analytic results have been exploited to produce a computer code rdml1 to generate these solutions and to obtain precise global error bounds

an alternative order parameter for the 4 state potts model abstract we have investigated the dynamic critical behavior of the two dimensional 4 state potts model using an alternative order parameter first used by vanderzande j phys a 20 1987 l549 in the study of the z 5 model we have estimated the global persistence exponent g by following the time evolution of the probability p t that the considered order parameter does not change its sign up to time t we have also obtained the critical exponents z and using this alternative definition of the order parameter and our results are in complete agreement with available values found in literature

discrete inequalities orthogonal polynomials and the spectral theory of difference operators in a recent paper the first two authors studied a class of series inequalities associated with a three term recurrence relation which includes a well known inequality of copson s it was shown that the validity of the inequality and the value of the best constant are determined in terms of the so called hellinger nevanlinna m function the theory is the discrete analogue of that established by everitt for a class of integro differential inequalities in this paper the properties of the m function are investigated and connections with the theory of orthogonal polynomials and the hamburger moment problem are explored the results are applied to give examples of the series inequalities associated with the classical orthogonal polynomials

symmetry properties of one and two dimensional fokker planck equations the authors study symmetry properties of some fokker planck fp equations in the one dimensional case when symmetry groups turn out to be six parameter ones this allows us to find changes of variables to reduce such fp equations to the one dimensional heat equation they present seven well known fp equations which are reduced to the heat equation they also study the symmetry and obtain some exact solutions of the kramers equation

random lie symmetries of ito stochastic differential equations lie point symmetries of stochastic differential equations sdes which include transformations depending on the brownian motion are considered the corresponding lie group transformations acting in the space of the independent variable time the dependent variables and the brownian motion preserve the differential form of sdes and the properties of the brownian motion such symmetries are called random symmetries the focus of the paper is a definition of the random symmetries and their properties it is shown that the random symmetries form a lie algebra we also investigate relations between the random symmetries and the conserved quantities of sdes for several stochastic differential equations we find their random symmetries and compare them with the deterministic symmetries which correspond to lie group transformations in the space of the independent and dependent variables

lie symmetry analysis of the lundgren monin novikov equations for multi point probability density functions of turbulent flow the problem of turbulence statistics described by the lundgren monin novikov lmn hierarchy of integro differential equations is studied in terms of its group properties for this we perform a lie group analysis of a truncated lmn chain which presents the first two equations in an infinite set of integro differential equations for the multi point probability density functions pdf s of velocity a complete set of point transformations is derived for the one point pdf s and the independent variables sample space of velocity space and time for this purpose we use a direct method based on the canonical lie backlund operator due to the one way coupling of correlation equations the present results are complete in the sense that no additional symmetries exist for the first leading equation even if the full infinite hierarchy is considered

short time critical dynamics of the baxter wu model we study the early time behavior of the baxter wu model an ising model with three spin interactions on a triangular lattice our estimates for the dynamic exponent z are compatible with results recently obtained for two models which belong to the same universality class of the baxter wu model the two dimensional four state potts model and the ising model with three spin interactions in one direction however our estimates for the dynamic exponent ensuremath theta of the baxter wu model are completely different from the values obtained for those models this discrepancy could be related to the absence of a marginal operator in the baxter wu model

bulk and surface critical behaviour at the conformal invariant point in the z5 model the author determines the bulk thermal and magnetic exponents at the conformal invariant point in the z5 model with a monte carlo based finite size calculation the results are in good agreement with the predictions of zamolodchikov and fateev 1985 the author also argues that the n 1 surface magnetic exponents xh j in the zn model are given by j n j n j 1 n 1 this prediction is also verified in a monte carlo finite size calculation for the z5 model

symmetries of stochastic differential equations a geometric approach a new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations sdes the correspondence between an algebra of weak symmetries for a given sde and an algebra of strong symmetries for a modified sde is proved under suitable regularity assumptions this general approach is applied to a stochastic version of a two dimensional symmetric ordinary differential equation and to the case of two dimensional brownian motion

on inequalities for powers of linear operators and for quadratic forms let h be a hilbert space in which a symmetric operator s with a dense domain d s is given and let s have a finite deficiency index r s this paper contains a necessary and sufficient condition for validity of the following inequalities of kolmogorov type and a method for calculating the best possible constants c n m s moreover let be a symmetric bilinear functional with a dense domain d such that d s d and f g sf g for all f d s g d a necessary and sufficient condition for validity of the inequality as well as a method for calculating the best possible constant k are obtained then an analogous approach is worked out in order to obtain the best possible additive inequalities of the form the paper is concluded by establishing the best possible constants in the inequalities where t is an arbitrary dissipative operator the theorems are extensions of the results of ju i ljubic w n everitt and t kato

on an inequality of the kolmogorov type for a second order differential expression in this paper we discuss an integro differential inequality formed from the square of a second order differential expression a connection between the existence of the inequality and the titchmarsh weyl m function is established and it is shown that the best constant in the inequality is determined by the behaviour of the m function analytic results are established for specific classes of problems a numerical treatment of the inequality is also discussed and estimates of the best constant in specific examples are given

