constructing robust good lattice rules for computational finance the valuation of many financial derivatives leads to high dimensional integrals the constructions of robust or universal good lattice rules for financial applications are both important and challenging an important common feature of the integrands in computational finance is that they can often be well approximated by a sum of low dimensional functions i e functions that depend on only a small number of variables usually just two variables for numerical integration of such functions the quality of the low order i e low dimensional projections of the node set is crucial in this paper we propose methods to construct good lattice points with optimal low order projections the quality of a point set is measured by a new measure called elementary order ell discrepancy which measures the quality of all order ell projections and is more informative than usual measures two constructions namely the korobov and the component by component constructions are studied such that the low order projections are optimized numerical experiments demonstrate that even in high dimensions it is possible to construct new good lattice points with order 2 projections that are better than those of the sobol points and random points and with higher order projections that are no worse while the sobol points lost the advantage over random points in order 2 projections on the average the new lattice rules have the potential to improve upon the accuracy for favorable functions while doing no harm for unfavorable ones their applications for pricing path dependent options and american options based on the least square monte carlo method are studied and their high efficiency is demonstrated a nice surprise revealed is the robustness property of such lattice rules the good projection property and the suitability for a large range of problems the potential possibility and limitations of good lattice points in achieving good quality of moderate and high order projections is investigated the reason why classical lattice rules may not be efficient for high dimensional finance problems is also discussed

pair correlation densities of inhomogeneous quadratic forms under explicit diophantine conditions on 2 w eprove that the local two point correlations of the sequence given by the values m 2 n 2 with m n 2 are those of a poisson process this partly confirms a conjecture of berry and tabor 2 on spectral statistics of quantized integrable systems and also establishes a particular case of the quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms of signature 2 2 the proof uses theta sums and ratner s classification of measures invariant under unipotent flows

analytical solution of the linear fractional system of commensurate order a useful representation of fractional order systems is the state space representation for the linear fractional systems of commensurate order the state space representation is defined as for regular integer state space representation with the state vector differentiated to a real order this paper presents a solution of the linear fractional order systems of commensurate order in the state space the solution is obtained using a technique based on functions of square matrices and the cayley hamilton theorem the technique developed for linear systems of integer order is extended to derive analytical solutions of linear fractional systems of commensurate order the basic ideas and the derived formulations of the technique are presented both homogeneous and inhomogeneous cases with usual input functions are solved the solution is calculated in the form of a linear combination of suitable fundamental functions the presented results are illustrated by analyzing some examples to demonstrate the effectiveness of the presented analytical approach

synchronization in fractional order differential systems an symmetrically coupled system consisting of identical fractional order differential systems including chaotic and non chaotic systems is investigated in this paper such a coupled system has in its synchronous state a mode decomposition by which the linearized equation can be decomposed into motions transverse to and parallel to the synchronous manifold furthermore the decomposition can induce a sufficient condition on synchronization of the overall system which guarantees if satisfied that a group synchronization is achieved two typical numerical examples fractional brusselators and the fractional rossler system are used to verify the theoretical prediction the theoretical analysis and numerical results show that the lower the order of the fractional system the longer the time for achieving synchronization at a fixed coupling strength

the mobius function and distal flows we prove that the mobius function is linearly disjoint from an analytic skew product on the 2 torus these flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points the previous cases for which such disjointness has been proved are all regular we also establish the linear disjointness of the mobius function from various distal homogeneous flows

equidistribution of expanding translates of curves and dirichlet s theorem on diophantine approximation we show that for almost all points on any analytic curve on k which is not contained in a proper affine subspace the dirichlet s theorem on simultaneous approximation as well as its dual result for simultaneous approximation of linear forms cannot be improved the result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in k 1 the proof involves ratner s theorem on ergodic properties of unipotent flows on homogeneous spaces

on korobov lattice rules in weighted spaces this paper studies the error bounds of multivariate integration in weighted function spaces using lattice rules of the korobov form in which the generating vector for an n point rule with n prime has the form 1 a ad 1 mod n with the parameter a chosen optimally we establish new error bounds for korobov lattice rules in weighted korobov spaces in particular we prove that if the weights decay sufficiently fast the optimal korobov lattice rule has an error bound of order o n alpha 2 delta for arbitrary delta 0 with the implied constant depending at worst polynomially on the dimension here alpha 1 is the smoothness parameter of the weighted korobov spaces we generalize the construction to the case where n is a product of arbitrary distinct prime numbers with the purpose of reducing the construction cost without sacrificing much of the quality of the lattice rules a corresponding result is deduced for weighted sobolev spaces of nonperiodic functions using randomly shifted optimal korobov lattice rules a comparison of the worst case errors for korobov lattice rules and the recent component by component constructions is presented the investigation establishes the usefulness of shifted optimal korobov lattice rules for integration even in high dimensions if the weights which characterize the weighted spaces are suitably chosen

