finite arbitrage times and the volatility smile abstract extending previous work on non equilibrium option pricing theory eur phys j 14 2000 383 394 a mean field approach is developed to understand the curvature of implied by black scholes bs volatility surfaces curves as a function of moneyness strike price divided by price the previously developed hypothesis of a finite arbitrage time during which fluctuations around the equilibrium state absence of arbitrage are allowed to occur is generalized as follows instead of a unique arbitrage time independant of moneyness a distribution of arbitrage times will be assumed where the mean arbitrage time will be a function of moneyness this hypothesis is motivated by the fact that the trading volume is the largest for at the money options assuming now the arbitrage time to be inversely proportional to trading volume naturally leads to our generalized hypothesis on the mean arbitrage time consequences on plain vanilla option prices will be studied

on the group of homotopy equivalences of a manifold we consider the group of homotopy equivalences s m of a simply connected manifold m which is part of the fundamental extension of groups due to barcus barratt we show that the kernel of this extension is always a finite group and we compute this kernel for various examples this leads to computations of the group s m for special manifolds m for example if m is a connected sum of products s x sm of spheres in particular the group s s x s is determined completely also the connection of s m with the group of isotopy classes of diffeomorphisms of m is studied the group x of homotopy equivalences of a space x is the set of homotopy classes of homotopy equivalences x x the group structure is induced by mapcomposition the group x i e the group of automorphisms of the homotopy type of x can be regarded as the homotopy symmetry group of the space x in the literature there has been a lot of interest in the computation of such groups compare for example the excellent survey article of m arkowitz 2 this paper is concerned with the structure of m in case m is a closed compact oriented manifold or more generally a poincare complex the computation of this group is an important step for the diffeomorphism classification of manifolds by surgery 40 the group m is also important for cooke s theory 17 of replacing homotopy actions by actions for a differential manifold m the group m is comparable with the group hodiff m of isotopy classes of diffeomorphisms of m in fact via the j homomorphism there is a striking similarity between these groups as is shown in 10 below still there is little known on the group m in the literature only very specific examples are computed see 2 this paper contains on the one hand general results on the structure of the group m see 1 5 on the other hand our methods are used for explicit computations see 6 10 let e be a small open cell of the simply connected m dimensional manifold m and let m m e be the complement the inclusion f sm 1 c m of the boundary is the attaching map for m we consider the fundamental extension of groups due to barcus barratt 42 m i m m 3 s m f here m i m is the subgroup of m consisting of all elements which can be represented by an orientation preserving map m m under m this group is received by the editors august 17 1994 1991 mathematics subject classification primary 55010 57050 1996 american mathematical society 4737 this content downloaded from 157 55 39 197 on wed 26 apr 2017 17 53 44 utc all use subject to http about jstor org terms

derivative pricing with non linear fokker planck dynamics abstract we examine how the black scholes derivative pricing formula is modified when the underlying security obeys non extensive statistics and fokker planck dynamics an unusual feature of such securities is that the volatility in the underlying ito langevin equation depends implicitly on the actual market rate of return this complicates most approaches to valuation here we show that progress is possible using variations of the cox ross valuation technique

abelianisation of orthogonal groups and the fundamental group of modular varieties abstract we study the commutator subgroup of integral orthogonal groups belonging to indefinite quadratic forms we show that the index of this commutator is 2 for many groups that occur in the construction of moduli spaces in algebraic geometry in particular the moduli of k3 surfaces we give applications to modular forms and to computing the fundamental groups of some moduli spaces

the anomaly line bundle of the self dual field theory in this work we determine explicitly the anomaly line bundle of the abelian self dual field theory over the space of metrics modulo diffeomorphisms including its torsion part inspired by the work of belov and moore we propose a non covariant action principle for a pair of euclidean self dual fields on a generic oriented riemannian manifold the corresponding path integral allows one to study the global properties of the partition function over the space of metrics modulo diffeomorphisms we show that the anomaly bundle for a pair of self dual fields differs from the determinant bundle of the dirac operator coupled to chiral spinors by a flat bundle that is not trivial if the underlying manifold has middle degree cohomology and whose holonomies are determined explicitly we briefly sketch the relevance of this result for the computation of the global gravitational anomaly of the self dual field theory that will appear in another paper

