the solution of tingley s problem for the operator norm unit sphere of complex n n matrices abstract in this paper we study geometric structures and isometries of the unit sphere of n n complex matrices we introduce the notion of proper exposed points of the unit ball of a banach space which provides a refinement of that of exposed point provided that the space is separable as a new geometric interpretation of unitary matrices it is shown that the set of all proper exposed points of the unit ball of m n c coincides with its unitary group u n where m n c denotes the algebra of all n n complex matrices moreover we show that every maximal convex subset of the unit sphere of m n c can be identified with the unit ball of m n 1 c by an affine bijection preserving the unitary matrices this map is isometric with respect to any unitarily invariant norm as an application we completely determine the forms of surjective isometries on the unit sphere of m n c with respect to the operator norm which provides an affirmative answer to tingley s problem on m n c

option pricing stochastic volatility singular dynamics and constrained path integrals abstract stochastic volatility models have been widely studied and used in the financial world the heston model heston 1993 7 is one of the best known models to deal with this issue these stochastic volatility models are characterized by the fact that they explicitly depend on a correlation parameter which relates the two brownian motions that drive the stochastic dynamics associated to the volatility and the underlying asset solutions to the heston model in the context of option pricing using a path integral approach are found in lemmens et al 2008 21 while in baaquie 2007 1997 12 13 propagators for different stochastic volatility models are constructed in all previous cases the propagator is not defined for extreme cases 1 it is therefore necessary to obtain a solution for these extreme cases and also to understand the origin of the divergence of the propagator in this paper we study in detail a general class of stochastic volatility models for extreme values 1 and show that in these two cases the associated classical dynamics corresponds to a system with second class constraints which must be dealt with using dirac s method for constrained systems dirac 1958 1967 22 23 in order to properly obtain the propagator in the form of a euclidean hamiltonian path integral henneaux and teitelboim 1992 25 after integrating over momenta one gets an euclidean lagrangian path integral without constraints which in the case of the heston model corresponds to a path integral of a repulsive radial harmonic oscillator in all the cases studied the price of the underlying asset is completely determined by one of the second class constraints in terms of volatility and plays no active role in the path integral

on the extremal structure of least upper bound norms and their dual abstract given finite dimensional real or complex banach spaces e and f with norms e r and f r we denote by n the least upper bound norm induced on l e f some results are given on the extremal structures of b the unit ball of n of its polar b and of b which is the polar of the unit ball of the least upper bound norm n the exposed faces the extreme points and a large family of other faces of b and b are presented it turns out that b is a subset of b the set of tangency points of the surfaces of b and b is completely determined and represented as the union of the exposed faces of b which are normal to rank one mappings we determine sharp bounds on the ranks of mappings in these exposed faces

the numerically stable reconstruction of jacobi matrices from spectral data we present an expose of the elementary theory of jacobi matrices and in particular their reconstruction from the gaussian weights and abscissas many recent works propose use of the diagonal hermitian lanczos process for this purpose we show that this process is numerically unstable we recall rutishauser s elegant and stable algorithm of 1963 based on plane rotations implement it efficiently and discuss our numerical experience we also apply rutishauser s algorithm to reconstruct a persymmetric jacobi matrix from its spectrum in an efficient and stable manner

isometries of grassmann spaces abstract botelho jamison and molnar have recently described the general form of surjective isometries of grassmann spaces on complex hilbert spaces under certain dimensionality assumptions in this paper we provide a new approach to this problem which enables us first to give a shorter proof and second to remove dimensionality constraints completely in one of the low dimensional cases which was not covered by botelho jamison and molnar an exceptional possibility occurs as a byproduct we are able to handle the real case as well furthermore in finite dimensions we remove the surjectivity assumption a variety of tools is used in order to achieve our goal such as topological geometrical and linear algebra techniques the famous two projections theorem for two finite rank projections will be re proven using linear algebraic methods a theorem of gyory and the second author on orthogonality preservers on grassmann spaces will be strengthened as well this latter result will be obtained by using chow s fundamental theorem of geometry of grassmannians

