growth and poles of meromorphic solutions of some difference equations abstract this paper is devoted to studying the growth property and the pole distribution of meromorphic solutions f of some complex difference equations with all coefficients being rational functions or of growth s r f we find the lower bound of the lower order or the relation between lower order and the convergence exponent of poles of meromorphic solutions of such equations

stock return distributions tests of scaling and universality from three distinct stock markets define the log return of stock price st over a time interval t rrtt log st tlog st1 analyses of returns of individual stocks 2 3 and stock indices 4 have shown that the cumulative distribution of returns is well fit by a power law asymptotic behavior with tail exponent r pr x xr 1

homogeneous einstein randers metrics on symplectic groups abstract in this paper we prove that lie group sp n n 3 admits at least one family of invariant einstein randers metrics which are not naturally reductive we also give the classification of these metrics under isometries and determine their group of isometries moreover we prove that if homogeneous randers space m f with navigation data h w is berwald then m f is naturally reductive if and only if the riemannian metric m h is naturally reductive

first passage and risk evaluation under stochastic volatility we solve the first passage problem for the heston random diffusion model we obtain exact analytical expressions for the survival and the hitting probabilities to a given level of return we study several asymptotic behaviors and obtain approximate forms of these probabilities which prove among other interesting properties the nonexistence of a mean first passage time one significant result is the evidence of extreme deviations which implies a high risk of default when certain dimensionless parameter related to the strength of the volatility fluctuations increases we confront the model with empirical daily data and we observe that it is able to capture a very broad domain of the hitting probability we believe that this may provide an effective tool for risk control which can be readily applicable to real markets both for portfolio management and trading strategies

curvatures of homogeneous randers spaces abstract we study curvatures of homogeneous randers spaces after deducing the coordinate free formulas of the flag curvature and ricci scalar of homogeneous randers spaces we give several applications we first present a direct proof of the fact that a homogeneous randers space is ricci quadratic if and only if it is a berwald space we then prove that any left invariant randers metric on a non commutative nilpotent lie group must have three flags whose flag curvature is positive negative and zero respectively this generalizes a result of j a wolf on riemannian metrics we prove a conjecture of j milnor on the characterization of central elements of a real lie algebra in a more generalized sense finally we study homogeneous finsler spaces of positive flag curvature and particularly prove that the only compact connected simply connected lie group admitting a left invariant finsler metric with positive flag curvature is su 2

multiple closed geodesics on bumpy finsler n spheres abstract in this paper we prove that for every bumpy finsler metric f on every rationally homological n dimensional sphere s n with n 2 there exist always at least two distinct prime closed geodesics

on dually flat randers metrics abstract the notion of dually flat finsler metrics arise from information geometry in this paper we will study a special class of finsler metrics called randers metrics to be dually flat a simple characterization is provided and some non trivial explicit examples are constructed in particular we will show that the dual flatness of a randers metric always arises from that of some riemannian metric by doing some special deformations

effect of nonlinear correlations on the statistics of return intervals in multifractal data sets we study the statistics of return intervals between events above a certain threshold in multifractal data sets without linear correlations we find that nonlinear correlations in the record lead to a power law i decay of the autocorrelation function of the return intervals ii increase in the conditional return period and iii decay in the probability density function of the return intervals we show explicitly that all the observed quantities depend both on the threshold value and system size and hence there is no simple scaling observed we also demonstrate that this type of behavior can be observed in real economic records and can be used to improve considerably risk estimation

hitting time distributions in financial markets we analyze the hitting time distributions of stock price returns in different time windows characterized by different levels of noise present in the market the study has been performed on two sets of data from us markets the first one is composed by daily price of 1071 stocks trade for the 12 year period 1987 1998 the second one is composed by high frequency data for 100 stocks for the 4 year period 1995 1998 we compare the probability distribution obtained by our empirical analysis with those obtained from different models for stock market evolution specifically by focusing on the statistical properties of the hitting times to reach a barrier or a given threshold we compare the probability density function pdf of three models namely the geometric brownian motion the garch model and the heston model with that obtained from real market data we will present also some results of a generalized heston model r 2007 elsevier b v all rights reserved