an ℏ brownian motion and the existence of stochastic option prices abstract we introduce a particular type of brownian motion i e a brownian motion with a diffusion coefficient containing we show that under classical black scholes methodology we can obtain a pde with a stochastic rate of return in this environment of non classical uncertainty valued preferences for risk may exist

semi implicit integration factor methods on sparse grids for high dimensional systems numerical methods for partial differential equations in high dimensional spaces are often limited by the curse of dimensionality though the sparse grid technique based on a one dimensional hierarchical basis through tensor products is popular for handling challenges such as those associated with spatial discretization the stability conditions on time step size due to temporal discretization such as those associated with high order derivatives in space and stiff reactions remain here we incorporate the sparse grids with the implicit integration factor method iif that is advantageous in terms of stability conditions for systems containing stiff reactions and diffusions we combine iif in which the reaction is treated implicitly and the diffusion is treated explicitly and exactly with various sparse grid techniques based on the finite element and finite difference methods and a multi level combination approach the overall method is found to be efficient in terms of both storage and computational time for solving a wide range of pdes in high dimensions in particular the iif with the sparse grid combination technique is flexible and effective in solving systems that may include cross derivatives and non constant diffusion coefficients extensive numerical simulations in both linear and nonlinear systems in high dimensions along with applications of diffusive logistic equations and fokker planck equations demonstrate the accuracy efficiency and robustness of the new methods indicating potential broad applications of the sparse grid based integration factor method

infinite loop spaces and positive scalar curvature we study the homotopy type of the space of metrics of positive scalar curvature on high dimensional compact spin manifolds hitchin used the fact that there are no harmonic spinors on a manifold with positive scalar curvature to construct a secondary index map from the space of positive scalar metrics to a suitable space from the real k theory spectrum our main results concern the nontriviality of this map we prove that for 2n ge 6 the natural ko orientation from the infinite loop space of the madsen tillmann weiss spectrum factors up to homotopy through the space of metrics of positive scalar curvature on any 2n dimensional spin manifold for manifolds of odd dimension 2n 1 ge 7 we prove the existence of a similar factorisation when combined with computational methods from homotopy theory these results have strong implications for example the secondary index map is surjective on all rational homotopy groups we also present more refined calculations concerning integral homotopy groups to prove our results we use three major sets of technical tools and results the first set of tools comes from riemannian geometry we use a parameterised version of the gromov lawson surgery technique which allows us to apply homotopy theoretic techniques to spaces of metrics of positive scalar curvature secondly we relate hitchin s secondary index to several other index theoretical results such as the atiyah singer family index theorem the additivity theorem for indices on noncompact manifolds and the spectral flow index theorem finally we use the results and tools developed recently in the study of moduli spaces of manifolds and cobordism categories the key new ingredient we use in this paper is the high dimensional analogue of the madsen weiss theorem proven by galatius and the third named author

word maps isotopy and entropy on trouve des diffeomorphismes de basse entropie dans chaque classe d isotopie sur s 3 s 3

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

a high order boundary integral method for surface diffusions on elastically stressed axisymmetric rods many applications in materials involve surface diffusion of elastically stressed solids study of singularity formation and long time behavior of such solid surfaces requires accurate simulations in both space and time here we present a high order boundary integral method for an elastically stressed solid with axi symmetry due to surface diffusions in this method the boundary integrals for isotropic elasticity in axi symmetric geometry are approximated through modified alternating quadratures along with an extrapolation technique leading to an arbitrarily high order quadrature in addition a high order temporal integration factor method based on explicit representation of the mean curvature is used to reduce the stability constraint on time step to apply this method to a periodic in axial direction and axi symmetric elastically stressed cylinder we also present a fast and accurate summation method for the periodic green s functions of isotropic elasticity using the high order boundary integral method we demonstrate that in absence of elasticity the cylinder surface pinches in finite time at the axis of the symmetry and the universal cone angle of the pinching is found to be consistent with the previous studies based on a self similar assumption in the presence of elastic stress we show that a finite time geometrical singularity occurs well before the cylindrical solid collapses onto the axis of symmetry and the angle of the corner singularity on the cylinder surface is also estimated

