hamiltonian formulation of systems with linear velocities within riemann liouville fractional derivatives the link between the treatments of constrained systems with fractional derivatives by using both hamiltonian and lagrangian formulations is studied it is shown that both treatments for systems with linear velocities are equivalent

dynamics of the fractional oscillator abstract the integral equation of motion of a simple harmonic oscillator is generalized by taking the integral to be of arbitrary order according to the methods of fractional calculus to yield the equation of motion of a fractional oscillator the solution is obtained in terms of mittag leffler functions using laplace transforms the expressions for the generalized momentum and the total energy of the fractional oscillator are also obtained numerical application and the phase plane representation of the dynamics are discussed

the incidental parameter problem since 1948 this paper was written to mark the 50th anniversary of neyman and scott s econometrica paper defining the incidental parameter problem it surveys the history both of the paper and of the problem in the statistics and econometrics literature

fractional radiative transfer equation within chebyshev spectral approach in this work we report the convergence of the chebyshev polynomials combined with the s n method for the steady state transport equation using the fractional derivative the procedure is based on the expansion of the angular flux in a truncated series of orthogonal polynomials that results in the transformation of the multidimensional problem into a system of fractional differential equations the convergence of this approach is studied in the context of the multidimensional discrete ordinates equations

variational problems with fractional derivatives invariance conditions and nother s theorem abstract a variational principle for lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases necessary and sufficient conditions for an infinitesimal transformation group basic nother s identity are obtained these conditions extend the classical results valid for integer order derivatives a generalization of nother s theorem leading to conservation laws for fractional euler lagrangian equation is obtained as well results are illustrated by several concrete examples finally an approximation of a fractional euler lagrangian equation by a system of integer order equations is used for the formulation of an approximated invariance condition and corresponding conservation laws

forecasting using principal components from a large number of predictors this article considers forecasting a single time series when there are many predictors n and time series observations t when the data follow an approximate factor model the predictors can be summarized by a small number of indexes which we estimate using principal components feasible forecasts are shown to be asymptotically efficient in the sense that the difference between the feasible forecasts and the infeasible forecasts constructed using the actual values of the factors converges in probability to 0 as both n and t grow large the estimated factors are shown to be consistent even in the presence of time variation in the factor model

table errata higher transcendental functions vol iii mcgraw hill new york 1955 mr 16 586 by a erdelyi w magnus f oberhettinger and f g tricomi

mechanics with fractional derivatives lagrangian and hamiltonian mechanics can be formulated to include derivatives of fractional order f riewe phys rev 53 1890 1996 lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction the present work continues the development of fractional derivative mechanics by deriving a modified hamilton s principle introducing two types of canonical transformations and deriving the hamilton jacobi equation using generalized mechanics with fractional and higher order derivatives the method is illustrated with a frictional force proportional to velocity in contrast to conventional mechanics with integer order derivatives quantization of a fractional derivative hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives instead a quantum mechanical wave equation is proposed that follows from the hamilton jacobi equation by application of the correspondence principle

on the rest state stability of an objective fractional derivative viscoelastic fluid model the one dimensional fractional derivative maxwell model has been found very useful in modeling the linear viscoelastic response in the glass transition and relaxation regions that motivated further work on generalizations to nonlinear viscoelastic fractional constitutive equations in the work of palade et al int j eng sci 37 315 1999 a fully objective constitutive equation for an incompressible fluid reducible to the linear fractional derivative maxwell model under the small deformations hypothesis is given together with a state of rest stability analysis in the stability section of the aforementioned work for some physically aceptable perturbation inputs it has been shown that no continuously differentiable solutions may be found to the corresponding equations of motion in this work we extend the stability analysis framework to l2 hilbert functional space and prove the existence of solutions to the corresponding equations of motions