pricing and hedging with discontinuous functions quasi monte carlo methods and dimension reduction quasi monte carlo qmc methods are important numerical tools in the pricing and hedging of complex financial instruments the effectiveness of qmc methods crucially depends on the discontinuity and the dimension of the problem this paper shows how the two fundamental limitations can be overcome in some cases we first study how path generation methods pgms affect the structure of the discontinuities and what the effect of discontinuities is on the accuracy of qmc methods the insight is that the discontinuities can be qmc friendly i e aligned with the coordinate axes or not depending on the pgm the pgms that offer the best performance in qmc methods are those that make the discontinuities qmc friendly the structure of discontinuities can affect the accuracy of qmc methods more significantly than the effective dimension this insight motivates us to propose a novel way of handling the discontinuities the basic idea is to align the discontinuities with the coordinate axes by a judicious design of a method for simulating the underlying processes numerical experiments demonstrate that the proposed method leads to dramatic variance reduction in qmc methods for pricing options and for estimating greeks it also reduces the effective dimension of the problem this paper was accepted by assaf zeevi stochastic models and simulation

representations of integers by an invariant polynomial and unipotent flows we study a refined version of linnik s problem on the asymptotic behavior of the number of representations of integers m by an integral polynomial as m tends to infinity assuming that the polynomials arise from invariant theory we reduce the question to the study of limiting behavior of measures invariant under unipotent flows our main tool is then ratner s theorem on the uniform distribution of unipotent flows in a form refined by dani and margulis dm2

analytical and numerical solutions of a nonlinear alcoholism model via variable order fractional differential equations abstract in this paper we analyze an alcoholism model which involves the impact of twitter via liouville caputo and atangana baleanu caputo fractional derivatives with constant and variable order two fractional mathematical models are considered with and without delay special solutions using an iterative scheme via laplace and sumudu transform were obtained we studied the uniqueness and existence of the solutions employing the fixed point postulate the generalized model with variable order was solved numerically via the adams method and the adams bashforth moulton scheme stability and convergence of the numerical solutions were presented in details numerical examples of the approximate solutions are provided to show that the numerical methods are computationally efficient therefore by including both the fractional derivatives and finite time delays in the alcoholism model studied we believe that we have established a more complete and more realistic indicator of alcoholism model and affect the spread of the drinking

free path lengths in quasicrystals previous studies of kinetic transport in the lorentz gas have been limited to cases where the scatterers are distributed at random e g at the points of a spatial poisson process or at the vertices of a euclidean lattice in the present paper we investigate quasicrystalline scatterer configurations which are non periodic yet strongly correlated a famous example is the vertex set of a penrose tiling our main result proves the existence of a limit distribution for the free path length which answers a question of wennberg the limit distribution is characterised by a certain random variable on the space of higher dimensional lattices and is distinctly different from the exponential distribution observed for random scatterer configurations the key ingredients in the proofs are equidistribution theorems on homogeneous spaces which follow from ratner s measure classification

limiting distributions of curves under geodesic flow on hyperbolic manifolds we consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic n manifold under the geodesic flow suppose that the curve is not contained in a stable leaf of the flow it is shown that under the geodesic flow the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural riemannian measure on the unit tangent bundle of a closed totally geodesically immersed submanifold moreover if this immersed submanifold is a proper subset then a lift of the curve to the universal covering space t 1 h n is mapped into a proper subsphere of the ideal boundary sphere h n under the visual map this proper subsphere can be realized as the ideal boundary of an isometrically embedded hyperbolic subspace in h n covering the closed immersed submanifold in particular if the visual map does not send a lift of the curve into a proper subsphere of h n then under the geodesic flow the curve gets asymptotically equidistributed on the unit tangent bun dle of the manifold with respect to the normalized natural rie mannian measure the proof uses dynamical properties of unipotent flows on finite volume homogeneous spaces of so n 1