higher order exponential time differencing scheme for system of coupled nonlinear schrodinger equations the coupled nonlinear schrodinger equations are highly used in modeling the various phenomena in nonlinear fiber optics like propagation of pulses efficient and reliable numerical schemes are required for analysis of these models and for improvement of the fiber communication system in this paper we introduce a new version of the cox and matthews third order exponential time differencing runge kutta etd3rk scheme based on the 1 2 pade approximation to the exponential function in addition we present its local extrapolation form to improve temporal accuracy to the fourth order the developed scheme and its extrapolation are seen to be strongly stable which have ability to damp spurious oscillations caused by high frequency components in the solution a computationally efficient algorithm of the new scheme based on a partial fraction splitting technique is presented in order to investigate the performance of the novel scheme we considered the system of two and four coupled nonlinear schrodinger equations and performed several numerical experiments on them the numerical experiments showed that the developed numerical scheme provide an efficient and reliable way for computing long range solitary solutions given by coupled nonlinear schrodinger equations and conserved the conserved quantities mass and energy exactly to at least five decimal places

the numerical study for the ground and excited states of fractional bose einstein condensates abstract in this paper we study the ground and first excited states of the fractional bose einstein condensates bec which is modeled by fractional gross pitaevskii gp equation we first introduce the normalized gradient flow method and prove its energy diminishing property then the weighted shifted grunwald letnikov difference wsgd method is used to discretize the gross pitaevskii equation the corresponding normalization and energy diminishing property for the semi discrete scheme are proved for the time discretization we use the implicit integration factor iif method which decouples the diffusion and nonlinear terms separately finally the numerical methods are applied to compute the ground and first excited states of fractional bec with harmonic oscillator harmonic plus optical lattice and box potential our numerical results show that the ground and excited states in fractional gp equation differ from those of the standard non fractional gp equation

an implicit explicit runge kutta chebyshev scheme for diffusion reaction equations an implicit explicit imex extension of the explicit runge kutta chebyshev rkc scheme designed for parabolic pdes is proposed for diffusion reaction problems with severely stiff reaction terms the imex scheme treats these reaction terms implicitly and diffusion terms explicitly within the setting of linear stability theory the new imex scheme is unconditionally stable for reaction terms having a jacobian matrix with a real spectrum for diffusion terms the stability characteristics remain unchanged a numerical comparison for a stiff nonlinear radiation diffusion problem between an rkc solver an imex rkc solver and the popular implicit bdf solver vodpk using the krylov solver gmres illustrates the excellent performance of the new scheme

finite group actions on kervaire manifolds abstract let m k 4 k 2 be the kervaire manifold a closed piecewise linear pl manifold with kervaire invariant 1 and the same homology as the product s 2 k 1 s 2 k 1 of spheres we show that a finite group of odd order acts freely on m k 4 k 2 if and only if it acts freely on s 2 k 1 s 2 k 1 if m k is smoothable then each smooth structure on m k admits a free smooth involution if k 2 j 1 then m k 4 k 2 does not admit any free top involutions free exotic pl involutions are constructed on m k 30 m k 62 and m k 126 each smooth structure on m k 30 admits a free z 2 z 2 action

local discontinuous galerkin methods for parabolic interface problems with homogeneous and non homogeneous jump conditions abstract in this paper we present the local discontinuous galerkin ldg methods for solving second order parabolic interface problems in two dimensional convex polygonal domains with homogeneous and non homogeneous jump conditions respectively we start analyzing from the homogeneous problems after giving the semi discrete ldg method we derive its stability and a prior error estimate then after a series of derivation we obtain the ldg method for the non homogeneous problems which takes the same formulation as the homogeneous case with a special choice of numerical fluxes that incorporate the jump conditions across interface so its analysis can be easily obtained within the framework developed for the homogeneous case it is an advantage of the ldg method that it provides a natural framework to enforce the discontinuities in both the potential and the flux weakly in the discrete formulation provided that the triangulation of the domain is fitted to the interface numerical experiments are conducted to confirm our theoretical analysis with the help of the explicit temporal discretization and show that the ldg methods have the optimal and suboptimal convergence rate in the l 2 norms for the potential and the flux respectively in addition they also indicate that the ldg methods possess the same convergence properties in the l norms what is more to avoid the strict time step restriction of explicit schemes the implicit integration factor iif method is employed which not only relaxes the time step to t o h m i n but also remains the important property of ldg method that the computation can proceed element by element and avoid solving a global system of algebraic equations as the standard implicit schemes do we provide numerical examples to illustrate that the numerical scheme combined ldg method with iif method indeed reduces the computational cost greatly

