extensions of the first and second complex step derivative approximations a general framework for the first and second complex step derivative approximation to compute numerical derivatives is presented for first derivatives the complex step approach does not suffer roundoff errors as in standard numerical finite difference approaches therefore since an arbitrarily small step size can be chosen the complex step approach can achieve near analytical accuracy however for second derivatives straight implementation of the complex step approach does suffer from roundoff errors therefore an arbitrarily small step size cannot be chosen in this paper the standard complex step approach is expanded by using general complex step sizes to provide a wider range of accuracy for both the first and second derivative approximations even higher accuracy formulations are obtained by repetitively applying richardson extrapolations the new extensions can allow the use of one step size to provide optimal accuracy for both derivative approximations

on the solutions of the cauchy problem for a class of partial differential equations with double characteristic at a point abstract we give an explicit representation of the solutions of the cauchy problem in terms of series of hypergeometric functions for the following class of partial differential equations with double characteristic at the origin x k t a x x k t b x u c x k 1 t u 0 u 0 x u 0 x t u 0 x u 1 x we show that the solutions are holomorphic ramified around the characteristic surfaces k k 1 k 2 k 3 with k 1 a k 1 t x k 1 0 k 2 b k 1 t x k 1 0 k 3 x 0

the t analogs of q characters at roots of unity for quantum affine algebras and beyond abstract the q characters were introduced by frenkel and reshetikhin the q characters of representations of quantum affine algebras and deformations of w algebras in recent developments in quantum affine algebras and related topics in contemp math vol 248 amer math soc providence ri 1999 pp 163 205 to study finite dimensional representations of the untwisted quantum affine algebra u q g for q generic the e characters at roots of unity were constructed by frenkel and mukhin the q characters at roots of unity adv math 171 1 2002 139 167 to study finite dimensional representations of various specializations of u q g at q s 1 in the finite simply laced case nakajima t analogue of the q characters of finite dimensional representations of quantum affine algebras in physics and combinatorics proc nagoya 2000 internat workshop world scientific singapore 2001 pp 181 212 quiver varieties and t analogs of q characters of quantum affine algebras ann of math in press preprint arxiv math qa 0105173 defined deformations of q characters called q t characters for q generic and also at roots of unity the definition is combinatorial but the proof of the existence uses the geometric theory of quiver varieties which holds only in the simply laced case in algebraic approach to q t characters adv math in press preprint arxiv math qa 0212257 we proposed an algebraic general non necessarily simply laced new approach to q t characters for q generic in this paper we propose two developments of this approach we treat the root of unity case and the case of a larger class of generalized cartan matrices including finite and affine cases except a 1 1 a 2 2 in particular we generalize the construction of analogs of kazhdan lusztig polynomials at roots of unity of h nakajima quiver varieties and t analogs of q characters of quantum affine algebras ann of math in press preprint arxiv math qa 0105173 to those cases we also study properties of various objects used in this article deformed screening operators at roots of unity t deformed polynomial algebras bicharacters arising from symmetrizable cartan matrices deformation of the frenkel mukhin s algorithm

on the cauchy problem for a class of iterated fuchsian partial differential equations abstract in this paper we consider the cauchy problem with ramified data for a class of iterated fuchsian partial differential equations we give an explicit representation of the solution in terms of gauss hypergeometric functions our results are illustrated through some examples

finite dimensional representations of quantum affine algebras at roots of unity we describe explicitly the canonical map x spec ue b spec ze where u b is a quantum loop algebra at an odd root of unity e here ze is the center of ue g and spec r stands for the set of all finite dimensional irreducible representations of an algebra r we show that spec z is a poisson proalgebraic group which is essentially the group of points of g over the regular adeles concentrated at 0 and oo our main result is that the image under x of spec u b is the subgroup of principal adeles department of mathematics harvard university cambridge massachusetts 02138 e mail address beckfmath harvard edu department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address kacimath mit edu this content downloaded from 207 46 13 124 on wed 22 jun 2016 06 11 29 utc all use subject to http about jstor org terms

a linear fuchsian equation with variable indices abstract we construct three types of solutions for a fuchsian equation with variable indices 1 branched solutions involving logarithms of the time variable t 2 solutions involving tx where x is a space variable and 3 for a model case exact solutions involving hypergeometric functions these three solutions have completely different singularities the constructions are given in a form suitable for application to more general equations as an illustration we resolve in particular an apparent discrepancy between two recent results on this problem