finsler manifolds with nonpositive flag curvature and constant s curvature the flag curvature is a natural extension of the sectional curvature in riemannian geometry and the s curvature is a non riemannian quantity which vanishes for riemannian metrics there are incomplete non riemannian finsler metrics on an open subset in rn with negative flag curvature and constant s curvature in this paper we are going to show a global rigidity theorem that every finsler metric with negative flag curvature and constant s curvature must be riemannian if the manifold is compact we also study the nonpositive flag curvature case

deformations and hilbert s fourth problem in this paper a special class of finsler metrics defined by a riemannian metric and an 1 form is studied the projectively flat metrics in dimension n ge 3 are classified by a new class of metric deformations in riemann geometry the results show that the projective flatness of such finsler metrics always arises from that of some riemannian metric

first passage and arrival time densities for levy flights and the failure of the method of images we discuss the first passage time problem in the semi infinite interval for homogeneous stochastic markov processes with levy stable jump length distributions x l x 1 x l namely levy flights lfs in particular we demonstrate that the method of images leads to a result which violates a theorem due to sparre andersen according to which an arbitrary continuous and symmetric jump length distribution produces a first passage time density fptd governed by the universal long time decay t 3 2 conversely we show that for lfs the direct definition known from gaussian processes in fact defines the probability density of first arrival which for lfs differs from the fptd our findings are corroborated by numerical results

on dually flat α β metrics abstract the dual flatness for riemannian metrics in information geometry has been extended to finsler metrics the aim of this paper is to study the dual flatness of the so called metrics in finsler geometry by doing some special deformations we will show that the dual flatness of an metric always arises from that of some riemannian metric in dimensional n 3

finsler geodesics in the presence of a convex function and their applications in this paper we obtain a result about the existence of only a finite number of geodesics between two fixed non conjugate points in a finsler manifold endowed with a convex function we apply it to randers and zermelo metrics as a by product we also get a result about the finiteness of the number of lightlike and timelike geodesics connecting an event to a line in a standard stationary spacetime

time minimal control of dissipative two level quantum systems the generic case the objective of this article is to complete preliminary results from the work of bonnard and sugny 2009 and sugny et al 2007 concerning the time minimal control of dissipative two level quantum systems whose dynamics is governed by the lindblad equation the extremal system is described by a 3d hamiltonian depending upon three parameters we combine geometric techniques with numerical simulations to deduce the optimal solutions

scaling properties and universality of first passage time probabilities in financial markets financial markets provide an ideal frame for the study of crossing or first passage time events of non gaussian correlated dynamics mainly because large data sets are available tick by tick data of six futures markets are herein considered resulting in fat tailed first passage time probabilities the scaling of the return with its standard deviation collapses the probabilities of all markets examined and also for different time horizons into single curves suggesting that first passage statistics is market independent at least for high frequency data on the other hand a very closely related quantity the survival probability shows away from the center and tails of the distribution a hyperbolic t ensuremath 1 2 decay typical of a markovian dynamics albeit the existence of memory in markets modifications of the weibull and student distributions are good candidates for the phenomenological description of first passage time properties under certain regimes the scaling strategies shown may be useful for risk control and algorithmic trading

homogeneous randers spaces with isotropic s curvature and positive flag curvature in this paper we will give a complete classification of homogeneous randers spaces with isotropic s curvature and positive flag curvature this results in a large class of finsler spaces with non constant positive flag curvature at the final part of the paper we prove a rigidity result asserting that a homogeneous randers space with almost isotropic s curvature and negative ricci scalar must be riemannian

finite order solutions of second order linear differential equations on considere l equation differentielle f a z f b z f 0 ou a z et b z sont des fonctions entieres on trouve des conditions sur a z et b z qui garantissent que chaque solution f 0 de l equation a un ordre fini