on the kodaira dimension of orthogonal modular varieties we prove that up to scaling there are only finitely many integral lattices l of signature 2 n with n ge 21 or n 17 such that the modular variety defined by the orthogonal group of l is not of general type in particular when n ge 108 every modular variety defined by an arithmetic group for a rational quadratic form of signature 2 n is of general type we also obtain similar finiteness in n ge 9 for the stable orthogonal groups as a byproduct we derive finiteness of lattices admitting reflective modular form of bounded vanishing order which proves a conjecture of gritsenko and nikulin

the picard group of the moduli space of curves with level structures for 4 l and g large we calculate the integral picard groups of the moduli spaces of curves and principally polarized abelian varieties with level l structures in particular we determine the divisibility properties of the standard line bundles over these moduli spaces and we calculate the second integral cohomology group of the level l subgroup of the mapping class group in a previous paper the author determined this rationally this entails calculating the abelianization of the level l subgroup of the mapping class group generalizing previous results of perron sato and the author finally along the way we calculate the first homology group of the mod l symplectic group with coefficients in the adjoint representation

on the mapping class groups of r sp sp for p 3 7 for m r sharp r s p times s p p 3 7 we calculate pi 0 rm diff m r theta 2p 1 and mathcal e m r respectively the group of isotopy classes of orientation preserving diffeomorphisms of mr modulo isotopy classes with representatives which are the identity outside a 2p disc and the group of homotopy classes of orientation preserving homotopy equivalences of mr

unconditionally energy stable large time stepping method for the l2 gradient flow based ternary phase field model with precise nonlocal volume conservation abstract we consider numerical approximations for solving a new allen cahn type ternary phase field model we first formulate a phase field model from an energetic variational formulation based on the l 2 gradient flow and add three nonlocal lagrange multipliers to the system to conserve the volume for each phase then by combining the sav approach with the stabilization technique where a crucial linear stabilization term is added to enhance the stability and keep the required accuracy thus allowing large time steps we arrive at a decoupled linear non iterative and volume conserved scheme this scheme is unconditionally energy stable and requires solving only four linear second order elliptic equations with constant coefficients we further prove energy stability and present numerous 2d and 3d numerical simulations to show accuracy and stability

fast implicit integration factor method for nonlinear space riesz fractional reaction diffusion equations abstract after spatially discretizing the nonlinear space riesz fractional reaction diffusion equations by the fractional centered difference formula the resulting semi discrete equations lead to a nonlinear ordinary differential equation system in order to obtain good stability and robustness the implicit integration factor iif method that treats the reaction term implicitly and the diffusion term exactly is employed to solve the system to deal with the extremely expensive cost in iif we also propose the shift invert lanczos method based on the gohberg semencul formula to compute the matrix exponential by using the particular property that the coefficient matrix is symmetric positive definite toeplitz the proposed fast method requires only o m memory storage and o m log m computation cost in each iterative step compared to the o m 2 storage and o m 3 computational complexity of the direct solution method where m is the size of the spatial grid some numerical experiments are performed to verify the correctness of the theoretical results and demonstrate the efficiency of our fast solution algorithm

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

on the adaptive numerical solution of nonlinear partial differential equations in wavelet bases this work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the formut lu nf u where l and n are linear differential operators andf u is a nonlinear function these equations are adaptively solved by projecting the solutionuand the operators l and n into a wavelet basis vanishing moments of the basis functions permit a sparse representation of the solution and operators using these sparse representations fast and adaptive algorithms that apply operators to functions and evaluate nonlinear functions are developed for solving evolution equations for a wavelet representation of the solutionuthat containsnssignificant coefficients the algorithms update the solution usingo ns operations the approach is applied to a number of examples and numerical results are given

theta functions and a presentation of 21 2g 1 sp 2g 2 abstract we investigate a curtis tits style group presentation based on the dynkin diagram c g in which the short root generators have order two while the long root generators have order four we prove that this describes a finite group with an almost extraspecial normal subgroup of order 2 2 g 2 and quotient isomorphic to the symplectic group sp 2 g 2 such an extension has to be non split if g 3 and was proved to exist in papers of bolt room and wall from 1961 2 and griess from 1973 our presentation proves that it is a double cover of a finite quotient of sp 2 g z we investigate a 2 g dimensional complex representation on a suitable space of theta functions and produce some consequences for the signatures of 4 manifolds described as surface bundles over surfaces in particular we prove that if the monodromy is contained in the theta subgroup sp q 2 g z sp 2 g z then the signature of the 4 manifold is divisible by eight

