lower bounds for geometrical and physical problems motion planning involving arbitrarily many degrees of freedom is known to be pspace hard in this paper we examine the complexity of generalized motion planning problems for planar mechanisms consisting of independently movable objects our constructions constitute a general framework for reducing problems in information processing to motion planning leading to easy proofs of known pspace hardness results and to exponential lower bounds for geometrical problems related to motion planning particulalrly we show that the problem of deciding whether a given mechanism a can always avoid a collision with another mechanism b is expspace hard new lower bounds are also obtained for the problem of planning under given physical side conditions we consider the case that certain motions require forces e g to subdue friction and ask for motions that stay under a given energy limit within our framework we show that such shortest path problems are exptime hard if we use number representations by mantissa and exponent and even undecidable if we allow that some motions require no force or an infinite amount the proof consists of a simulation of turing machines with infinite tape and shows that the notion of turing computability can be interpreted in purely geometrical terms the geometrical model obtained is capable of expressing a variety of physical planning problems

maps on idempotent matrices over division rings abstract let d be an arbitrary division ring and p n d the set of all n n idempotent matrices over d under some mild conditions we give a complete description of maps on p n d that preserve either commutativity or order or orthogonality we give examples showing that our assumptions cannot be relaxed much further as an application we will prove a quaternionic analogue of ovchinnikov s result that is important in quantum mechanics other applications of our theorems include results on automorphisms of operator and matrix semigroups local automorphisms linear preserver problems and geometry of matrices and grassmannians

on the glivenko cantelli problem in stochastic programming linear recourse and extensions integrals of optimal values of random optimization problems depending on a finite dimensional parameter are approximated by using empirical distributions instead of the original measure under fairly broad conditions it is proved that uniform convergence of empirical approximations of the right hand sides of the constraints implies uniform convergence of the optimal values in the linear and convex case

mappings that preserve pairs of operators with zero triple jordan product abstract let f be a field and n 3 suppose s 1 s 2 m n f contain all rank one idempotents the structure of surjections s 1 s 2 satisfying aba 0 a b a 0 is determined similar results are also obtained for a subsets of bounded operators acting on a complex or real banach space x b the space of hermitian matrices acting on n dimensional vectors over a skew field d c subsets of self adjoint bounded linear operators acting on an infinite dimensional complex hilbert space it is then illustrated that the results can be applied to characterize mappings on matrices or operators such that f aba f a b a for all a b for functions f such as the spectral norm schatten p norm numerical radius and numerical range etc

nonlinear maps preserving numerical radius of indefinite skew products of operators let h be a complex hilbert space of dimension greater than 2 and j b h be an invertible self adjoint operator denote by a j 1a j the indefinite conjugate of a b h with respect to j and denote by w a the numerical radius of a let w and v be subsets of b h which contain all rank one operators and let w v be a surjective map we show that satisfies w ab w a b and w a b w a b for all a b w if and only if there exist scalars i 1 1 i 1 2 unitary or conjugate unitary operators u v on h satisfying u u 1i v v 2i and a functional w c with a 1 such that a a uav for all a w satisfies w ab a w a b a for all a b w if and only if either there exist 1 1 a unitary or conjugate unitary operator u on h satisfying u u i and a functional w c with a 1 such that a a uau for all a w or there exist a nonzero real number b a unitary or conjugate unitary operator u on h satisfying u ju bj 1 and a functional w c with a 1 such that a a ua u for all a w

ring isomorphisms and linear or additive maps preserving zero products on nest algebras it is shown that every ring isomorphism between norm closed standard subalgebras of nest algebras acting on infinite dimensional banach spaces is either a spatial isomorphism or a spatial conjugate isomorphism this is then used to characterize the linear or additive maps between nest algebras which preserve zero products

nonlinear preservers of pseudospectral radius on banach spaces abstract given a complex banach space x we denote by l x the algebra of all linear bounded operators on it for e 0 let r e t denote the e pseudospectral radius of t l x for an infinite dimensional space x we characterize surjective maps l x l x such that r e t 1 t 2 r e t 1 t 2 for all t 1 t 2 l x an analogous result is obtained in the finite dimensional case with no surjectivity assumption on the map