quasi monte carlo methods in financial engineering an equivalence principle and dimension reduction quasi monte carlo qmc methods are playing an increasingly important role in the pricing of complex financial derivatives for models in which the prices of the underlying assets are driven by brownian motions the performance of qmc methods is known to depend crucially on the construction of brownian motions this paper focuses on the impact of various constructions although the brownian bridge bb construction often yields very good results as papageorgiou pointed out there are financial derivatives for which the bb construction performs badly papageorgiou a 2002 the brownian bridge does not offer a consistent advantage in quasi monte carlo integration j complexity18 1 171 186 in this paper we first extend papageorgiou s analysis to establish an equivalence principle if the bb construction or any other construction is the preferred construction for a particular financial derivative then for any other construction there is another financial derivative for which the latter construction is the preferred one in this sense all methods of construction are equivalent and no method is consistently superior to others it all depends on the particular financial derivative we then show how to find a good construction for a particular class of financial derivatives in practice our strategy is to find a good construction for an easy problem and then apply it to more complicated problems related to the easy one this strategy is applied to the arithmetic asian options including bermudan asian options based on the weighted average of the stock prices we do this by studying a simpler problem namely the geometric asian option for which the best construction is easily available and applying it to the arithmetic asian option numerical experiments confirm the success of this strategy whereas in qmc all the common methods the standard method bb and principal component analysis may lose their power in some situations the new method behaves very well in all cases further large variance reduction can be achieved in combination with a control variate the new method can be interpreted as a practical way of reducing the effective dimension for some class of functions

rigidity of measures invariant under semisimple groups in positive characteristic m ratner has conjectured a positive characteristic version of her seminal results classifying orbit closures and invariant measures of unipotent flows on homogeneous spaces in this paper we provide a partial answer by establishing a positive characteristic version of her classification result for measures invariant under semisimple groups

quadrature error bounds with applications to lattice rules reproducing kernel hilbert spaces are used to derive error bounds and worst case integrands for a large family of quadrature rules in the case of lattice rules applied to periodic integrands these error bounds resemble those previously derived in the literature however the theory developed here does not require periodicity and is not restricted to lattice rules an analysis of variance anova decomposition is employed in defining the inner product it is shown that imbedded rules are superior when integrating functions with large high order anova effects

on the stability and convergence of the time fractional variable order telegraph equation in this work we have generalized the time fractional telegraph equation using the concept of derivative of fractional variable order the generalized equation is called time fractional variable order telegraph equation this new equation was solved numerically via the crank nicholson scheme stability and convergence of the numerical solution were presented in details numerical simulations of the approximate solution of the time fractional variable order telegraph equation were presented for different values of the grid point

completely periodic directions and orbit closures of many pseudo anosov teichmueller discs in q 1 1 1 1 in this paper we investigate the closure of a large class of teichmuller discs in the stratum mathcal q 1 1 1 1 or equivalently in a rm gl 2 mathbb r invariant locus mathcal l of translation surfaces of genus three we describe a systematic way to prove that the rm gl 2 mathbb r orbit closure of a translation surface in mathcal l is the whole locus mathcal l the strategy of the proof is an analysis of completely periodic directions on such a surface and an iterated application of ratner s theorem to unipotent subgroups acting on an adequate splitting this analysis applies for example to all teichmuller discs obtained by the thurston veech s construction with a trace field of degree three which are moreover obviously not veech we produce an infinite series of such examples and show moreover that the favourable splitting situation does not arise everywhere on mathcal l contrary to the situation in genus two we also study completely periodic directions on translation surfaces in mathcal l for instance we prove that completely periodic directions are dense on surfaces obtained by the thurston veech s construction

chaos and multiple attractors in a fractal fractional shinriki s oscillator model abstract in this paper we obtain novel chaotic behaviors using differential and integral operators with power law exponential decay and mittag leffler law for the shinriki s oscillator model we studied the uniqueness and existence of the solutions employing the fixed point postulate also we consider fractal fractional operators to capture self similarities for this chaotic attractor these novel operators predict chaotic behaviors involving the fractal derivative in convolution with power law exponential decay law and the mittag leffler function the generalized model with power law and mittag leffler kernel was solved numerically via the adams bashforth moulton and adams moulton scheme respectively for another cases the numerical schemes are based on the fundamental theorem of fractional calculus and the lagrange polynomial interpolation numerical simulations for the symmetric and asymmetric cases are obtained to show the applicability and computational efficiency of these methods

stationary measures and invariant subsets of homogeneous spaces ii let g be a real lie group be a lattice in g and be a compactly generated closed subgroup of g if the zariski closure of the group ad is semisimple with no compact factor we prove that every orbit closure in g is a nite volume homogeneous space we also establish related equidistribution properties

low discrepancy sequences in high dimensions how well are their projections distributed quasi monte carlo qmc methods have been successfully used to compute high dimensional integrals arising in many applications especially in finance to understand the success and the potential limitation of qmc this paper focuses on quality measures of point sets in high dimensions we introduce the email protected superposition and truncation discrepancies which measure the quality of selected projections of a point set on lower dimensional spaces these measures are more informative than the classical ones we study their relationships with the integration errors and study the tractability issues we present efficient algorithms to compute these discrepancies and perform computational investigations to compare the performance of the sobol nets with that of the sets of latin hypercube sampling and random points numerical results show that in high dimensions the superiority of the sobol nets mainly derives from the one dimensional projections and the projections associated with the earlier dimensions for order 2 and higher order projections all these point sets have similar behavior on the average in weighted cases with fast decaying weights the sobol nets have a better performance than the other two point sets the investigation enables us to better understand the properties of qmc and throws new light on when and why qmc can have a better or no better performance than monte carlo for multivariate integration in high dimensions