moduli spaces of curves and representation theory we establish a canonical isomorphism between the second cohomology of the lie algebra of regular differential operators on x of degree 1 and the second singular cohomology of the moduli space hat f g 1 of quintuples c p z l wherec is a smooth genusg riemann surface p a point onc z a local parameter atp l a degreeg 1 line bundle onc and a class of local trivializations ofl atp which differ by a non zero factor the construction uses an interplay between various infinite dimensional manifolds based on the topological spaceh of germs of holomorphic functions in a neighborhood of 0 in x and related topological spaces the basic tool is a canonical map from hat f g 1 to the infinite dimensional grassmannian of subspaces ofh which is the orbit of the subspaceh of holomorphic functions on x vanishing at under the group auth as an application we give a lie algebraic proof of the mumford formula n 6n2 6n 1 1 where n is the determinant line bundle of the vector bundle on the moduli space of curves of genusg whose fiber overc is the space of differentials of degreen onc

complex step derivative approximation in noisy environment abstract the complex step derivative approximation is a powerful method for derivative approximations which has been successfully implemented in deterministic numerical algorithms we explore and analyze its implementation in noisy environment through examples error analysis and application to optimization methods numerical results show a promising performance of the complex step gradient approximation in noisy environment

geometry of the analytic loop group abstract we introduce and study a notion of analytic loop group with a riemann hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity u g with non trivial central charge we introduce a poisson structure and study properties of its poisson dual group we prove that the hopf poisson structure is isomorphic to the semi classical limit of the center of u g it is a geometric realization of the center then the symplectic leaves and corresponding equivalence classes of central characters of u g are parameterized by certain g bundles on an elliptic curve

on the generalization of the complex step method we generalize the well known complex step method for computing derivatives by introducing a complex step in a strict sense exploring different combinations of terms we derive 52 approximations for computing the first order derivatives and 43 for the second order derivatives for an appropriate combination of terms and appropriate choice of the step size in the real and imaginary directions fourth order accuracy can be achieved in a very simple and efficient scheme on a compact stencil new different ways of computing second order derivatives in one single step are shown many of the first order derivative approximations avoid the problem of subtractive cancellation inherent to the classic finite difference approximations for real valued steps and the superior accuracy and stability of the generalized complex step approximations are demonstrated for an analytic test function

localization of modules for a semisimple lie algebra in prime characteristic with an appendix by r bezrukavnikov and s riche computation for sl 3 we show that on the level of derived categories representations of the lie algebra of a semisimple algebraic group over a field of finite characteristic with a given generalized regular central character are the same as coherent sheaves on the formal neighborhood of the corresponding generalized springer fiber the first step is to observe that the derived functor of global sections provides an equivalence between the derived category of d modules with no divided powers on the flag variety and the appropriate derived category of modules over the corresponding lie algebra thus the derived version of the beilinson bernstein localization theorem holds in sufficiently large positive characteristic next one finds that for any smooth variety this algebra of differential operators is an azumaya algebra on the cotangent bundle in the case of the flag variety it splits on springer fibers and this allows us to pass from d modules to coherent sheaves the argument also generalizes to twisted d modules as an application we prove lusztig s conjecture on the number of irreducible modules with a fixed central character we also give a formula for behavior of dimension of a module under translation functors and reprove the kac weisfeiler conjecture the sequel to this paper bmr2 treats singular infinitesimal characters

using complex variables to estimate derivatives of real functions a method to approximate derivatives of real functions using complex variables which avoids the subtractive cancellation errors inherent in the classical derivative approximations is described numerical examples illustrating the power of the approximation are presented

parameterized reduced order models from a single mesh using hyper dual numbers abstract in order to assess the predicted performance of a manufactured system analysts must consider random variations both geometric and material in the development of a model instead of a single deterministic model of an idealized geometry with idealized material properties the incorporation of random geometric variations however potentially could necessitate the development of thousands of nearly identical solid geometries that must be meshed and separately analyzed which would require an impractical number of man hours to complete this research advances a recent approach to uncertainty quantification by developing parameterized reduced order models these parameterizations are based upon taylor series expansions of the system s matrices about the ideal geometry and a component mode synthesis representation for each linear substructure is used to form an efficient basis with which to study the system the numerical derivatives required for the taylor series expansions are obtained via hyper dual numbers and are compared to parameterized models constructed with finite difference formulations the advantage of using hyper dual numbers is two fold accuracy of the derivatives to machine precision and the need to only generate a single mesh of the system of interest the theory is applied to a stepped beam system in order to demonstrate proof of concept the results demonstrate that the hyper dual number multivariate parameterization of geometric variations which largely are neglected in the literature are accurate for both sensitivity and optimization studies as model and mesh generation can constitute the greatest expense of time in analyzing a system the foundation to create a parameterized reduced order model based off of a single mesh is expected to reduce dramatically the necessary time to analyze multiple realizations of a component s possible geometry