sharp uncertainty principles on general finsler manifolds the paper is devoted to sharp uncertainty principles heisenberg pauli weyl caffarelli kohn nirenberg and hardy inequalities on forward complete finsler manifolds endowed with an arbitrary measure under mild assumptions the existence of extremals corresponding to the sharp constants in the heisenberg pauli weyl and caffarelli kohn nirenberg inequalities fully textit characterizes the nature of the finsler manifold in terms of three non riemannian quantities namely its textit reversibility and the vanishing of the textit flag curvature and s textit curvature induced by the measure respectively it turns out in particular that the busemann hausdorff measure is the optimal one in the study of sharp uncertainty principles on finsler manifolds the optimality of our results are supported by randers type finslerian examples originating from the zermelo navigation problem

on the energy functional on finsler manifolds and applications to stationary spacetimes in this paper we first study some global properties of the energy functional on a non reversible finsler manifold in particular we present a fully detailed proof of the palais smale condition under the completeness of the finsler metric moreover we define a finsler metric of randers type which we call fermat metric associated to a conformally standard stationary spacetime we shall study the influence of the fermat metric on the causal properties of the spacetime mainly the global hyperbolicity moreover we study the relations between the energy functional of the fermat metric and the fermat principle for the light rays in the spacetime this allows one to obtain existence and multiplicity results for light rays using the finsler theory finally the case of timelike geodesics with fixed energy is considered

some einstein randers metrics on homogeneous spaces abstract in this paper we study invariant einstein randers metrics on some homogeneous riemannian manifolds of rank 2 by finding out all the invariant killing vector fields on the corresponding homogeneous einstein riemannian manifolds we obtain a complete description of all the invariant einstein randers metrics on those homogeneous manifolds since the underlying riemannian einstein metrics are generically not diagonal this presents a large number of non riemannian homogeneous einstein randers spaces of nonconstant flag curvature

minimal surfaces in a randers sphere with the rotational killing vector field abstract the minimal surfaces in finsler geometry with respect to the busemann hausdorff measure and the holmes thompson measure are called bh minimal and ht minimal surfaces respectively let p 1 p 2 p 3 p 4 be the coordinates of r 4 and s 3 f be a randers sphere of flag curvature k 1 with the navigation data h w where h is the standard sphere metric and w e 0 0 p 4 p 3 0 e 1 is a killing vector field in this paper we study the rotationally invariant minimal surface in s 3 f generated by rotating the curve x s y s z s 0 in the upper half sphere of s 2 around the p 1 p 2 plane s r we first show that such a rotational bh minimal surface in s 3 f is either a great 2 sphere or the catenoid in s 3 h then we give a classification of the rotational ht minimal surfaces where we use the angle data to analyze the solutions of the system of ode that characterizes the ht minimality and prove that such a rotational ht minimal surface must be a great 2 sphere an ht minimal torus or a rotational surface of unduloid type as a special case we obtain a distinguished embedded compact ht minimal torus depending on e the completeness of these surfaces is also studied

extreme times for volatility processes extreme times techniques generally applied to nonequilibrium statistical mechanical processes are also useful for a better understanding of financial markets we present a detailed study on the mean first passage time for the volatility of return time series the empirical results extracted from daily data of major indices seem to follow the same law regardless of the kind of index thus suggesting an universal pattern the empirical mean first passage time to a certain level l is fairly different from that of the wiener process showing a dissimilar behavior depending on whether l is higher or lower than the average volatility all of this indicates a more complex dynamics in which a reverting force drives volatility toward its mean value we thus present the mean first passage time expressions of the most common stochastic volatility models whose approach is comparable to the random diffusion description we discuss asymptotic approximations of these models and confront them to empirical results with a good agreement with the exponential ornstein uhlenbeck model

s curvature of cohomogeneity one randers spaces abstract in this paper we study s curvature of cohomogeneity one randers spaces with orbit space s 1 in particular we present an explicit coordinate free formula of s curvature in terms of some quantities which is determined by the metric and the structure of the manifold as an application we give a sufficient and necessary condition for a cohomogeneity one randers space to have almost isotropy s curvature furthermore we give some simple criterions for a cohomogeneity one randers space to be a berwald space or a douglas space