nonequilibrium critical relaxation of the order parameter and energy in the two dimensional ferromagnetic potts model the static and dynamic critical properties of the ferromagnetic q state potts models on a square lattice with q 2 and 3 are numerically studied via the nonequilibrium relaxation method the relaxation behavior of both the order parameter and energy as well as that of the second moments are investigated from which static and dynamic critical exponents can be obtained we find that the static exponents thus obtained from the relaxation of the order parameter and energy together with the second moments of the order parameter exhibit a close agreement with the exact exponents especially for the case of the q 2 ising model when care is taken in the choice of the initial states for the relaxation of the second moments as for the case of q 3 the estimates for the static exponents become less accurate but still exhibit reasonable agreement with the exactly known static exponents the dynamic critical exponent for the q 2 ising model is estimated from the relaxation of the second moments of the order parameter with mixed initial conditions to give z q 2 approximately 2 1668 19

persistence in an antiferromagnetic ising system with conserved magnetisation abstract we obtain the persistence exponents for an antiferromagnetic ising system in which the magnetisation is kept constant this system belongs to model c system with non conserved order parameter with a conserved density and is expected to have persistence exponents different from that of model a system with no conservation but independent of the conserved density our numerical results for both local persistence at zero temperature and global persistence at the critical temperature however indicate that the exponents are dependent on the conserved magnetisation in both two and three dimensions this non universal feature is attributed to the presence of the conserved field and is special to the persistence phenomena

conformal invariance studies of the baxter wu model and a related site colouring problem the partition function of the baxter wu model is exactly related to the generating function of a site colouring problem on a hexagonal lattice we extend the original bethe ansatz solution of these models in order to obtain the eigenspectra of their transfer matrices in finite geometries and general toroidal boundary conditions the operator content of these models is studied by solving numerically the bethe ansatz equations and by exploring conformal invariance since the eigenspectra are calculated for large lattices the corrections to finite size scaling are also calculated

a numerical method for the determination of the titchmarsh weyl m coefficient a numerical method for determining the titchmarsh weyl m function for the singular eigenvalue equation py9 9 qy wy on a where a is finite is presented the algorithm based on weyl s theory utilizes a result first used by atkinson to map a point on the real line onto the weyl circle in the complex plane in the limit point case these circles nest and tend to the limit point m using weyl s result for the diameter of the circles error estimates for m are obtained in 1971 w n everitt obtained an extension of an integral inequality of hardy littlewood namely the help inequality he showed that the existence of that inequality is determined by the properties of the null set of im 2 m in view of the major difficulties in analysing m even in the rare cases when it is given explicitly very few examples of the help inequality are known the computational techniques discussed in this paper have been applied to the problem of finding best constants in these inequalities

numerical determination of the titchmarsh weyl m coefficient and its applications to help inequalities this paper is concerned with numerical methods for finding m lambda the titchmarsh weyl m coefficient for the singular eigenvalue equation y qy lambda y on 0 infty and the results are applied to the problem of finding best constants for everitt s extension to the hardy little wood polya help integral inequality

critical behaviour and logarithmic corrections of a quantum model with three spin interaction extended finite size scaling and series expansion methods are used to determine the critical properties of the one dimensional quantum ising model with three spin interaction by supposing the presence of logarithmic corrections the universality class of the model is investigated the results obtained by different methods support the conjecture that the model belongs to the same universality class as the four state potts model

on an extension of copson s inequality for infinite series in 1979 copson proved the following analogue of the hardy littlewood inequality if is a sequence of real numbers such that are convergent where a n a n 1 a n and 2 a n a n then is convergent and the constant 4 being best possible equality occurs if and only if a n 0 for all n in this paper we give a result that extends copson s result to inequalities of the form where mx n p n l x n l q n x n n 0 1 the validity of such an inequality and the best possible value of the constant k are determined in terms of the analogue of the titchmarsh weyl m function for the difference equation mx n w n x n n 0 1

on maximal lie point symmetry groups admitted by scalar stochastic differential equations it is proved that the lie point symmetry group admitted by a scalar stochastic differential equation sde of order n 3 is at most n 2 dimensional this result supplements those for first and second order sdes maximal dimensionality can be used to carry out lie group classifications lie point group classification of the third order sdes is provided

classification of the extended symmetries of fokker planck equations the authors obtain a complete classification of the possible types of the extended symmetry of any fokker planck equation together with a necessary and sufficient condition for each type they give also the most general form up to a change of variables of the lie generators of these symmetries and a comparison with the case of the heat equation

non markovian persistence and nonequilibrium critical dynamics the persistence exponent ensuremath theta for the global order parameter m t of a system quenched from the disordered phase to its critical point describes the probability p t ensuremath sim t ensuremath ensuremath theta that m t does not change sign in the time interval t following the quench we calculate ensuremath theta to o ensuremath epsilon 2 for model a of hohenberg and halperin rev mod phys 49 435 1977 and to order ensuremath epsilon for model c and show that at this order m t is a non markov process consequently to our knowledge ensuremath theta is a new exponent the calculation is performed by expanding around a markov process using a simplified version of the perturbation theory recently introduced by majumdar and sire phys rev lett 77 1420 1996