on the accuracy of the complex step finite difference method abstract the complex step finite difference method csfdm is a very simple methodology that can be implemented in well known numerical techniques helping to improve for instance in the wave propagation problem time and or space derivative based wavefields we clarify differences between the csfdm and previous implementations of the complex step cs derivative approximation in well known numerical techniques we study dispersion properties for the one way and two way wave equations using the finite difference method fdm the pseudospectral method psm the finite element method fem and the csfdm under the influence of a plane wave and ricker source time functions we show the gain in numerical accuracy offered by the methodology of the csfdm over the fdm psm and fem we finally discuss consequences of the csfdm in future scenarios and propose directions of study in this area

complex variable method for eigensolution sensitivity analysis the application of complex variable method cvm for eigenvalue and eigenvector sensitivity analysis is presented gradi ent based methods usually used in structural optimization require accurate sensitivity information cvm is accurate robust and easy to implement as compared to other approximate sensitivity analysis methods by cvm the first order modal sensitivity can b e calculated using an existing structural analysis package this is achieved by making a small complex perturbation to a design variable the resulting eigensolutions would be complex the sensitivity of eigensolutions with respect to that design variable can then be obtained by taking the complex part of the eigensolutions divided by the perturbation since there is no subtraction involved in the sensitivity calculation the approximation is not sensitive to step size thus unlike finite difference method s cvm does not suffer the effect of subtraction cancellation error when a very small step size is used the computation of 1 st order modal sensitivity is not hampered by repeated eigenvalue problem as is the case with other sensitivity analysis methods for repeated eigenvalues cvm automatically provides correct eigenvalue sensitivity and the associated eigenvector sensitivity for a differentiable mode a 5 dof system with an eigenvalue of multiplicity 3 is used to illustrate the salient features of cvm in eigen solution sensitivity analysis nomenclature k stiffness matrix for spring mass system m mass matrix for spring mass system eigenvalue for spring mass system eigenvector for spring mass system derivative of w r t design variable

on the extension of the complex step derivative technique to pseudospectral algorithms the complex step derivative csd technique is a convenient and highly accurate strategy to perform a linearized perturbation analysis to determine a directional derivative via a minor modification of an existing nonlinear simulation code the technique has previously been applied to nonlinear simulation codes such as finite element codes which employ real arithmetic only the present note examines the suitability of this technique for extension to efficient pseudospectral simulation codes which nominally use the fast fourier transform fft to convert back and forth between the physical and transformed representations of the system it is found that if used carefully this extension retains the remarkable accuracy of the csd approach however to perform this extension without sacrificing this accuracy particular care must be exercised specifically the state real and perturbation imaginary components of the complexified system must be transformed separately and arranged in such a manner that they are kept distinct during the process of differentiation in the transformed space in order to avoid the linear combination of the large and small quantities in the analysis it is shown that this is relatively straightforward to implement even in complicated nonlinear simulation codes thus positioning the csd approach as an attractive and relatively simple alternative to hand coding a perturbation a k a tangent linear code for determining the directional derivative even when pseudospectral algorithms are employed

sensitivity analysis for navier stokes equations on unstructured meshes using complex variables the use of complex variables for determining sensitivity derivatives for turbulent flows is examined although a step size parameter is required the numerical derivatives are not subject to subtractive cancellation errors and therefore exhibit true second order accuracy as the step size is reduced as a result this technique guarantees two additional digits of accuracy each time the step size is reduced one order of magnitude this behavior is in contrast to the use of finite differences which suffer from inaccuracies due to subtractive cancellation errors in addition the complex variable procedure is easily implemented into existing codes

nonlinear robust performance analysis using complex step gradient approximation in this paper the complex step method is applied in the setting of numerical optimisation problems involving dynamical systems modelled as nonlinear differential equations the main advantage of the complex step method for gradient approximation is that it entails no subtractive cancellation error and therefore the truncation error can be made arbitrarily to machine precision small the method is applied to two robust performance analysis problems the accuracy and convergence rate of the solutions computed using the proposed approach are seen to be significantly better than those achieved using standard gradient approximation methods

the q characters at roots of unity abstract we consider various specializations of the untwisted quantum affine algebras at roots of unity we define and study the q characters of their finite dimensional representations

hopf algebras with trace and representations we study the restriction of representations of cayley hamilton algebras to subalgebras this theory is applied to determine tensor products and branching rules for representations of quantum groups at roots of 1