projectively flat general α β metrics with constant flag curvature abstract in this paper we study the flag curvature of a new class of finsler metrics called general metrics which are defined by a riemannian metric and a 1 form the classification of such metrics with constant flag curvature are completely determined under some suitable conditions which make them locally projectively flat as a result we construct many new projectively flat finsler metrics with flag curvature 1 0 and 1 in section 9 all of which are of singularity at some directions

lower order and baker wandering domains of solutions to differential equations with coefficients of exponential growth abstract we investigate transcendental entire solutions of complex differential equations f a z f h z where the entire function a z has a growth property similar to the exponential functions and h z is an entire function of order less than that of a z we first prove that the lower order of the entire solution to the equation is infinity by using our result on the lower order we prove the entire solution does not bear any baker wandering domains

improved risk estimation in multifractal records application to the value at risk in finance we suggest a risk estimation method for financial records that is based on the statistics of return intervals between events above below a certain threshold q and is particularly suited for multifractal records the method is based on the knowledge of the probability w q t ensuremath delta t that within the next ensuremath delta t units of time at least one event above q occurs if the last event occurred t time units ago we propose an analytical estimate of w q and show explicitly that the proposed method is superior to the conventional precursory pattern recognition technique widely used in signal analysis which requires considerable fine tuning and is difficult to implement we also show that the estimation of the value at risk which is a standard tool in finances can be improved considerably by the method

on the flag curvature of a class of randers metric generated from the navigation problem abstract let f x y be a randers metric of douglas type on a manifold m and v be a conformal vector field on m f let f x y be a randers metric generated from f v by solving the navigation problem f x y f x y v x 1 in this paper we establish the relationship among the flag curvature of f the flag curvature of f and the conformal vector field v see theorem 1 1

distribution of first passage times to specific targets on compactly explored fractal structures the distribution of the first passage times fpt of a one dimensional random walker to a target site follows a power law f t t ensuremath 3 2 we generalize this result to another situation pertinent to compact exploration and consider the fpt of a random walker with specific source and target points on an infinite fractal structure with spectral dimension d s l2 we show that the probability density of the first return to the origin has the form f t t d s 2 ensuremath 2 and the fpt to a specific target at distance r follows the law f mathbf r t r d w ensuremath d f t d s 2 ensuremath 2 where d w and d f are the walk dimension and the fractal dimension of the structure respectively the distance dependence of f mathbf r t reproduces the one of the mean fpt of a random walk in a confined domain

projectively flat randers metrics with constant flag curvature we classify locally projectively flat randers metrics with constant ricci curvature and obtain a new family of randers metrics of negative constant flag curvature

on general α β metrics with isotropic s curvature abstract the s curvature is an important non riemannian quantity in finsler geometry in this paper we study a class of finsler metrics called general metrics which are defined by a riemannian metric and a 1 form we characterize all of general metrics with isotropic s curvature

time optimal control of a self propelled particle in a spatiotemporal flow field abstractwe address a minimum time problem that constitutes an extension of the classical zermelo navigation problem in higher dimensions in particular we address the problem of steering a self propelled particle to a prescribed terminal position with free terminal velocity in the presence of a spatiotemporal flow field furthermore we assume that the norm of the rate of change of the particle s velocity relative to the flow is upper bounded by an explicit upper bound to address the problem we first employ pontryagin s minimum principle to parameterise the set of candidate time optimal control laws in terms of a parameter vector that belongs to a compact set subsequently we develop a simple numerical algorithm for the computation of the minimum time to come function that is tailored to the particular parametrisation of the set of the candidate time optimal control laws of our problem the proposed approach bypasses the task of converting the optimal control problem to a parameter optimisation proble

zermelo navigation on riemannian manifolds in this paper we study zermelo navigation on riemannian manifolds and use that to solve a long standing problem in finsler geometry namely the complete classification of strongly convex randers metrics of constant flag curvature