unitary operators in real von neumann algebras we prove that the open unit ball of any von neumann algebra a is contained in the sequentially convex hull of ua the set of unitary elements of a therefore the closed unit ball of a coincides with the closed convex hull of ua in the complex case this statement is actually valid for any unital c algebra it is the well known russo dye theorem however for real scalars the results we are presenting provide new information even for the algebra l h of bounded linear operators from an infinite dimensional hilbert space h into itself let us say in this sense that the possibility of rebuilding the unit ball of l h through the closed convex hull of the unitary elements appears in the literature as an open problem we also obtain some results about the extremal structure of these spaces

abstract perturbed krylov methods abstract we introduce the framework of abstract perturbed krylov methods this is a new and unifying point of view on krylov subspace methods based solely on the matrix equation aq k f k q k 1 c k q k c k q k 1 c k 1 k e k t and the assumption that the matrix ck is unreduced hessenberg we give polynomial expressions relating the ritz vectors quasi orthogonal residual iterates and quasi minimal residual iterates to the starting vector q1 and the perturbation term fk the properties of these polynomials and similarities between them are analyzed in some detail the results suggest the interpretation of abstract perturbed krylov methods as additive overlay of several abstract exact krylov methods

hamiltonian and potentials in derivative pricing models exact results and lattice simulations the pricing of options warrants and other derivative securities is one of the great success of financial economics these financial products can be modeled and simulated using quantum mechanical instruments based on a hamiltonian formulation we show here some applications of these methods for various potentials which we have simulated via lattice langevin and monte carlo algorithms to the pricing of options we focus on barrier or path dependent options showing in some detail the computational strategies involved

a schur method for the square root of a matrix abstract a fast and stable method for computing the square root x of a given matrix a x 2 a is developed the method is based on the schur factorization a qsq h and uses a fast recursion to compute the upper triangular square root of s it is shown that if x 2 a is not large then the computed square root is the exact square root of a matrix close to a the method is extended for computing the cube root of a matrices exist for which the square root computed by the schur method is ill conditioned but which nonetheless have well conditioned square roots an optimization approach is suggested for computing the well conditioned square roots in these cases

exposed and smooth points of some classes of operation in l lp abstract let x be a banach space and b 1 x the unit ball of x a point x of b 1 x is called an exposed point if there exists f x such that f f x 1 and f y y x in b 1 x it is called smooth if there exists a unique f x such that f f x 1 the object of this paper is to study the exposed and smooth points of b 1 l l p and b 1 k l p where l l p is the space of bounded linear operators on l p 1 p k l p is the class of compact operators in l l p indeed we characterize smooth points and exposed points of b 1 l l p and b 1 k l p for 1 p p 2 b 1 l l p is exposed we also characterize the exposed points of b 1 l l 1 l p 1 p

on an inverse eigenvalue problem for unitary hessenberg matrices abstract we show that a unitary upper hessenberg matrix with positive subdiagonal elements is uniquely determined by its eigenvalues and the eigenvalues of a modified principal submatrix this provides an analog of a well known result for jacobi matrices

on the closed representation for the inverses of hessenberg matrices the general representation for the elements of the inverse of any hessenberg matrix of finite order is here extended to the reduced case with a new proof those entries are given with proper hessenbergians from the original matrix it justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate order and some elementary properties of the inverse these results are applied on the resolvent matrix associated to a finite hessenberg matrix in standard form two examples on the unit disk are given

exposed faces and duality for symmetric and unitarily invariant norms abstract let be a unitarily invariant norm on the space of real or complex n m matrices and g the associated symmetric gauge function thus a g s a where s a is the decreasing sequence of singular values of a denote by b and bg the closed unit balls of and g in a previous paper we showed a close relationship between the faces of b and those of bg in particular to each face of b we associated a standard face of bg and we used this association to completely describe the matrices that are members of an individual face of in the present paper we consider the duality operator that transforms each exposed face of b into an exposed face of the unit ball of s dual and the duality operator that does the same with exposed faces of bg we show that these two operators are very nicely related among other results we prove the following given an exposed face e of b the standard face associated with the dual of e is precisely the dual of the standard face of bg associated with e e is an exposed face of b if and only if its associated standard face is an exposed face of bg as a by product we completely determine the subdifferential of in terms of the subdifferential of g and we completely characterize the matrices that are dual to a given matrix a with respect to

