feedback affinization of nonlinear control systems abstract in this paper conditions for the nonlinear control system x f x u x m u r m to have a nonlinear feedback control u x v v r m m m 0 such that the nonlinear system takes a form of an affine system x a x b x v are presented all results require algebraic operations and differentiation of functions only

operator splitting methods for pricing american options under stochastic volatility we consider the numerical pricing of american options under heston s stochastic volatility model the price is given by a linear complementarity problem with a two dimensional parabolic partial differential operator we propose operator splitting methods for performing time stepping after a finite difference space discretization the idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps with this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint our analysis suggests that the crank nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy the numerical experiments show that the operator splitting methods have comparable discretization errors they also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations

nonlinear decoupling via feedback a differential geometric approach the paper deals with the nonlinear decoupling and noninteracting control problems a complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems the paper also includes algorithms concerned with the actual construction of the appropriate control laws

rational irreducible characters and rational conjugacy classes in finite groups we prove that a finite group g has two rational valued irreducible characters if and only if it has two rational conjugacy classes and determine the structure of any such group along the way we also prove a conjecture of gow stating that any finite group of even order has a non trivial rational valued irreducible character of odd degree

on the rationality of knorr s virtually irreducible lattices abstract it was established by reinhard knorr that the half of brauer s height zero conjecture which assumes an abelian defect group would follow from the nonexistence of certain integral representations of abelian p groups we establish nonexistence of such representations of rank less than 14 for the elementary abelian group of order 8 and believe these to be the first results of this kind the result is made possible by imposing a simple rationality condition

projective modules and involutions let g be a finite group and let t g t2 1 then is a g set under conjugation let k be an algebraically closed field of characteristic 2 it is shown that each projective indecomposable summand of the g permutation module k is irreducible and self dual whence it belongs to a real 2 block of defect zero this together with the fact that each irreducible kg module that belongs to a real 2 block of defect zero occurs with multiplicity 1 as a direct summand of k establishes a bijection between the projective components of k and the real 2 blocks of g of defect zero

a novel pricing method for european options based on fourier cosine series expansions here we develop an option pricing method for european options based on the fourier cosine series and call it the cos method the key insight is in the close relation of the characteristic function with the series coefficients of the fourier cosine expansion of the density function in most cases the convergence rate of the cos method is exponential and the computational complexity is linear its range of application covers underlying asset processes for which the characteristic function is known and various types of option contracts we will present the method and its applications in two separate parts the first one is this paper where we deal with european options in particular in a follow up paper we will present its application to options with early exercise features

spitzer identity wiener hopf factorization and pricing of discretely monitored exotic options the wiener hopf factorization of a complex function arises in a variety of fields in applied mathematics such as probability finance insurance queuing theory radio engineering and fluid mechanics the factorization fully characterizes the distribution of functionals of a random walk or a levy process such as the maximum the minimum and hitting times here we propose a constructive procedure for the computation of the wiener hopf factors valid for both single and double barriers based on the combined use of the hilbert and the z transform the numerical implementation can be simply performed via the fast fourier transform and the euler summation given that the information in the wiener hopf factors is strictly related to the distributions of the first passage times as a concrete application in mathematical finance we consider the pricing of discretely monitored exotic options such as lookback and barrier options when the underlying price evolves according to an exponential levy process we show that the computational cost of our procedure is independent of the number of monitoring dates and the error decays exponentially with the number of grid points

the loewy structure of certain fixpoint algebras part i abstract we investigate the loewy structure of the fixpoint algebra of the group algebra of the additive group of a finite field f under the action of a subgroup of the multiplicative group of f

quadratic geometries projective modules and idempotents abstract let g be a finite group and let k be a field we say that a kg module v has a quadratic geometry or is of quadratic type if there exists a non degenerate equivalently non singular g invariant quadratic form on v if v is irreducible or projective indecomposable and k of odd characteristic the existence of a quadratic geometry can be read off from invariants of representation theory in terms of the group algebra the existence of such a geometry is characterized by the existence of an idempotent e which corresponds to v and is invariant under the antiautomorphism on kg given by g g 1 for g g in characteristic two the problem is more delicate for an irreducible module the existence of a quadratic geometry is partly a question in cohomology unfortunately a satisfactory answer does not seem to be known however if v is projective liftability of the module to characteristic zero may help to find the geometry more easily this is the subject of the paper the notation which we shall use throughout the paper is standard and may be taken from huppert and blackburn finite groups ii springer verlag berlin 1982 and isaacs character theory of finite groups academic press new york 1976

knorr lattices their characters and some subsets of fpd examples of positive height cyclic knorr lattices for elementary abelian p groups are constructed and bounds relating the height of such lattices to the size of the group are given the problem is formulated in terms of finite geometry when p 2 certain examples are shown to arise from difference sets

an iterative splitting method for pricing european options under the heston model abstract in this paper we propose an iterative splitting method to solve the partial differential equation pde in option pricing problems we focus on the heston stochastic volatility model and the derived two dimensional pde we take the european option as an example and conduct numerical experiments using different boundary conditions the iterative splitting method transforms the two dimensional equation into two quasi one dimensional equations with the variable on the other dimension fixed which helps to lower the computational cost numerical results show that the iterative splitting method together with an artificial boundary condition abc based on the method by li and huang 2019 gives the most accurate option price and greeks compared to the classic finite difference method with the commonly used boundary conditions in heston 1993

condensed forms of linear control system under output feedback abstract we present two complementary condensed forms of linear control system with output x a x b u x r n u r m y c x y r r under transformation of the state coordinates from them we derive two complementary condensed forms under output feedback as an application of the condensed forms we show that the output feedback control of the linear control systems generically reduces to a matrix completion problem and we provide a partial solution of that problem based on existing methods