a fractional dirac equation and its solution this paper presents a fractional dirac equation and its solution the fractional dirac equation may be obtained using a fractional variational principle and a fractional klein gordon equation both methods are considered here we extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of caputo derivatives by applying the variational principle to a fractional action s we obtain the fractional euler lagrange equations of motion we present a lagrangian and a hamiltonian for the fractional dirac equation of order we also use a fractional klein gordon equation to obtain the fractional dirac equation which is the same as that obtained using the fractional variational principle eigensolutions of this equation are presented which follow the same approach as that for the solution of the standard dirac equation we also provide expressions for the path integral quantization for the fractional dirac field which in the limit 1 approaches to the path integral for the regular dirac field it is hoped that the fractional dirac equation and the path integral quantization of the fractional field will allow further development of fractional relativistic quantum mechanics

hamilton jacobi fractional mechanics abstract as a continuation of rabei et al work eqab m rabei khaled i nawafleh raed s hijjawi sami i muslih dumitru baleanu the hamilton formalism with fractional derivatives j math anal appl 327 2007 891 897 the hamilton jacobi partial differential equation is generalized to be applicable for systems containing fractional derivatives the hamilton jacobi function in configuration space is obtained in a similar manner to the usual mechanics two problems are considered to demonstrate the application of the formalism the result is found to be in exact agreement with agrawal s formalism

fractional oscillator driven by a gaussian noise abstract the time dependent response is studied of a fractional oscillator driven by a gaussian white noise applying the method of characteristic functionals combined with the technique of integral transforms we derive explicit formulas for the variances of position and velocity and the asymptotic power spectrum of the oscillator the effects of the order of fractional derivative on statistical characteristics of the oscillator are analyzed numerically

maximum likelihood estimation and inference for high dimensional generalized factor models with application to factor augmented regressions abstract this paper reestablishes the main results in bai 2003 and bai and ng 2006 for generalized factor models with slightly stronger conditions on the relative magnitude of n number of subjects and t number of time periods convergence rates of the estimated factor space and loading space and asymptotic normality of the estimated factors and loadings are established under mild conditions that allow for linear logit probit tobit poisson and some other single index nonlinear models the probability density mass function is allowed to vary across subjects and time thus mixed models are also allowed for for factor augmented regressions this paper establishes the limit distributions of the parameter estimates the conditional mean and the forecast when factors estimated from nonlinear mixed data are used as proxies for the true factors

a fractional calculus of variations for multiple integrals with application to vibrating string we introduce a fractional theory of the calculus of variations for multiple integrals our approach uses the recent notions of riemann liouville fractional derivatives and integrals in the sense of jumarie the main results provide fractional versions of the theorems of green and gauss fractional euler lagrange equations and fractional natural boundary conditions as an application we discuss the fractional equation of motion of a vibrating string

determining the number of factors in approximate factor models in this paper we develop some econometric theory for factor models of large dimensions the focus is the determination of the number of factors r which is an unresolved issue in the rapidly growing literature on multifactor models we first establish the convergence rate for the factor estimates that will allow for consistent estimation of r we then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria the theory is developed under the framework of large cross sections n and large time dimensions t no restriction is imposed on the relation between n and t simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice

solving fractional integral equations by the haar wavelet method haar wavelets for the solution of fractional integral equations are applied fractional volterra and fredholm integral equations are considered the proposed method also is used for analysing fractional harmonic vibrations the efficiency of the method is demonstrated by three numerical examples

multiple bayes factors for testing hypotheses abstract partial and multiple bayes factors are introduced to obtain pairwise comparisons of hypotheses in a statistical experiment with a partition on the parameter space robust bayesian analyses are performed by introducing suitable classes of priors and by calculating lower and upper bounds of bayes factors and posterior probabilities classes of intuitively meaningful priors are introduced including unimodal densities without the constraint of symmetry for the case of precise hypotheses procedures for the corresponding optimizations are specified and examples are given