twistor geometry of a pair of second order odes we discuss the twistor correspondence between path geometries in three dimensions with vanishing wilczynski invariants and anti self dual conformal structures of signature 2 2 we show how to reconstruct a system of odes with vanishing invariants for a given conformal structure highlighting the ricci flat case in particular using this framework we give a new derivation of the wilczynski invariants for a system of odes whose solution space is endowed with a conformal structure we explain how to reconstruct the conformal structure directly from the integral curves and present new examples of systems of odes with point symmetry algebra of dimension four and greater which give rise to anti self dual structures with conformal symmetry algebra of the same dimension some of these examples are 2 2 analogues of plane wave space times in general relativity finally we discuss a variational principle for twistor curves arising from the finsler structures with scalar flag curvature

solution to the quantum zermelo navigation problem the solution to the problem of finding a time optimal control hamiltonian to generate a given unitary gate in an environment in which there exists an uncontrollable ambient hamiltonian e g a background field is obtained in the classical context finding the time optimal way to steer a ship in the presence of a background wind or current is known as the zermelo navigation problem whose solution can be obtained by working out geodesic curves on a space equipped with a randers metric the solution to the quantum zermelo problem which is shown here to take a remarkably simple form is likewise obtained by finding explicit solutions to the geodesic equations of motion associated with a randers metric on the space of unitary operators the result reveals that the optimal control in a sense goes along with the wind

ricci quadratic homogeneous randers spaces abstract a finsler space is called ricci quadratic if its ricci curvature r i c x y is quadratic in y it is called a berwald space if its chern connection defines a linear connection directly on the underlying manifold m in this article we prove that a homogeneous randers space is ricci quadratic if and only if it is of berwald type

some properties of meromorphic solutions of q difference equations abstract we investigate the growth of transcendental meromorphic solutions of some complex q difference equations and find lower bounds for nevanlinna lower order for meromorphic solutions of such equations we also obtain a q difference version of tumura clunie theorem

homogeneous einstein randers metrics on spheres abstract in this paper we study the homogeneous einstein randers metrics on spheres it turns out that we can find out all the homogeneous non riemannian einstein randers metrics on spheres furthermore we obtain a complete classification of such metrics under isometries using this we present a large number of homogeneous einstein randers metrics of non constant flag curvature

einstein randers metrics on some homogeneous manifolds abstract in this paper we get some new examples of non riemannian einstein randers metrics on homogeneous manifolds g h when g is either the compact simple lie group f 4 or g 2 based on the discussions associated with kahler c spaces we obtain some new einstein riemannian metrics on some homogeneous manifolds and prove that they admit non riemannian einstein randers metrics

return interval distribution of extreme events and long term memory the distribution of recurrence times or return intervals between extreme events is important to characterize and understand the behavior of physical systems and phenomena in many disciplines it is well known that many physical processes in nature and society display long range correlations hence in the last few years considerable research effort has been directed towards studying the distribution of return intervals for long range correlated time series based on numerical simulations it was shown that the return interval distributions are of stretched exponential type in this paper we obtain an analytical expression for the distribution of return intervals in long range correlated time series which holds good when the average return intervals are large we show that the distribution is actually a product of power law and a stretched exponential form we also discuss the regimes of validity and perform detailed studies on how the return interval distribution depends on the threshold used to define extreme events

einstein riemannian metrics and einstein randers metrics on a class of homogeneous manifolds abstract in this paper we get a class of homogeneous manifolds admitting einstein riemannian metrics moreover we prove that they admit non riemannian einstein randers metrics

time minimal control of dissipative two level quantum systems the integrable case the objective of this article is to apply recent developments in geometric optimal control to analyze the time minimum control problem of dissipative two level quantum systems whose dynamics is governed by the lindblad equation we focus our analysis on the case where the extremal hamiltonian is integrable