fractional adams bashforth moulton methods we first formulate a fractional class of explicit adams bashforth a b and implicit adams moulton a m methods of first and second order accuracy for the time integration of d t 0 c u x t g t u 0 1 where d t 0 c denotes the fractional derivative in the caputo sense in this fractional setting and in contrast to the standard adams methods an extra history load term emerges and the associated weight coefficients are dependent however when 1 the developed schemes reduce to the well known a b and a m methods with standard coefficients hence in terms of scientific computing our approach constitutes a minimal modification of the existing adams libraries next we develop an implicit explicit imex splitting scheme for linear and nonlinear fractional pdes of a general advection reaction diffusion type and we apply our scheme to the time space fractional keller segel chemotaxis system in this context we evaluate the nonlinear advection term explicitly employing the fractional a b method in the prediction step and we treat the corresponding diffusion term implicitly in the correction step using the fractional a m scheme moreover we perform the corresponding spatial discretization by employing an efficient and spectrally accurate fractional spectral collocation method our numerical experiments exhibit the efficiency of the proposed imex scheme in solving nonlinear fractional pdes

fractional bloch equation with delay in this paper we investigate a fractional generalization of the bloch equation that includes both fractional derivatives and time delays the appearance of the fractional derivative on the left side of the bloch equation encodes a degree of system memory in the dynamic model for magnetization the introduction of a time delay on the right side of the equation balances the equation by also adding a degree of system memory on the right side of the equation the analysis of this system shows different stability behavior for the t 1 and the t 2 relaxation processes the t 1 decay is stable for the range of delays tested 1 100 ms while the t 2 relaxation in this model exhibited a critical delay typically 6 ms above which the system was unstable delays are expected to appear in nmr systems in both the system model and in the signal excitation and detection processes therefore by including both the fractional derivative and finite time delays in the bloch equation we believe that we have established a more complete and more realistic model for nmr resonance and relaxation

uniformly distributed orbits of certain flows on homogeneous spaces is non compact for a co compact lattice this result was strengthened by f irstenberg 1 lj proving that every orbit is uniformly distributed with respect to a g invariant measure for non uniform lattices in sl 2 r using a classification of invariant measures obtained by dani in 21 dani and smillie 31 proved that every non periodic orbit is uniformly distributed there are also various results obtained on orbit closures and invariant measures etc of larger subgroups consisting of unipotent elements especially the horospherical subgroups recently there was a spurt in the area initiated by margulis proof cf 151 see also 7 of oppenheim conjecture on values of quadratic forms at integral points using the study of unipotent flows the reader is referred to the survey articles by dani 41 and margulis 14j for various related developments we now note some conjectures expected to hold for orbits of a unipotent flow namely the u action on

on measures invariant under tori on quotients of semisimple groups we classify invariant and ergodic probability measures on arithmetic homogeneous quotients of semisimple s algebraic groups invariant under a maximal split torus in at least one simple local factor and show that the algebraic support of such a measure splits into the product of four homogeneous spaces a torus a homogeneous space on which the measure is up to nite index the haar measure a product of homogeneous spaces on each of which the action degenerates to a rank one action and a homogeneous space in which every element of the action acts with zero entropy

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

invariant measures on g γ for split simple lie groups g we study the left action of a cartan subgroup on the space x g where is a lattice in a simple split connected lie group g of rank n 1 let be an invariant measure on x we give several conditions using entropy and conditional measures each of which characterizes the haar measure on x furthermore we show that the conditional measure on the foliation of unstable manifolds has the structure of a product measure the main new element compared to the previous work on this subject is the use of noncommutativity of root foliations to establish rigidity of invariant measures 2003 wiley periodicals inc

a solution to a problem of cassels and diophantine properties of cubic numbers we prove that almost any pair of real numbers satises the following inhomogeneous uniform version of littlewood s conjecture 8 2 r lim inf jnj 1 jnjhn ihn i 0 c1 wherehi denotes the distance from the nearest integer the existence of even a single pair that satises statement c1 solves a problem of cassels from the 50 s we then prove that if 1 span a totally real cubic number eld then satisfy c1 this generalizes a result of cassels and swinnerton dyer which says that such pairs satisfy littlewood s conjecture it is further shown that if are any two real numbers such that 1 are linearly dependent over q they cannot satisfy c1 the results are then applied to give examples of irregular orbit closures of the diagonal group of a new type the results are derived from rigidity results concerning hyperbolic actions of higher rank commutative groups on homogeneous spaces