individual and time effects in nonlinear panel models with large n t fixed eects estimators of nonlinear panel data models can be severely biased because of the well known incidental parameter problem we develop analytical and jackknife bias corrections for nonlinear models with both individual and time eects under asymptotic sequences where the time dimension t grows with the cross sectional dimension n the time eects introduce additional incidental parameter bias as the existing bias corrections apply to models with only individual eects we derive the appropriate corrections the basis for the corrections are general asymptotic expansions of xed eects estimators with incidental parameters in multiple dimensions we apply the expansions to m estimators with concave objective functions in parameters for panel models with additive individual and time eects these estimators cover xed eects estimators of the most popular limited dependent variable models such as logit probit ordered probit tobit and poisson models our analysis therefore extends the use of large t bias adjustments to an important class of models we also develop bias corrections for functions of the data parameters and individual and time eects including average partial eects in this case the incidental parameter bias is asymptotically of second order in general but the corrections improve nite sample properties

variational problems with fractional derivatives euler lagrange equations we generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized also for the standard case when these two bounds coincide we derive a new form of euler lagrange equations we use approximations for fractional derivatives in the lagrangian and obtain the euler lagrange equations which approximate the initial euler lagrange equations in a weak sense

fractional variational principles with delay the fractional variational principles within riemann liouville fractional derivatives in the presence of delay are analyzed the corresponding euler lagrange equations are obtained and one example is analyzed in detail

dynamics with low level fractionality abstract the notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order this type of equation appears in the description of chaotic dynamics wave propagation in fractal media and field theory for the fractional linear oscillator the physical meaning of the derivative of order 2 is dissipation in systems with many spacially coupled elements oscillators the fractional derivative along the space coordinate corresponds to a long range interaction we discuss a method of constructing a solution using an expansion in n with small and positive integer n the method is applied to the fractional linear and nonlinear oscillators and to fractional ginzburg landau or parabolic equations

testing a point null hypothesis the irreconcilability of p values and evidence abstract the problem of testing a point null hypothesis or a small interval null hypothesis is considered of interest is the relationship between the p value or observed significance level and conditional and bayesian measures of evidence against the null hypothesis although one might presume that a small p value indicates the presence of strong evidence against the null such is not necessarily the case expanding on earlier work especially edwards lindman and savage 1963 and dickey 1977 it is shown that actual evidence against a null as measured say by posterior probability or comparative likelihood can differ by an order of magnitude from the p value for instance data that yield a p value of 05 when testing a normal mean result in a posterior probability of the null of at least 30 for any objective prior distribution objective here means that equal prior weight is given the two hypotheses and that the prior is symmetric and nonincreasing away from the null other definiti

reconciling bayesian and frequentist evidence in the one sided testing problem abstract for the one sided hypothesis testing problem it is shown that it is possible to reconcile bayesian evidence against h 0 expressed in terms of the posterior probability that h 0 is true with frequentist evidence against h 0 expressed in terms of the p value in fact for many classes of prior distributions it is shown that the infimum of the bayesian posterior probability of h 0 is equal to the p value in other cases the infimum is less than the p value the results are in contrast to recent work of berger and sellke 1987 in the two sided point null case where it was found that the p value is much smaller than the bayesian infimum some comments on the point null problem are also given

the multi state latent factor intensity model for credit rating transitions a new empirical reduced form model for credit rating transitions is introduced it is a parametric intensity based duration model with multiple states and driven by exogenous covariates and latent dynamic factors the model has a generalized semi markov structure designed to accommodate many of the stylized facts of credit rating migrations parameter estimation is based on monte carlo maximum likelihood methods for which the details are discussed in this paper a simulation experiment is carried out to show the effectiveness of the estimation procedure an empirical application is presented for transitions in a 7 grade rating system the model includes a common dynamic component that can be interpreted as the credit cycle asymmetric effects of this cycle across rating grades and additional semi markov dynamics are found to be statistically significant finally we investigate whether the common factor model suffices to capture systematic risk in rating transition data by introducing multiple factors in the model