zermelo navigation in the quantum brachistochrone we analyse the optimal times for implementing unitary quantum gates in a constrained finite dimensional controlled quantum system the family of constraints studied is that the permitted set of time dependent hamiltonians is the unit ball of a norm induced by an inner product on we also consider a generalization of this to arbitrary norms we construct a randers metric by applying a theorem of shen on zermelo navigation the geodesics of which are the time optimal trajectories compatible with the prescribed constraint we determine all geodesics and the corresponding time optimal hamiltonian for a specific constraint on the control i e for any given value of some of the results of carlini et al are re derived using alternative methods a first order system of differential equations for the optimal hamiltonian is obtained and shown to be of the form of the euler poincar equations we illustrate that this method can form a methodology for determining which physical substrates are effective at supporting the implementation of fast quantum computation

nevanlinna theory for the q difference operator and meromorphic solutions of q difference equations it is shown that if f is a meromorphic function of order zero and q in mathbb c then begin equation label abstid m bigg r frac f qz f z bigg o t r f tag ddag end equation for all r on a set of logarithmic density 1 the remainder of the paper consists of applications of identity eqref abstid to the study of value distribution of zero order meromorphic functions and in particular zero order meromorphic solutions of q difference equations the results obtained include q shift analogues of the second main theorem of nevanlinna theory picard s theorem and clunie and mohon ko lemmas

nontrivial minimal surfaces in a class of finsler spheres of vanishing s curvature abstract the minimal surface theory in finsler geometry deserves to be well developed as that in riemannian geometry in this paper we derive the mean curvature of submanifolds in a general manifold by considering the busemann hausdorff measure and holmes thompson measure respectively we then study the rotationally invariant minimal surfaces in the finsler 3 sphere endowed with an metric f k k k k k 1 where is a smooth function s 3 k is the berger sphere e 4 k 1 and k is a killing one form of constant length along the hopf fibers of s 3 we define the energy of the minimal surfaces and by using the volume ratio function introduced by the author and y b shen we give the explicit local expressions of the rotationally invariant bh minimal and ht minimal surfaces in such sphere respectively as a special case we give a detailed study of the rotationally invariant ht minimal surfaces in the 3 sphere with square metric

on the nevanlinna characteristic of f qz and its applications in this paper we investigate the relation of the nevanlinna characteristic functions t r f qz and t r f z for a zero order meromorphic function f and a non zero constant q it is shown that t r f qz 1 o 1 t r f z for all r on a set of lower logarithmic density 1 this estimate is sharp in the sense that for any q c such that q 1 and 0 there exists a meromorphic function h of order such that t r h qz q o 1 t r h z as r outside of an exceptional set of finite linear measure as applications we give some results on zero order meromorphic solutions of q difference equations and on value distribution and uniqueness of certain types of q difference polynomials

geodesics in randers spaces of constant curvature geodesies in randers spaces of constant curvature are classified randers metrics have received much attention lately as solutions to zermelo s problem of navigation largely because this navigation structure provides the frame work for a complete classification of constant flag curvature randers spaces flag curvature is the finslerian analog of riemannian sectional curvature see br04 briefly a randers metric is of constant flag curvature if and only if it solves zer melo s problem of navigation on a riemannian manifold of constant sectional cur vature under the influence of an infinitesimal homothety w see subsection 1 1 for a sketch of the navigation problem and theorem 3 for an explicit statement of the classification result the aim of this paper is to develop a geometric description of the geodesies in these spaces of constant curvature intuitively these paths minimize travel time across a riemannian landscape under windy conditions presently we will show that these curves are given by composing geodesies of the riemannian metric with the flow generated by w this claim is formalized by theorem 2 geodesies on surfaces of constant nonpositive curvature are illustrated in section 3 we then turn in section 4 to the constant flag curvature k 1 randers metrics on sn the case of the sphere is especially interesting it is possible to endow this closed manifold with a metric whose geodesies display distinctly non riemannian behaviors for example 1 a metric is projectively flat if every point admits coordinates in which the geodesies are straight lines belt r ami s theorem assures us that a riemannian metric is of constant sectional curvature if and only if it is projectively flat in contrast few randers spaces of constant flag curvature are projectively flat there are infinitely many nonisometric randers metrics of constant positive