equidistribution of minimal hypersurfaces for generic metrics for almost all riemannian metrics in the c infty baire sense on a closed manifold m n 1 3 le n 1 le 7 we prove that there is a sequence of closed smooth embedded connected minimal hypersurfaces that is equidistributed in m this gives a quantitative version of the main result of irie et al ann math 187 3 963 972 2018 that established density of minimal hypersurfaces for generic metrics as in irie et al 2018 the main tool is the weyl law for the volume spectrum proven by liokumovich et al ann math 187 3 933 961 2018

fast algorithms for component by component construction of rank 1 lattice rules in shift invariant reproducing kernel hilbert spaces we reformulate the original component by component algorithm for rank 1 lattices in a matrix vector notation so as to highlight its structural properties for function spaces similar to a weighted korobov space we derive a technique which has construction cost o snlog n in contrast with the original algorithm which has construction cost o sn 2 herein s is the number of dimensions and n the number of points taken prime in contrast to other approaches to speed up construction our fast algorithm computes exactly the same quantity as the original algorithm the presented algorithm can also be used to construct randomly shifted lattice rules in weighted sobolev spaces

efficient weighted lattice rules with applications to finance good lattice rules are an important type of quasi monte carlo algorithm they are known to have good theoretical properties in the sense that they can achieve an error bound or optimal error bound that is independent of the dimension for weighted function spaces with suitably decaying weights to use the theory of weighted function spaces for practical applications one has to determine what weights should be used after explaining that lattice rules based on figures of merit with classical weights classical lattice rules may not give good results for high dimensional problems we propose a matching strategy which chooses the weights in such a way that the typical functions of a weighted korobov space are similar in the sense of similar relative sensitivity indices to those of the given function the matching strategy relates the weights of the spaces to the sensitivity indices of the given function we apply the korobov construction and the component by component construction of lattice rules with the suitably chosen weights to the valuation of high dimensional financial derivative securities and find that such weighted lattice rules improve dramatically on the results for the classical lattice rules moreover they are competitive with other types of low discrepancy sequences we also demonstrate that lattice rules combined with variance reduction techniques and dimension reduction techniques increase the efficiency significantly we show that an acceptance rejection approach to the construction of lattice rules can reduce the construction cost with almost no loss of accuracy

isolation equidistribution and orbit closures for the sl 2 r action on moduli space we prove results about orbit closures and equidistribution for the sl 2 r action on the moduli space of compact riemann surfaces which are analogous to the theory of unipotent ows the proofs of the main theorems rely on the measure classication theorem of the rst two authors and a certain isolation property of closed sl 2 r invariant manifolds developed in this paper

spatiotemporal patterns in the belousov zhabotinskii reaction systems with atangana baleanu fractional order derivative abstract in this paper a robust numerical simulation technique based on the fractional adams bashforth and the fourier spectral methods are formulated to explore some spatiotemporal patterns in a range of belousov zhabotinskii reaction systems the standard integer order time derivative is replaced with the atangana baleanu fractional order derivative in the sense of caputo details of existence and stability of positive solution are given numerical experiments are carried out at some instances of fractional power to demonstrate the suitability of the methods and to explore the dynamic richness in some chemical species when modelled with non integer order derivatives

finite difference methods with non uniform meshes for nonlinear fractional differential equations in this article finite difference methods with non uniform meshes for solving nonlinear fractional differential equations are presented where the non equidistant stepsize is non decreasing the rectangle formula and trapezoid formula are proposed based on the non uniform meshes combining the above two methods we then establish the predictor corrector scheme the error and stability analysis are carefully investigated at last numerical examples are carried out to verify the theoretical analysis besides the comparisons between non uniform and uniform meshes are given where the non uniform meshes show the better performance when dealing with the less smooth problems

the error bounds and tractability of quasi monte carlo algorithms in infinite dimension dimensionally unbounded problems are frequently encountered in practice such as in simulations of stochastic processes in particle and light transport problems and in the problems of mathematical finance this paper considers quasi monte carlo integration algorithms for weighted classes of functions of infinitely many variables in which the dependence of functions on successive variables is increasingly limited the dependence is modeled by a sequence of weights the integrands belong to rather general reproducing kernel hilbert spaces that can be decomposed as the direct sum of a series of their subspaces each subspace containing functions of only a finite number of variables the theory of reproducing kernels is used to derive a quadrature error bound which is the product of two terms the generalized discrepancy and the generalized variation tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor s is bounded by ce p for some exponent p and some positive constant c the e exponent of tractability is defined as the smallest power of e 1 in these bounds it is shown by using monte carlo quadrature that the e exponent is no greater than 2 for these weighted classes of integrands under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the halton sequence that the e exponent of tractability is 1 which implies that infinite dimensional integration is no harder than one dimensional integration