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

a conservative spectral collocation method for the nonlinear schrdinger equation in two dimensions in this study we present a conservative fourier spectral collocation fsc method to solve the two dimensional nonlinear schrdinger nls equation we prove that the proposed method preserves the mass and energy conservation laws in semi discrete formulations using the spectral differentiation matrices the nls equation is reduced to a system of nonlinear ordinary differential equations odes the compact implicit integration factor ciif method is later developed for the nonlinear odes in this approach the storage and cpu cost are significantly reduced such that the use of ciif method becomes attractive for two dimensional nls equation numerical results are presented to demonstrate the conservation accuracy and efficiency of the method

krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes application to discontinuous galerkin methods abstract integration factor methods are a class of exactly linear part time discretization methods in q nie y t zhang r zhao efficient semi implicit schemes for stiff systems journal of computational physics 214 2006 521 537 a class of efficient implicit integration factor iif methods were developed for solving systems with both stiff linear and nonlinear terms arising from spatial discretization of time dependent partial differential equations pdes with linear high order terms and stiff lower order nonlinear terms the tremendous challenge in applying iif temporal discretization for pdes on high spatial dimensions is how to evaluate the matrix exponential operator efficiently for spatial discretization on unstructured meshes to solve pdes on complex geometrical domains how to efficiently apply the iif temporal discretization was open in this paper we solve this problem by applying the krylov subspace approximations to the matrix exponential operator then we apply this novel time discretization technique to discontinuous galerkin dg methods on unstructured meshes for solving reaction diffusion equations numerical examples are shown to demonstrate the accuracy efficiency and robustness of the method in resolving the stiffness of the dg spatial operator for reaction diffusion pdes application of the method to a mathematical model in pattern formation during zebrafish embryo development shall be shown

the kodaira dimension of the moduli of k3 surfaces the global torelli theorem for projective k3 surfaces was first proved by piatetskii shapiro and shafarevich 35 years ago opening the way to treating moduli problems for k3 surfaces the moduli space of polarised k3 surfaces of degree 2d is a quasi projective variety of dimension 19 for general d very little has been known hitherto about the kodaira dimension of these varieties in this paper we present an almost complete solution to this problem our main result says that this moduli space is of general type for d 61 and for d 46 50 54 57 58 60

additive semi implicit runge kutta methods for computing high speed nonequilibrium reactive flows this paper is concerned with time stepping numerical methods for computing stiff semi discrete systems of ordinary differential equations for transient hypersonic flows with thermo chemical nonequilibrium the stiffness of the equations is mainly caused by the viscous flux terms across the boundary layers and by the source terms modeling finite rate thermo chemical processes implicit methods are needed to treat the stiff terms while more efficient explicit methods can still be used for the nonstiff terms in the equations this paper studies three different semi implicit runge kutta methods for additively split differential equations in the form ofu f u g u wherefis treated by explicit runge kutta methods andgis simultaneously treated by three implicit runge kutta methods a diagonally implicit runge kutta method and two linearized implicit runge kutta methods the coefficients of up to third order accurate additive semi implicit runge kutta methods have been derived such that the methods are both high order accurate and strongly a stable for the implicit terms the results of two numerical tests on the stability and accuracy properties of these methods are also presented in the paper

homological stability for moduli spaces of high dimensional manifolds i s galatius was partially supported by nsf grants dms 1105058 and dms 1405001 the european research council erc under the european union s horizon 2020 research and innovation pro gramme grant agreement no 682922 as well as the danish national research foundation through the centre for symmetry and deformation dnrf92 and erc 682992 o randal williams was partially supported by the herchel smith fund and epsrc grant ep m027783 1

a new nonlinear galerkin finite element method for the computation of reaction diffusion equations abstract to get the stationary patterns in the approximation of reaction diffusion systems the computation of evolution equations on large intervals of time is required in this paper a new nonlinear galerkin method based on finite element discretization is presented for the approximation of reaction diffusion equations on large intervals the new scheme is based on two different finite element spaces defined respectively on one subspace of lower degree shape function and one of higher degree shape function nonlinearity and time dependence are both treated on the lower degree space and only a fixed stationary equation needs to be solved on the higher degree space at each time level the proof of the stability and convergence of the method is presented in one dimension numerical solutions of some well studied reaction diffusion systems are presented to demonstrate the effectiveness of the nonlinear galerkin method