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

stochastic integer programming limit theorems and confidence intervals we consider empirical approximations sample average approximations of two stage stochastic mixed integer linear programs and derive central limit theorems for the objectives and optimal values the limit theorems are based on empirical process theory and the functional delta method we also show how these limit theorems can be used to derive confidence intervals for optimal values via resampling methods bootstrap subsampling

a stochastic approach to stability in stochastic programming abstract a random optimization problem p 0 min x 0 0 x is approximated by a sequence of random surrogate problems p n n n with p n min x n n x p a given probability space we investigate the convergence almost surely and in probability of the optimal values and the solution sets the results can be regarded as random versions of well known stability statements of parametric programming semicontinuous convergence almost surely in probability of sequences of random functions is a crucial assumption in this framework and will be investigated in more detail

unitary similarity invariant function preservers of skew products of operators abstract let b h denote the banach algebra of all bounded linear operators on a complex hilbert space h with dim h 3 and let a and b be subsets of b h which contain all rank one operators suppose f is a unitary invariant norm the pseudo spectra the pseudo spectral radius the c numerical range or the c numerical radius for some finite rank operator c the structure is determined for surjective maps a b satisfying f a b f a b for all a b a to establish the proofs some general results are obtained for functions f f 1 h 0 0 where f 1 h is the set of rank one operators in b h satisfying a f u a u f a for a complex unit a f 1 h and unitary u b h b for any rank one operator x f 1 h the map t f t x on 0 is strictly increasing and c the set f x x f 1 h and x 1 attains its maximum and minimum

schur product of matrices and numerical radius range preserving maps abstract let f a be the numerical range or the numerical radius of a square matrix a denote by a b the schur product of two matrices a and b characterizations are given for mappings on square matrices satisfying f a b f a b for all matrices a and b analogous results are obtained for mappings on hermitian matrices

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

on structure and stability in stochastic programs with random technology matrix and complete integer recourse for two stage stochastic programs with integrality constraints in the second stage we study continuity properties of the expected recourse as a function both of the first stage policy and the integrating probability measure

automorphisms of the poset of skew projections abstract any automorphism of the poset b of all idempotents of the algebra b h of all bounded linear operators in a hilbert space h dim h 3 is shown to be of the form p apa 1 or p ap a 1 with a c where c is the set of all semi linear bijections a h h if h is finite dimensional and the set of all continuous invertible linear or conjugate linear operators in h if h is infinite dimensional further applications of the method are outlined

orthogonality preserving transformations on indefinite inner product spaces generalization of uhlhorn s version of wigner s theorem abstract we present an analogue of uhlhorn s version of wigner s theorem on symmetry transformations for the case of indefinite inner product spaces this significantly generalizes a result of van den broek the proof is based on our main theorem which describes the form of all bijective transformations on the set of all rank one idempotents of a banach space which preserve zero products in both directions

condition spectra of special operators and condition spectra preservers abstract let h be a complex hilbert space and let l h be the algebra of all bounded linear operators on h descriptions of the condition spectra of various special classes of operators are established as application characterization is obtained for maps on l h leaving invariant the condition spectrum or the condition spectral radius of the product of operators analogous results are obtained for jordan triple product of operators

on frictional mechanical systems and their computational power in this paper we define a class of mechanical systems consisting of rigid objects defined by linear or quadratic surface patches connected by frictional contact linkages between surfaces this class of mechanisms is similar to the analytical engine developed by babbage in the 1800s except that we assume frictional surfaces instead of toothed gears we prove that a universal turing machine tm can be simulated by a universal frictional mechanical system in this class consisting of a constant number of parts our universal frictional mechanical system has the property that it can reach a distinguished final configuration through a sequence of legal movements if and only if the universal tm accepts the input string encoded by its initial configuration there are two implications from this result first the robotic mover s problem is undecidable when there are frictional linkages second a mechanical computer can be constructed that has the computational power of any conventional electronic computer and yet has only a constant number of mechanical parts previous constructions for mechanical computing devices such as babbage s analytical engine either provided no general construction for finite state control or the control was provided by electronic devices as was common in electromechanical computers such as mark i subsequent to turing s result our result seems to be the first to provide a general proof of the simulation of a universal tm via a purely mechanical mechanism in addition we discuss the universal frictional mechanical system in the context of an error model that allows an error up to epsilon in each mechanical operation we first show that for a universal tm m a frictional mechanical system in this epsilon error model can be constructed such that given any space bound s the system can simulate the computation of m on any input string omega if m decides omega in space bound s provided that epsilon 2 cs for some constant c we also show that for any universal tm m and space bound s there exists a frictional mechanical system in the epsilon error model with epsilon omega 1 it has o s parts and can simulate m on any input omega that m decides in space bound s