the mobius function is strongly orthogonal to nilsequences we show that the mobius function n is strongly asymptotically or thogonal to any polynomial nilsequence f g n n2n here g is a sim ply connected nilpotent lie group with a discrete and cocompact subgroup so g is a nilmanifold g z g is a polynomial sequence and f g r is a lipschitz function more precisely we show that j 1 n p n n 1 n f g n j f g a log a n for all a 0 in particu lar this implies the mobius and nilsequence conjecture mn s from our earlier paper for every positive integer s this is one of two major in gredients in our programme to establish a large number of cases of the generalised hardy littlewood conjecture which predicts how often a collec tion 1 t z d z of linear forms all take prime values the proof is a relatively quick application of the results in our recent companion paper we give some applications of our main theorem we show for exam ple that the mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c we also obtain a result about the distribution of nilse quences a n x n2n as n ranges only over the primes

handling discontinuities in financial engineering good path simulation and smoothing discontinuities are common in the pricing and hedging of complex financial derivatives quasi monte carlo qmc methods for high dimensional finance problems with discontinuities can be inefficient because of the lack of good smoothness and high dimensionality interestingly path simulation method psm may affect both factors implying its significance in qmc methods what defines a good psm for problems with discontinuities the ability to align the discontinuities with the coordinate axes is a desirable property for a psm we show that for an arbitrary psm there exists a class of options with discontinuous payoff functions such that the transformed functions have only axis parallel discontinuities for which good qmc performance can be expected in this sense any psm can be good in qmc methods for a specific class of problems we analyze the structure of discontinuities for digital options using the new approach and show the superiority and the uniqueness up to a permutation of the standard co

measure complexity and mobius disjointness abstract in this paper the notion of measure complexity is introduced for a topological dynamical system and it is shown that sarnak s mobius disjointness conjecture holds for any system for which every invariant borel probability measure has sub polynomial measure complexity we then apply this result to a number of situations including certain systems whose invariant measure don t all have discrete spectrum

geodesic planes in hyperbolic 3 manifolds this paper initiates the study of rigidity for immersed totally geodesic planes in hyperbolic 3 manifolds m of infinite volume in the case of an acylindrical 3 manifold whose convex core has totally geodesic boundary we show that the closure of any immersed geodesic plane is a properly immersed submanifold of m on the other hand we show that rigidity fails for quasifuchsian manifolds

mobius disjointness for homogeneous dynamics we prove sarnak s mobius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected lie groups namely we show that if g is any such group g a lattice and u g an ad unipotent element then for every x g and every function f continuous on the 1 point compactification of g the sequence f xun cannot correlate with the mobius function on average

maximal representations of uniform complex hyperbolic lattices let rho be a maximal representation of a uniform lattice gamma subset rm su n 1 n geq 2 in a classical lie group of hermitian type h we prove that necessarily h rm su p q with p geq qn and there exists a holomorphic or antiholomorphic rho equivariant map from complex hyperbolic space to the symmetric space associated to rm su p q this map is moreover a totally geodesic homothetic embedding in particular up to a representation in a compact subgroup of rm su p q the representation rho extends to a representation of rm su n 1 in rm su p q

why are high dimensional finance problems often of low effective dimension many problems in mathematical finance can be formulated as high dimensional integrals where the large number of dimensions arises from small time steps in time discretization and or a large number of state variables quasi monte carlo qmc methods have been successfully used for approximating such integrals to understand this success this paper focuses on investigating the special features of some typical high dimensional finance problems namely option pricing and bond valuation we provide new insight into the connection between the effective dimension and the efficiency of qmc and we present methods to analyze the dimension structure of a function we confirm the observation of caflisch morokoff and owen that functions from finance are often of low effective dimension in the sense that they can be well approximated by their low order anova analysis of variance terms usually just the order 1 and order 2 terms we explore why the effective dimension is small for many integrals from finance by deriving explicit forms of the anova terms in simple cases we find that the importance of each dimension is naturally weighted by certain hidden weights these weights characterize the relative importance of different variables or groups of variables and limit the importance of the higher order anova terms we study the variance ratios captured by low order anova terms and their asymptotic properties as the dimension tends to infinity and we show that with the increase of dimension the lower order terms continue to play a significant role and the higher order terms tend to be negligible this provides some insight into high dimensional problems from finance and explains why qmc algorithms are efficient for problems of this kind