jackknife and analytical bias reduction for nonlinear panel models fixed effects estimators of panel models can be severely biased because of the well known incidental parameters problem we show that this bias can be reduced by using a panel jackknife or an analytical bias correction motivated by large t we give bias corrections for averages over the fixed effects as well as model parameters we find large bias reductions from using these approaches in examples we consider asymptotics where t grows with n as an approximation to the properties of the estimators in econometric applications we show that if t grows at the same rate as n the fixed effects estimator is asymptotically biased so that asymptotic confidence intervals are incorrect but that they are correct for the panel jackknife we show t growing faster than n 1 3 suffices for correctness of the analytic correction a property we also conjecture for the jackknife

existence and uniqueness theorem for a class of delay differential equations with left and right caputo fractional derivatives the existence and uniqueness theorems for functional right left delay and left right advanced fractional functional differential equations with bounded delay and advance respectively are proved the continuity with respect to the initial function for these equations is also proved under some lipschitz kind conditions the q operator is used to transform the delay type equation to an advanced one and vice versa an example is given to clarify the results

a numerical technique for solving fractional optimal control problems this paper presents a numerical method for solving a class of fractional optimal control problems focps the fractional derivative in these problems is in the caputo sense the method is based upon the legendre orthonormal polynomial basis the operational matrices of fractional riemann liouville integration and multiplication along with the lagrange multiplier method for the constrained extremum are considered by this method the given optimization problem reduces to the problem of solving a system of algebraic equations by solving this system we achieve the solution of the focp illustrative examples are included to demonstrate the validity and applicability of the new technique

conserved quantities for hamiltonian systems on time scales abstract conserved quantities for hamiltonian systems on time scales with nabla derivatives and delta derivatives are presented first hamilton principle on time scales with nabla derivatives is established and hamilton canonical equation with nabla derivatives is obtained second noether identity and noether theorem for hamiltonian systems with nabla derivatives are achieved third hamilton canonical equation with delta derivatives noether identity and noether theorem for hamiltonian systems with delta derivatives are gotten through duality principle on the basis of the corresponding results with nabla derivatives fourth some special cases of noether identity and noether theorem are given and finally two examples are devoted to illustrate the methods and results

generalized high dimensional trace regression via nuclear norm regularization abstract we study the generalized trace regression with a near low rank regression coefficient matrix which extends notion of sparsity for regression coefficient vectors specifically given a matrix covariate x the probability density function of the response y is f y x c y exp 1 y b where tr t x this model accommodates various types of responses and embraces many important problem setups such as reduced rank regression matrix regression that accommodates a panel of regressors matrix completion among others we estimate through minimizing empirical negative log likelihood plus nuclear norm penalty we first establish a general theory and then for each specific problem we derive explicitly the statistical rate of the proposed estimator they all match the minimax rates in the linear trace regression up to logarithmic factors numerical studies confirm the rates we established and demonstrate the advantage of generalized trace regression over linear trace regression when the response is dichotomous we also show the benefit of incorporating nuclear norm regularization in dynamic stock return prediction and in image classification

fractional curve flows and solitonic hierarchies in gravity and geometric mechanics methods from the geometry of nonholonomic manifolds and lagrange finsler spaces are applied in fractional calculus with caputo derivatives and for elaborating models of fractional gravity and fractional lagrange mechanics the geometric data for such models are encoded into fractional bi hamiltonian structures and associated solitonic hierarchies the constructions yield horizontal vertical pairs of fractional vector sine gordon equations and fractional vector mkdv equations when the hierarchies for corresponding curve fractional flows are described in explicit forms by fractional wave maps and analogs of schrodinger maps

the hamilton formalism with fractional derivatives recently the traditional calculus of variations has been extended to be applicable for systems containing fractional derivatives in this paper the passage from the lagrangian containing fractional derivatives to the hamiltonian is achieved the hamilton s equations of motion are obtained in a similar manner to the usual mechanics in addition the classical fields with fractional derivatives are investigated using hamiltonian formalism two discrete problems and one continuous are considered to demonstrate the application of the formalism the results are obtained to be in exact agreement with agrawal s formalism

confidence intervals for diffusion index forecasts and inference for factor augmented regressions we consider the situation when there is a large number of series n e ach witht observations and each series has some predictive ability for some variable of interest a methodology of growing interest is first to estimate common factors from the panel of data by the method of principal components and then to augment an otherwise standard regression with the estimated factors in this paper we show that the least squares estimates obtained from these factor augmented regressions are t consistent and asymptotically normal if t n 0 the conditional mean predicted by the estimated factors is min t n consistent and asymptotically normal except when t n goes to zero inference should take into account the effect of estimated regressors on the estimated conditional mean we present analytical formulas for prediction intervals that are valid regardless of the magnitude of n t and that can also be used when the factors are nonstationary