analysis of sample path optimization sample path optimization is a method for optimizing limit functions occurring in stochastic modeling problems such as steady state functions in discrete event dynamic systems it is closely related to retrospective optimization techniques and to m estimation the method has been computationally tested elsewhere on problems arising in production and in project planning with apparent success in this paper we provide a mathematical justification for sample path optimization by showing that under certain assumptions which hold for the problems just mentioned the method will almost surely find a point that is in a specified sense sufficiently close to the set of optimizers of the limit function

conditional value at risk in stochastic programs with mixed integer recourse in classical two stage stochastic programming the expected value of the total costs is minimized recently mean risk models studied in mathematical finance for several decades have attracted attention in stochastic programming we consider conditional value at risk as risk measure in the framework of two stage stochastic integer programming the paper addresses structure stability and algorithms for this class of models in particular we study continuity properties of the objective function both with respect to the first stage decisions and the integrating probability measure further we present an explicit mixed integer linear programming formulation of the problem when the probability distribution is discrete and finite finally a solution algorithm based on lagrangean relaxation of nonanticipativity is proposed

asymptotic similarity preserving additive maps on b x abstract let x be an infinite dimensional complex banach space and denote b x the algebra of all bounded linear operators acting on x we show that an additive surjective map on b x preserves asymptotic similarity in both directions if and only if there exist a nonzero scalar c an invertible bounded linear or conjugate linear operator a and an asymptotic similarity invariant additive functional on b x such that either t cata 1 t i for all t or t cat a 1 t i for all t in the case that x has infinite multiplicity especially if x is an infinite dimensional hilbert space above asymptotic similarity invariant additive functional is always zero

on operators preserving james orthogonality abstract we characterise isometries and coisometries in b h in terms of james orthogonality as a consequence we obtain a characterisation of surjective linear or conjugate linear mappings b h b h that preserve james orthogonality in both directions

applying projective geometry to transformations on rank one idempotents abstract we present a short proof of molnar s characterization of bijective transformations on the set of all rank one idempotent operators on a banach space which preserve zero products in both directions an improvement in the finite dimensional case is given we apply these results to describe automorphisms of standard operator semigroups and to improve uhlhorn s version of wigner s theorem

linear maps transforming the higher numerical ranges abstract let k 1 n the k numerical range of a m n is the set w k a tr x ax k x is n k x x i k and the k numerical radius of a is the quantity w k a max z z w k a suppose k 1 k 1 n and n c n k min k n k it is shown that there is a linear map m n m n satisfying w k a w k a for all a m n if and only if n n k k or n n k n k is a positive integer moreover if such a linear map exists then there are unitary matrix u m n and nonnegative integers p q with p q n n such that has the form a u a a p a t a t q u or a u a a p a t a t q u where m n m n has the form a tr a i n n k a k linear maps m n m n satisfying w k a w k a for all a m n are also studied furthermore results are extended to triangular matrices

applying the minimax criterion in stochastic recourse programs we consider an optimization problem in which some uncertain parameters are replaced by random variables the minimax approach to stochastic programming concerns the problem of minimizing the worst expected value of the objective function with respect to the set of probability measures that are consistent with the available information on the random data only very few practicable solution procedures have been proposed for this problem and the existing ones rely on simplifying assumptions in this paper we establish a number of stability results for the minimax stochastic program justifying in particular the approach of restricting attention to probability measures with support in some known finite set following this approach we elaborate solution procedures for the minimax problem in the setting of two stage stochastic recourse models considering the linear recourse case as well as the integer recourse case since the solution procedures are modifications of well known algorithms their efficacy is immediate from the computational testing of these procedures and we do not report results of any computational experiments