on accurate product integration rules for linear fractional differential equations this paper addresses the numerical solution of linear fractional differential equations with a forcing term competitive and highly accurate product integration rules are derived by starting from an equivalent formulation in terms of a volterra integral equation with a generalized mittag leffler function in the kernel the error analysis is reported and aspects related to the computational complexity are treated numerical tests confirming the theoretical findings are presented

random weights robust lattice rules and the geometry of the cbcrc algorithm in this paper we study lattice rules which are cubature formulae to approximate integrands over the unit cube 0 1 s from a weighted reproducing kernel hilbert space we assume that the weights are independent random variables with a given mean and variance for two reasons stemming from practical applications i it is usually not known in practice how to choose the weights thus by assuming that the weights are random variables we obtain robust constructions with respect to the weights of lattice rules this to some extend removes the necessity to carefully choose the weights ii in practice it is convenient to use the same lattice rule for many different integrands the best choice of weights for each integrand may vary to some degree hence considering the weights random variables does justice to how lattice rules are used in applications in this paper the worst case error is therefore a random variable depending on random weights we show how one can construct lattice rules which perform well for weights taken from a set with large measure such lattice rules are therefore robust with respect to certain changes in the weights the construction algorithm uses the component by component cbc idea based on two criteria one using the mean of the worst case error and the second criterion using a bound on the variance of the worst case error we call the new algorithm the cbc2c component by component with 2 constraints algorithm we also study a generalized version which uses r constraints which we call the cbcrc component by component with r constraints algorithm we show that lattice rules generated by the cbcrc algorithm simultaneously work well for all weights in a subspace spanned by the chosen weights 1 r thus in applications instead of finding one set of weights it is enough to find a convex polytope in which the optimal weights lie the price for this method is a factor r in the upper bound on the error and in the construction cost of the lattice rule thus the burden of determining one set of weights very precisely can be shifted to the construction of good lattice rules numerical results indicate the benefit of using the cbc2c algorithm for certain choices of weights

orbits of discrete subgroups on a symmetric space and the furstenberg boundary let x be a symmetric space of noncompact type and let be a lattice in the isometry group of x we study the distribution of orbits of acting on the symmetric space x and its geometric boundary x generalizing the main equidistribution result of margulis s thesis m theorem 6 to higher rank symmetric spaces more precisely for any y x and b x we investigate the distribution of the set y b 1 in x x it is proved in particular that the orbits of in the furstenberg boundary are equidistributed and that the orbits of in x are equidistributed in sectors defined with respect to a cartan decomposition our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces which we obtain using shah s result s corollary 1 2 based on ratner s measure classification theorem r1 theorem 1

constructing randomly shifted lattice rules in weighted sobolev spaces shifted rank 1 lattice rules a special class of quasi monte carlo methods have recently been proposed by the present authors for the integration of functions belonging to certain weighted sobolev spaces the shifts in these rules were generated in a deterministic manner in contrast in this paper we generate these shifts randomly this allows probabilistic estimates for the error in a given integral it also reduces the number of operations required to find the generating vectors for the underlying lattice rules component by component the rules thus constructed achieve a worst case strong tractability error bound in an average or probabilistic sense

variance reduction via lattice rules this is a review article on lattice methods for multiple integration over the unit hypercube with a variance reduction viewpoint it also contains some new results and ideas the aim is to examine the basic principles supporting these methods and how they can be used effectively for the simulation models that are typically encountered in the area of management science these models can usually be reformulated as integration problems over the unit hypercube with a large sometimes infinite number of dimensions we examine selection criteria for the lattice rules and suggest criteria which take into account the quality of the projections of the lattices over selected low dimensional subspaces the criteria are strongly related to those used for selecting linear congruential and multiple recursive random number generators numerical examples illustrate the effectiveness of the approach

numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion in this paper we study the time space fractional order fractional for simplicity nonlinear subdiffusion and superdiffusion equations which can relate the matter flux vector to concentration gradient in the general sense describing for example the phenomena of anomalous diffusion fractional brownian motion and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the space riemann liouville fractional derivative with order 1 b 1 2 and the finite difference scheme for the time caputo derivative with order a 0 1 for subdiffusion and 1 2 for superdiffusion are analyzed respectively results on the existence and uniqueness of the weak solutions the numerical stability and the error estimates are presented numerical examples are included to confirm the theoretical analysis during our simulations an interesting diffusion phenomenon of particles is observed that is on average the diffusion velocity for 0 a 1 is slower than that for a 1 but the diffusion velocity for 1 a 2 is faster than that for a 1 for the spatial diffusion we have a similar observation

an extension of quantitative nondivergence and applications to diophantine exponents we present a sharpening of nondivergence estimates for unipotent or more gener ally polynomial like flows on homogeneous spaces applied to metric diophantine approxima tion it yields precise formulas for diophantine exponents of affine subspaces ofr n and their nondegenerate submanifolds