a compact fourth order l stable scheme for reaction diffusion systems with nonsmooth data exponential time differencing runge kutta etdrk schemes based on diagonal pade approximations for the numerical solution of reaction diffusion systems containing nonsmooth data have the disadvantage of producing poor numerical results when the time steps are too large relative to the spatial steps on the other hand methods based on sub diagonal pade approximations do not suffer from this bottleneck in this paper we present a novel modified version of cox and matthews fourth order etdrk scheme based on sub diagonal pade approximation to matrix exponential functions in combination with compact fourth order finite difference scheme in space for direct integration of reaction diffusion problems with nonsmooth data for an efficient implementation of the scheme a partial fraction splitting technique is utilized in which it is required to solve several backward euler type linear systems at each time step moreover the design of the algorithm offers parallel implementation so we implement the proposed algorithm in parallel on two processors utilizing matlabmpi a parallel message passing version of matlab and obtain that the parallel version is computationally more efficient than the existing schemes considered in this paper we investigate the amplification factor of the scheme and plot its boundaries of stability regions which give an indication of the stability of the scheme calculation of the local truncation error and an empirical convergence analysis demonstrates the fourth order accuracy of the proposed scheme accuracy computational efficiency and reliability of the new scheme are demonstrated with numerical examples and comparing it with existing schemes

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

a note on black scholes implied volatility abstract an approximate formula for the black scholes implied volatility is given by means of an asymptotic representation of the black scholes formula this representation is based on a variable change that reduces the number of meaningful variables from five to three it is stated clearly which is the family of functions we are going to work specially the inverse of the normal accumulative function estimates for the error in the resulting approximate formulas for both the option value and the volatility are obtained as well

an integration factor method for stochastic and stiff reaction diffusion systems stochastic effects are often present in the biochemical systems involving reactions and diffusions when the reactions are stiff existing numerical methods for stochastic reaction diffusion equations require either very small time steps for any explicit schemes or solving large nonlinear systems at each time step for the implicit schemes here we present a class of semi implicit integration factor methods that treat the diffusion term exactly and reaction implicitly for a system of stochastic reaction diffusion equations our linear stability analysis shows the advantage of such methods for both small and large amplitudes of noise direct use of the method to solving several linear and nonlinear stochastic reaction diffusion equations demonstrates good accuracy efficiency and stability properties this new class of methods which are easy to implement will have broader applications in solving stochastic reaction diffusion equations arising from models in biology and physical sciences

on the jacobi group and the mapping class group of the paper contains a proof that the mapping class group of the manifold s 3 x s 3 is isomorphic to a central extension of the full jacobi group j by the group of 7 dimensional homotopy spheres using a presentation of the group j and the p invariant of the homotopy spheres we give a presentation of this mapping class group with generators and defining relations we also compute the cohomology of the group j and determine 2 cocycles that correspond to the mapping class group of s 3 s 3

efficient semi implicit schemes for stiff systems when explicit time discretization schemes are applied to stiff reaction diffusion equations the stability constraint on the time step depends on two terms the diffusion and the reaction the part of the stability constraint due to diffusion can be totally removed if the linear diffusions are treated exactly using integration factor if or exponential time differencing etd methods for systems with severely stiff reactions those methods are not efficient because the reaction terms in if or etd are still approximated with explicit schemes in this paper we introduce a new class of semi implicit schemes which treats the linear diffusions exactly and explicitly and the nonlinear reactions implicitly a distinctive feature of the scheme is the decoupling between the exact evaluation of the diffusion terms and implicit treatment of the nonlinear reaction terms as a result the size of the nonlinear system arising from the implicit treatment of the reactions is independent of the number of spatial grid points it only depends on the number of original equations unlike the case in which standard implicit temporal schemes are directly applied to the reaction diffusion system the stability region for this class of methods is much larger than existing methods using an explicit treatment of reaction terms in particular the one with second order accuracy is unconditionally linearly stable with respect to both diffusion and reaction direct numerical simulations on test equations as well as morphogen systems from developmental biology show the new semi implicit schemes are efficient robust and accurate