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

generalized latent trait models in this paper we discuss a general model framework within which manifest variables with different distributions in the exponential family can be analyzed with a latent trait model a unified maximum likelihood method for estimating the parameters of the generalized latent trait model will be presented we discuss in addition the scoring of individuals on the latent dimensions the general framework presented allows not only the analysis of manifest variables all of one type but also the simultaneous analysis of a collection of variables with different distributions the approach used analyzes the data as they are by making assumptions about the distribution of the manifest variables directly

an intrinsic limiting procedure for model selection and hypotheses testing abstract improper priors typically arise in default bayesian estimation problems in the bayesian approach to model selection or hypothesis testing the main tool is the bayes factor when improper priors for the parameters appearing in the models are used the bayes factor is not well defined the intrinsic bayes factor introduced by berger and pericchi is an interesting method for overcoming that difficulty that method is of particular interest as a means for generating proper prior distributions intrinsic priors for model comparison from the improper priors typically used in estimation the goal of this article is to develop a limiting procedure that provides a solid justification for the use of bayes factor with intrinsic priors the procedure is formalized and discussed for nested and nonnested models illustrations and comparisons with other approximations to bayes factors such as the bayesian information criterion of schwarz and the fractional bayes factor of o hagan are provided

fractional variational calculus in terms of riesz fractional derivatives this paper presents extensions of traditional calculus of variations for systems containing riesz fractional derivatives rfds specifically we present generalized euler lagrange equations and the transversality conditions for fractional variational problems fvps defined in terms of rfds we consider two problems a simple fvp and an fvp of lagrange results of the first problem are extended to problems containing multiple fractional derivatives functions and parameters and to unspecified boundary conditions for the second problem we present lagrange type multiplier rules for both problems we develop the euler lagrange type necessary conditions which must be satisfied for the given functional to be extremum problems are considered to demonstrate applications of the formulations explicitly we introduce fractional momenta fractional hamiltonian fractional hamilton equations of motion fractional field theory and fractional optimal control the formulations presented and the resulting equations are similar to the formulations for fvps given in agrawal 2002 j math anal appl 272 368 2006 j phys a math gen 39 10375 and to those that appear in the field of classical calculus of variations these formulations are simple and can be extended to other problems in the field of fractional calculus of variations

fractional langevin equation overdamped underdamped and critical behaviors the dynamical phase diagram of the fractional langevin equation is investigated for a harmonically bound particle it is shown that critical exponents mark dynamical transitions in the behavior of the system four different critical exponents are found i ensuremath alpha c 0 402 ifmmode pm else textpm fi 0 002 marks a transition to a nonmonotonic underdamped phase ii ensuremath alpha r 0 441 dots marks a transition to a resonance phase when an external oscillating field drives the system and iii ensuremath alpha ensuremath chi 1 0 527 dots and iv ensuremath alpha ensuremath chi 2 0 707 dots mark transitions to a double peak phase of the loss when such an oscillating field present as a physical explanation we present a cage effect where the medium induces an elastic type of friction phase diagrams describing over and underdamped regimes with or without resonances show behaviors different from normal

nonconservative lagrangian and hamiltonian mechanics traditional lagrangian and hamiltonian mechanics cannot be used with nonconservative forces such as friction a method is proposed that uses a lagrangian containing derivatives of fractional order a direct calculation gives an euler lagrange equation of motion for nonconservative forces conjugate momenta are defined and hamilton s equations are derived using generalized classical mechanics with fractional and higher order derivatives the method is applied to the case of a classical frictional force proportional to velocity textcopyright 1996 the american physical society