real subpairs and frobenius schur indicators of characters in 2 blocks abstract let b be a real 2 block of a finite group g a defect couple of b is a certain pair d e of 2 subgroups of g such that d is a defect group of b and d e the block b is principal if e d otherwise e d 2 we show that d e determines which b subpairs are real the involution module of g arises from the conjugation action of g on its involutions we outline how d e influences the vertices of components of the involution module that belong to b these results allow us to enumerate the frobenius schur indicators of the irreducible characters in b when b has a dihedral defect group the answer depends both on the decomposition matrix of b and on a defect couple of b we also determine the vertices of the components of the involution module of b

duality in g algebras keywords finite group complete discrete valuation ring symmetric g algebra symmetric bilinear form projective ideal symmetric algebra rg lattice unimodular symmetric g algebra almost projective ideal almost split sequences block defect groups virtually irreducible lattices splitting trace lattices reference ctg article 1988 005doi 10 1007 bf01161746 record created on 2008 12 16 modified on 2017 05 12

projective indecomposable modules scott modules and the frobenius schur indicator abstract let be a principal indecomposable character of a finite group g in characteristic 2 the frobenius schur indicator of is shown to equal the rank of a bilinear form defined on the span of the involutions in g moreover if the principal indecomposable module corresponding to affords a quadratic geometry then 0 this result is used to prove a more precise form of a theorem of benson and carlson on the existence of scott components in the endomorphism ring of an indecomposable g module in case the module affords a g invariant symmetric form

some cyclic virtually irreducible lattices abstract virtually irreducible lattices which satisfy a simple rationality condition and have positive height are constructed for elementary abelian 2 groups the non existence of such lattices would have implied part of brauer s height zero conjecture a characterization of rank 6 lattices for the elementary abelian group of size 16 is given

linearization and input output decoupling for general nonlinear systems necessary and sufficient conditions are obtained for the linearization and input decoupling by state feedback of general nonlinear systems it is shown how these conditions can be derived from the already known conditions for affine nonlinear systems thereby also elucidating the existing theory for affine systems

symmetric bilinear forms and vertices in characteristic 2 let g be a finite group and let k be an algebraically closed field of characteristic 2 suppose that m is an indecomposable kg module which affords a non degenerate g invariant symmetric bilinear form we assign to m a collection of 2 subgroups of g called its symmetric vertices each of which contains a green vertex of m with index at most 2 if m is irreducible then its symmetric vertices are uniquely determined up to g conjugacy if b is the real 2 block of g containing m we show that each symmetric vertex of m is contained in an extended defect group of b moreover we characterise the extended defect groups in terms of symmetric vertices in order to prove these results we develop the theory of involutary g algebras this allows us to translate questions about symmetric kg modules into questions about projective modules of quadratic type

involutions cohomology and metabolic spaces abstract we prove that over a field of characteristic two the involution module of a finite group allows a natural splitting into a direct sum of cohomology modules

sufficient conditions for linear control system decoupling by static state feedback sufficient conditions for solvability of linear control system decoupling by static state feedback are given in terms of integer relations it is shown that these integer relations in the generic case are very simple if the decoupling conditions are satisfied a given algorithm for decoupling based on solving systems of linear equations and spending controls can be applied

characters bilinear forms and solvable groups abstract we prove a number of results about the ordinary and brauer characters of finite solvable groups in characteristic 2 by defining and using the concept of the extended nucleus of a real irreducible character in particular we show that the isaacs canonical lift of a real irreducible brauer character has frobenius schur indicator 1 we also show that the principal indecomposable module corresponding to a real irreducible brauer character affords a quadratic geometry if and only if each extended nucleus is a split extension of a nucleus

on the involution module of gln 2f for any group g the set of involutions i in g that is the set of group elements that have order two forms a g set under conjugation the corresponding kg permutation module ki is the involution module of g here k is an algebraically closed field of characteristic two in this paper we discuss aspects of the involution module of the general linear group gln 2f we determine almost all components of this module furthermore we present a vertex and the green correspondent of each component

blocks of defect zero and products of elements of orderp abstract suppose thatgis a finite group and thatfis a field of characteristicp 0 which is a splitting field for all subgroups ofg lete0be the sum of the block idempotents of defect zero infg and let be the set of solutions togp 1 ing we show thate0 2 whenpis odd ande0 3 whenp 2 in the latter case 2 r whereris the set of real elements of 2 defect zero soe0 r r 2 we also show thate0 4 4 2 whenp 2 where 4is the set of solutions tog4 1 these results give us various criteria for the existence ofp blocks of defect zero

the evaluation of barrier option prices under stochastic volatility this paper considers the problem of numerically evaluating barrier option prices when the dynamics of the underlying are driven by stochastic volatility following the square root process of heston 1993 7 we develop a method of lines approach to evaluate the price as well as the delta and gamma of the option the method is able to efficiently handle both continuously monitored and discretely monitored barrier options and can also handle barrier options with early exercise features in the latter case we can calculate the early exercise boundary of an american barrier option in both the continuously and discretely monitored cases

g algebras jacobson radical and almost split sequences keywords finite group indecomposable kg module sas sequence almost split exact sequences restriction relative jacobson radicals local criterion multiplicity module vertex source indecomposable projective module twisted group algebra induction reference ctg article 1988 006doi 10 1007 bf01393690 record created on 2008 12 16 modified on 2017 05 12

cross characteristic representations of even characteristic symplectic groups we classify the small irreducible representations of sp 2n q with q even in odd characteristic this improves even the known results for complex representations the smallest representation for this group is much larger than in the case when q is odd this makes the problem much more difficult