convergence theory for nonconvex stochastic programming with an application to mixed logit monte carlo methods have extensively been used and studied in the area of stochastic programming their convergence properties typically consider global minimizers or first order critical points of the sample average approximation saa problems and minimizers of the true problem and show that the former converge to the latter for increasing sample size however the assumption of global minimization essentially restricts the scope of these results to convex problems we review and extend these results in two directions we allow for local saa minimizers of possibly nonconvex problems and prove under suitable conditions almost sure convergence of local second order solutions of the saa problem to second order critical points of the true problem we also apply this new theory to the estimation of mixed logit models for discrete choice analysis new useful convergence properties are derived in this context both for the constrained and unconstrained cases and associated estimates of the simulation bias and variance are proposed

the lower semicontinuity of optimal solution sets abstract we provide a sufficient condition which guarantees the lower semicontinuity of the optimal solution set for a general nonlinear programming problem it is shown that the same condition is both sufficient and necessary for convex programming problems that satisfy the slater condition

maps preserving the local spectrum of skew product of operators abstract let h and k be infinite dimensional complex hilbert spaces and let b h be the algebra of all bounded linear operators on h let t h denote the local spectrum of an operator t b h at any vector h h and fix two nonzero vectors h 0 h and k 0 k we show that if a map b h b k has a range containing all operators of rank at most two and satisfies t s h 0 t s k 0 for all t s b h then there exist two unitary operators u and v in b h k such that u h 0 k 0 for some nonzero c and t u t v for all t b h we also described maps b h b k satisfying t s t h 0 t s t k 0 for all t s b h and with the range containing all operators of rank at most four

isometries for unitarily invariant norms a survey of linear isometries for unitarily invariant norms on real or complex rectangular matrices is given which includes some latest development on the topic a result on isometries for unitarily invariant norms without the linearity assumption is presented related results and problems are discussed

pseudospectra of special operators and pseudospectrum preservers abstract denote by b h the banach algebra of all bounded linear operators on a complex hilbert space h let a b h and denote by a the spectrum of a for e 0 define the e pseudospectrum e a of a as e a z a e e b h e e in this paper the pseudospectra of several special classes of operators are characterized as an application complete descriptions are given of the maps of b h leaving invariant the pseudospectra of a b for different kinds of binary operations on operators such as the difference a b the operator product ab and the jordan product a b b a

non linear numerical radius isometries on atomic nest algebras and diagonal algebras abstract a nonlinear map between operator algebras is said to be a numerical radius isometry if w t s w t s for all t s in its domain algebra where w t stands for the numerical radius of t let n and m be two atomic nests on complex hilbert spaces h and k respectively denote alg n the nest algebra associated with n and d n alg n alg n the diagonal algebra we give a thorough classification of weakly continuous numerical radius isometries from alg n onto alg m and a thorough classification of numerical radius isometries from d n onto d m

preservers of radial unitary similarity functions on products of operators abstract let s be a subset of the algebra l h of all bounded linear operators on an infinite dimensional complex hilbert space h and let f s 0 be a radial unitary similarity invariant function under some assumption on s and f characterizations are obtained for surjective maps on s satisfying f t s f t s t s s where the binary operation stands for the product or the jordan semi triple product on operators analogous descriptions are obtained for the finite dimensional case without the surjectivity assumption on

maps on matrix spaces it is well known that every automorphism of the full matrix algebra is inner we give a short proof of this statement and discuss several extensions of this theorem including structural results for multiplicative maps on matrix algebras characterizations of monotone and orthogonality preserving maps on idempotent matrices some nonlinear preserver results and some recent theorems concerning geometry of matrices we show that all these topics are closely related and point out the connections with physics and geometry several open problems are posed