hessenberg eigenvalue eigenmatrix relations abstract explicit relations between eigenvalues eigenmatrix entries and matrix elements of unreduced hessenberg matrices are derived the main result is based on the taylor expansion of the adjugate of zi h on the one hand and inherent properties of hessenberg matrix structure on the other hand this result is utilized to construct computable relations between eigenvalues eigenvector components eigenvalues of principal submatrices and products of lower diagonal elements generalizing similar identities for jacobi matrices

hyman s method revisited the qr algorithm is considered one of the most reliable methods for computing matrix eigenpairs however it is unable to detect multiple eigenvalues and jordan blocks matlab s eigensolver returns heavily perturbed eigenvalues and eigenvectors in such cases and there is no hint for possible principal vectors this paper calls attention to hyman s method as it is applicable for computing principal vectors and higher derivatives of the characteristic polynomial that may help to estimate multiplicity an important information for more reliable computation we suggest a test matrix collection for jordan blocks the first numerical tests with these matrices reveal that the computational problems are deeper than expected at the beginning of this work

quantum finance hamiltonian for coupon bond european and barrier options coupon bond european and barrier options are financial derivatives that can be analyzed in the hamiltonian formulation of quantum finance forward interest rates are modeled as a two dimensional quantum field theory and its hamiltonian and state space is defined european and barrier options are realized as transition amplitudes of the time integrated hamiltonian operator the double barrier option for a financial instrument is knocked out terminated with zero value if the price of the underlying instrument exceeds or falls below preset limits the barrier option is realized by imposing boundary conditions on the eigenfunctions of the forward interest rates hamiltonian the price of the european coupon bond option and the zero coupon bond barrier option are calculated it is shown that is general the constraint function for a coupon bond barrier option can to a good approximation be linearized a calculation using an overcomplete set of eigenfunctions yields an approximate price for the coupon bond barrier option which is given in the form of an integral of a factor that results from the barrier condition times another factor that arises from the payoff function

tingley s problem on finite von neumann algebras abstract in this paper we give an affirmative answer to tingley s problem between finite von neumann algebras to do this we see that every norm closed proper face of the unit ball of a unital c algebra can be represented as the intersection of a family of maximal proper faces

birkhoff james orthogonality and smoothness of bounded linear operators abstract we present a sufficient condition for smoothness of bounded linear operators on banach spaces for the first time let t a b x y where x is a real banach space and y is a real normed linear space we find sufficient condition for t b a t x b a x for some x s x with t x t and use it to show that t is a smooth point in b x y if t attains its norm at unique upto multiplication by scalar vector x s x tx is a smooth point of y and sup y c t y t for all closed subsets c of s x with d x c 0 for operators on a hilbert space h we show that t b a t x b a x for some x s h with t x t if and only if the norm attaining set m t x s h t x t s h 0 for some finite dimensional subspace h 0 and t h o t we also characterize smoothness of compact operators on normed spaces and bounded linear operators on hilbert spaces

characterization of k smooth operators between banach spaces abstract we study k smoothness of bounded linear operators defined between arbitrary banach spaces as an application we characterize k smooth operators defined from l 1 n to an arbitrary banach space we also completely characterize k smooth operators defined between arbitrary two dimensional banach spaces

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

pseudo hermitian formulation of the quantum black scholes hamiltonian abstract we show that the non hermitian black scholes hamiltonian and its various generalizations are pseudo hermitian the metric operator is explicitly constructed for this class of hamiltonians it is also shown that the effective black scholes hamiltonian and its partner form a pseudo supersymmetric system