stochastic resonance in a fractional oscillator with random mass and random frequency for a fractional linear oscillator subjected to two multiplicative dichotomous noises and a additive fractional gaussian noise and driven by a periodic signal we study the stochastic resonance sr in this paper using fractional shapiro loginov formula and the laplace transformation technique we acquire the exact expression of the first order moment of the system s steady response meanwhile we discuss the evolutions of the output amplitude with frequency of the periodic signal noise parameters fractional order and friction coefficient we find that sr in the wide sense existing in this system specially the evolution of the output amplitude with frequency of the periodic signal presents one peak oscillation and two peak oscillation moreover the friction coefficient can induce stochastic multi resonance

a numerical scheme for two dimensional optimal control problems with memory effect a new formulation for multi dimensional fractional optimal control problems is presented in this article the fractional derivatives which are coming from the formulation of the problem are defined in the riemann liouville sense some terminal conditions are imposed on the state and control variables whose dimensions need not be the same a numerical scheme is described by using the grunwald letnikov definition to approximate the riemann liouville fractional derivatives the set of fractional differential equations which are obtained after the discretization of the time domain are solved within the grunwald letnikov approximation to obtain the state and the control variable numerically a two dimensional fractional optimal control problem is studied as an example to demonstrate the performance of the scheme

sufficient forecasting using factor models we consider forecasting a single time series when there is a large number of predictors and a possible nonlinear effect the dimensionality was first reduced via a high dimensional approximate factor model implemented by the principal component analysis using the extracted factors we develop a novel forecasting method called the sufficient forecasting which provides a set of sufficient predictive indices inferred from high dimensional predictors to deliver additional predictive power the projected principal component analysis will be employed to enhance the accuracy of inferred factors when a semi parametric approximate factor model is assumed our method is also applicable to cross sectional sufficient regression using extracted factors the connection between the sufficient forecasting and the deep learning architecture is explicitly stated the sufficient forecasting correctly estimates projection indices of the underlying factors even in the presence of a nonparametric forecasting function the proposed method extends the sufficient dimension reduction to high dimensional regimes by condensing the cross sectional information through factor models we derive asymptotic properties for the estimate of the central subspace spanned by these projection directions as well as the estimates of the sufficient predictive indices we further show that the natural method of running multiple regression of target on estimated factors yields a linear estimate that actually falls into this central subspace our method and theory allow the number of predictors to be larger than the number of observations we finally demonstrate that the sufficient forecasting improves upon the linear forecasting in both simulation studies and an empirical study of forecasting macroeconomic variables

inferential theory for factor models of large dimensions this paper develops an inferential theory for factor models of large dimensions the principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator if normality is assumed we derive the rate of convergence and the limiting distributions of the estimated factors factor loadings and common components the theory is developed within the framework of large cross sections n and a large time dimension t to which classical factor analysis does not apply we show that the estimated common components are asymptotically normal with a convergence rate equal to the minimum of the square roots of n and t the estimated factors and their loadings are generally normal although not always so the convergence rate of the estimated factors and factor loadings can be faster than that of the estimated common components these results are obtained under general conditions that allow for correlations and heteroskedasticities in both dimensions stronger results are obtained when the idiosyncratic errors are serially uncorrelated and homoskedastic a necessary and sufficient condition for consistency is derived for large n but fixed t

default bayes factors for nonnested hypothesis testing abstract bayesian hypothesis testing for nonnested hypotheses is studied using various default bayes factors such as the fractional bayes factor the median intrinsic bayes factor and the encompassing and expected intrinsic bayes factors the different default methods are first compared with each other and with the p value in normal one sided testing to illustrate the basic issues general results for one sided testing in location and scale models are then presented the default bayes factors are also studied for specific models involving multiple hypotheses in particular a multiple hypothesis testing example involving a sequential clinical trial is discussed in most of the examples presented we also derive the intrinsic prior this is the prior distribution which if used directly would yield answers asymptotically equivalent to those for the given default bayes factor