endomorphisms of the poset of idempotent matrices abstract let d be a division ring n 3 an integer and p n d the poset of all n n idempotent matrices over d with the partial order defined by p q if p q q p p let t m n d be an invertible matrix and d d an endomorphism d d an anti endomorphism for any p p n d we denote by p p the idempotent matrix obtained from p by applying entrywise the map p n d p n d defined by p t p t 1 p p n d p t t p t 1 p p n d is an injective map preserving order in both directions every such map is called a standard map it has been known before that if d is an eas division ring then every injective order preserving map p n d p n d is either standard or of a very special degenerate form in this paper we use some ideas from geometry of algebraic homogeneous spaces and elementary field theory to give examples showing that the eas assumption is indispensable then we define generalized standard maps and using them we describe the general form of injective order preservers on p n d for an arbitrary division ring d our proof is shorter than the original one for the special case of eas division rings under somewhat stronger assumptions we get two characterizations of standard maps all the results are optimal as shown by counterexamples

twistor space minkowski space and the conformal group it is shown that the conformal group of compactified minkowski space is isomorphic to a group of rays of semilinear transformations of twistor space the action of the conformal group on twistor space is given by an explicit realisation of this isomorphism in this way we determine the transformation of twistor space under space inversion and time inversion

on boundaries of highly visible spaces and applications the purpose of this paper is to investigate the properties of a certain class of highly visible spaces for a given geometric space c containing obstacles specified by disjoint subsets of c the free space f is defined to be the portion of c not occupied by these obstacles the space is said to be highly visible if at each point in f a viewer can see at least an e fraction of the entire f this assumption has been used for robotic motion planning in the analysis of random sampling of points in the robot s configuration space as well as the upper bound of the minimum number of guards needed for art gallery problems however there is no prior result on the implication of this assumption to the geometry of the space under study for the two dimensional case with the additional assumptions that c is bounded within a rectangle of constant aspect ratio and that the volume ratio between f and c is a constant we use the proof technique of charging each obstacle boundary segment by a certain portion of c to show that the total length of all obstacle boundaries in c is o n f e if f contains polygonal obstacles with a total of n boundary edges or o n f e if c contains n convex obstacles that are piecewise smooth in both cases f is the volume of f for the polygonal case this bound is tight as we can construct a space whose boundary size is n f e these results can be partially extended to three dimensions we show that these results can be applied to the analysis of certain probabilistic roadmap planners as well as a variant of the art gallery problem we also propose a number of conjectures on the properties of these highly visible spaces

asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems we study the asymptotic behavior of the statistical estimators that maximize a not necessarily dieren tiable criterion function possibly subject to side constraints equalities and inequalities the consistency results generalize those of wald and huber conditions are also given under which one is still able to obtain asymptotic normality the analysis brings to the fore the relationship between the problem of nding statistical estimators and that of nding the optimal solutions of stochastic optimization problems with partial information the last section is devoted to the properties of the saddle points of the associated lagrangians

quantitative stability of variational systems ii a framework for nonlinear conditioning stability results of lipschitz and holder type are obtained for the solutions and optimal values of optimization problems when perturbations are measured in terms of the rho epi distance

maps preserving a ky fan norm of jordan product abstract given n 2 let 2 k n be fixed we study mappings on m n the algebra of n n complex matrices that satisfy n k a b b a n k a b b a for all a b m n where n k c denotes the ky fan k norm the sum of k greatest singular values of the matrix c if n 3 we additionally suppose either 1 i 2 i for some unimodular complex 1 2 or that is surjective and when n 2 the complete description is obtained without additional assumptions

generalized symmetry transformations on quaternionic indefinite inner product spaces an extension of quaternionic version of wigner s theorem uhlhorn s extension of wigner s unitary antiunitary theorem has been recently generalized by molnar to indefinite inner product spaces we present the quaternionic version of this result

maps leaving functional values of operator products invariant abstract let a be a standard operator algebra on a complex hilbert space h of dimension greater than 2 by invariants of certain functional values of operator products we characterize some surjective maps on a furthermore several kinds of general preserver problems on standard operator algebras are solved when we take respectively the functional as for example k numerical radius k 1 operator norm ky fan k norm schatten p norm 1 p and so on

local epi continuity and local optimization one of the fundamental questions in nonlinear optimization is how optimization problems behave when the functions defining them change e g by continuous deformation recently the study of epi continuity has somewhat unified the results in this area here we show how to localize the concept of epi continuity and how to apply these localized ideas to ensure persistence and stability of local optimizing sets we also show how these conditions follow from known properties of nonlinear programming problems