homological stability and stable moduli of flat manifold bundles abstract we prove that the group homology of the diffeomorphism group of g s n s n int d 2 n as a discrete group is independent of g in a range provided that n 2 this answers the high dimensional version of a question posed by morita about surface diffeomorphism groups made discrete the stable homology is isomorphic to the homology of a certain infinite loop space related to the haefliger s classifying space of foliations one geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized mumford morita miller classes lift to a secondary r z invariants for flat g s n s n bundles provided g 0

stable moduli spaces of high dimensional manifolds we prove an analogue of the madsen weiss theorem for high dimensional manifolds in particular we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of g copies of sn sn in the limit g to infty rationally it is a polynomial ring in certain explicit generators giving a high dimensional analogue of mumford s conjecture

periods of double epw sextics we study the period map for double epw sextics which are varieties making up a locally versal family of polarized hyperkahler fourfolds double epw sextics are parametrized by lagrangian subspaces of the third wedge product of a 6 dimensional complex vector space we prove that lagrangians in the indeterminacy locus of the period map contain a decomposable 3 vector enjoying very special properties in addition we prove a result about periods of double epw sextics which are either smooth or have isolated singular points with projectivized tangent cone isomorphic to the incidence variety in mathbb p 2 times mathbb p 2 vee

modular lagrangians and the theta multiplier let v be a free module over z 4 of rank 2 n and let be a non singular skew form on v let f denote the reduction of v modulo 2 and let q be a hyperbolic i e arf invariant zero quadratic form on p with as associated bilinear form we let a v denote the set of oriented lagrangians i e free totally isotropic submodules of rank n in v and ao v ca v the subset of those l in a v whose reductions modulo 2 are totally isotropic for q we call such an l an oriented isotropic lagrangian we will study a certain function m on ao v x ao v with values in the group of fourth roots of unity the function m is skew symmetric m m l re l m t and is a 1 cocycle by this

infinite loop spaces from operads with homological stability abstract motivated by the operad built from moduli spaces of riemann surfaces we consider a general class of operads in the category of spaces that satisfy certain homological stability conditions we prove that such operads are infinite loop space operads in the sense that the group completions of their algebras are infinite loop spaces the recent strong homological stability results of galatius and randal williams for moduli spaces of even dimensional manifolds can be used to construct examples of operads with homological stability as a consequence diffeomorphism groups and mapping class groups are shown to give rise to infinite loop spaces furthermore the map to k theory defined by the action of the diffeomorphisms on the middle dimensional homology is shown to be a map of infinite loop spaces

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

abelian quotients of mapping class groups of highly connected manifolds we compute the abelianisations of the mapping class groups of the manifolds w g 2n g s n times s n for n ge 3 and g ge 5 the answer is a direct sum of two parts the first part arises from the action of the mapping class group on the middle homology and takes values in the abelianisation of the automorphism group of the middle homology the second part arises from bordism classes of mapping tori and takes values in the quotient of the stable homotopy groups of spheres by a certain subgroup which in many cases agrees with the image of the stable j homomorphism we relate its calculation to a purely homotopy theoretic problem

black scholes model under subordination abstract in this paper we consider a new mathematical extension of the black scholes bs model in which the stochastic time and stock share price evolution is described by two independent random processes the parent process is brownian and the directing process is inverse to the totally skewed strictly stable process the subordinated process represents the brownian motion indexed by an independent continuous and increasing process this allows us to introduce the long term memory effects in the classical bs model

a new class of time discretization schemes for the solution of nonlinear pdes we consider issues of stability of time discretization schemes withexacttreatment of thelinear part elp schemes for solving nonlinear pdes a distinctive feature of elp schemes is the exact evaluation of the contribution of the linear term that is if the nonlinear term of the equation is zero then the scheme reduces to the evaluation of the exponential function of the operator representing the linear term computing and applying the exponential or other functions of operators with variable coefficients in the usual manner requires evaluating dense matrices and is highly inefficient it turns out that computing the exponential of strictly elliptic operators in the wavelet system of coordinates yields sparse matrices for a finite but arbitrary accuracy this observation makes our approach practical in a number of applications in particular we consider applications of elp schemes to advection diffusion equations we study the stability of these schemes and show that both explicit and implicit elp schemes have distinctly different stability properties if compared with known implicit explicit schemes for example we describe explicit schemes with stability regions similar to those of typical implicit schemes used for solving advection diffusion equations