quasi random sequences and their discrepancies quasi random also called low discrepancy sequences are a deterministic alternative to random sequences for use in monte carlo methods such as integration and particle simulations of transport processes the error in uniformity for such a sequence of n points in the s dimensional unit cube is measured by its discrepancy which is of size log n s n 1 for large n as opposed to discrepancy of size log log n 1 2 n 1 2 for a random sequence i e for almost any randomly chosen sequence several types of discrepancies one of which is new are defined and analyzed a critical discussion of the theoretical bounds on these discrepancies is presented computations of discrepancies are presented for a wide choice of dimension s number of points n and different quasi random sequences in particular for moderate or large s there is an intermediate regime in which the discrepancy of a quasi random sequence is almost exactly the same as that of a randomly chosen sequence a simplified p

on spectral disjointness of powers for rank one transformations and mobius orthogonality abstract we study the spectral disjointness of the powers of a rank one transformation for a large class of rank one constructions including those for which the cutting and stacking parameters are bounded and other examples such as rigid generalized chacon s maps and katok s map we prove that different positive powers of the transformation are pairwise spectrally disjoint on the continuous part of the spectrum our proof involves the existence in the weak closure of u t k k z of sufficiently many analytic functions of the operator u t then we apply these disjointness results to prove sarnak s conjecture for the possibly non uniquely ergodic symbolic models associated to these rank one constructions all sequences realized in these models are orthogonal to the mobius function

the distribution of periodic torus orbits on homogeneous spaces we prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces with particular focus on the case of the diagonal torus acting on quotients of pgln r after attaching to each periodic orbit an integral invariant the discriminant our results have the following flavour certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most o orbits of discriminant the proof relies on the well separatedness of periodic orbits together with measure rigidity for torus actions we also give examples of sequences of periodic orbits of this action that fail to become equidistributed even in higher rank we give an application of our results to sharpen a theorem of minkowski on ideal classes in totally real number fields of cubic and higher degrees

on the fractional adams method the generalized adams bashforth moulton method often simply called the fractional adams method is a useful numerical algorithm for solving a fractional ordinary differential equation d ay t f t y t y k 0 y 0 k k 0 1 n 1 where a 0 email protected email protected is the first integer not less than a and d ay t is the ath order fractional derivative of y t in the caputo sense although error analyses for this fractional adams method have been given for a 0 1 email protected c 1 a 0 t c 0 1 email protected c 3 g there are still some unsolved problems i the error estimates for email protected 0 1 email protected c 3 g ii the error estimates for email protected 0 1 email protected c 2 g iii the solution y t having some special forms in this paper we mainly study the error analyses of the fractional adams method for the fractional ordinary differential equations for the three cases i iii numerical simulations are also included which are in line with the theoretical analysis

good lattice rules in weighted korobov spaces with general weights we study the problem of multivariate integration and the construction of good lattice rules in weighted korobov spaces with general weights these spaces are not necessarily tensor products of spaces of univariate functions sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found these conditions are also necessary if the weights are such that the reproducing kernel of the weighted korobov space is pointwise non negative the existence of a lattice rule which achieves the nearly optimal convergence order is proven a component by component cbc algorithm that constructs good lattice rules is presented the resulting lattice rules achieve tractability or strong tractability error bounds and achieve nearly optimal convergence order for suitably decaying weights we also study special weights such as finite order and order dependent weights for these special weights the cost of the cbc algorithm is polynomial numerical computations show that the lattice rules constructed by the cbc algorithm give much smaller worst case errors than the mean worst case errors over all quasi monte carlo rules or over all lattice rules and generally smaller worst case errors than the best korobov lattice rules in dimensions up to hundreds numerical results are provided to illustrate the efficiency of cbc lattice rules and korobov lattice rules with suitably chosen weights in particular for high dimensional finance problems

a numerical solution for a variable order reaction diffusion model by using fractional derivatives with non local and non singular kernel abstract a reaction diffusion system can be represented by the gray scott model the reaction diffusion dynamic is described by a pair of time and space dependent partial differential equations pdes in this paper a generalization of the gray scott model by using variable order fractional differential equations is proposed the variable orders were set as smooth functions bounded in 0 1 and specifically the liouville caputo and the atangana baleanu caputo fractional derivatives were used to express the time differentiation in order to find a numerical solution of the proposed model the finite difference method together with the adams method were applied the simulations results showed the chaotic behavior of the proposed model when different variable orders are applied

mixing of all orders of lie groups actions we show that for lie groups whose adjoint representation reflects their topology mixing implies mixing of all orders in particular we prove that mixing actions of a semisimple lie group are mixing of all orders answering a conjecture of b marcus