additive preservers of numerical range abstract we characterize all additive mappings preserving classical numerical range on various subalgebras of m n the subalgebra of all diagonal matrices d n the subalgebra of all upper triangular matrices t n and subalgebras of block diagonal and block triangular matrices characterizations obtained show that in some cases m n for example all such preservers are linear on the other hand we show that the group of all additive preservers on t n contains some additional mappings

preservers of pseudo spectral radius of operator products abstract let h be an infinite dimensional complex hilbert space and let l h be the algebra of all bounded linear operators on h for e 0 and t l h let r e t denote the e pseudo spectral radius of t we characterize surjective maps on l h which satisfy r e t s r e t s for all t s l h we also obtain analogous result for the finite dimensional case without the surjectivity assumption on

c numerical radius isometries on matrix algebras and triangular matrix algebras abstract let c c 1 c n t r n and m n be the set of n n complex matrices for any a m n define the c numerical range and the c numerical radius of a by w c a i 1 n c i a x i x i x 1 x n is an orthonormal set in c n and w c a max z z w c a respectively let t n be the set of n n upper triangular matrices when w c is a norm on m n mappings t on m n or t n satisfying w c t a t b w c a b for all a b are characterized as an intermediate step we also characterize additive c numerical range preservers on m n or t n

maps preserving numerical radius and cross norms of operator products let h be a complex hilbert space with dim h 3 and x1d4b2 x1d4b1 h be subsets that contain all rank one idempotents let f stand for numerical radius w or a cross operator norm n all the general surjective maps x1d4b2 x1d4b1 satisfying f ab f a b resp f aba f a b a for all a b x1d4b2 are characterized

additive mappings between hermitian matrix spaces preserving rank not exceeding one abstract let k be a field of characteristic not two or three with an automorphism of order two let f a k a a we characterize additive mappings t from one f vector space of hermitian matrices over k to another that preserve rank less than or equal to one for the following cases i the image of t contains a matrix of rank at least three ii every nonzero endomorphism of f is surjective

risk aversion via excess probabilities in stochastic programs with mixed integer recourse we consider linear two stage stochastic programs with mixed integer recourse instead of basing the selection of an optimal first stage solution on expected costs alone we include into the objective a risk term reflecting the probability that a preselected cost threshold is exceeded after we have put the resulting mean risk model into perspective with stochastic dominance we study further structural properties of the model and derive some basic stability results in the algorithmic part of the paper we propose a scenario decomposition method and report initial computational experience

wigner s theorem on grassmann spaces wigner s celebrated theorem which is particularly important in the mathematical foundations of quantum mechanics states that every bijective transformation on the set of all rank one projections of a complex hilbert space which preserves the transition probability is induced by a unitary or an antiunitary operator this vital theorem has been generalised in various ways by several scientists in 2001 molnar provided a natural generalisation namely he provided a characterisation of not necessarily bijective maps which act on the grassmann space of all rank n projections and leave the system of jordan principal angles invariant see 17 and 20 in this paper we give a very natural joint generalisation of wigner s and molnar s theorems namely we prove a characterisation of all not necessarily bijective transformations on the grassmann space which fix the quantity trpq i e the sum of the squares of cosines of principal angles for every pair of rank n projections p and q

stability and sensitivity of optimization problems with first order stochastic dominance constraints we analyze the stability and sensitivity of stochastic optimization problems with stochastic dominance constraints of first order we consider general perturbations of the underlying probability measures in the space of regular measures equipped with a suitable discrepancy distance we show that the graph of the feasible set mapping is closed under rather general assumptions we obtain conditions for the continuity of the optimal value and upper semicontinuity of the optimal solutions as well as quantitative stability estimates of lipschitz type furthermore we analyze the sensitivity of the optimal value and obtain upper and lower bounds for the directional derivatives of the optimal value the estimates are formulated in terms of the dual utility functions associated with the dominance constraints

generalization of wigner s unitary antiunitary theorem for indefinite inner product spaces we present a generalization of wigner s unitary antiunitary theorem for pairs of ray transformations as a particular case we get a new wigner type theorem for non hermitian indefinite inner product spaces