a simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension it has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplex type algorithms for linear programming in this paper the average number of steps performed by a simplex algorithm the so called self dual method is analyzed the algorithm is not started at the traditional point 1 l t but points of the form 1 e e 2 t with e sufficiently small are used the result is better in two respects than those of the previous analyses first it is shown that the expected number of steps is bounded between two quadratic functions c 1 min m n 2 and c 2 min m n 2 of the smaller dimension of the problem this should be compared with the previous two major results in the field borgwardt proves an upper bound of o n 4 m 1 n 1 under a model that implies that the zero vector satisfies all the constraints and also the algorithm under his consideration solves only problems from that particular subclass smale analyzes the self dual algorithm starting at 1 1 t he shows that for any fixed m there is a constant c m such the expected number of steps is less than c m ln n m m 1 megiddo has shown that under smale s model an upper bound c m exists thus for the first time a polynomial upper bound with no restrictions except for nondegeneracy on the problem is proved and for the first time a nontrivial lower bound of precisely the same order of magnitude is established both borgwardt and smale require the input vectors to be drawn from spherically symmetric distributions in the model in this paper invariance is required only under certain

polynomial series expansions for confluent and gaussian hypergeometric functions based on the hadamard product of power series polynomial series expansions for confluent hypergeometric functions m a c and for gaussian hypergeometric functions f a b c are introduced and studied it turns out that the partial sums provide an interesting alternative for the numerical evaluation of the functions m a c and f a b c in particular if the parameters are also viewed as variables

bisection is optimal we seek an approximation to a zero of a continuous functionf a b such thatf a 0 andf b 0 it is known that the bisection algorithm makes optimal use ofn function evaluations i e yields the minimal error which is b a 2 n 1 see e g kung 2 traub and wozniakowski 5 proposed using more general information onf by permitting the adaptive evaluations ofn arbitrary linear functionals they conjectured 5 p 170 that the bisection algorithm remains optimal even if these general evaluations are permitted this paper affirmatively proves this conjecture in fact we prove optimality of the bisection algorithm even assuming thatf is infinitely many times differentiable on a b and has exactly one simple zero

conjugacy of finite biprefix codes two languages x and y are called conjugates if they satisfy the conjugacy equation xz zy for some non empty language z we will compare solutions of this equation with those of the corresponding equation of words and study the case of finite biprefix codes x and y we show that the maximal z in this case is rational we will also characterize x and y in the case where they are both finite biprefix codes this yields the decidability of the conjugacy of two finite biprefix codes

on hypergeometric functions and function spaces the aim of this paper is to discuss the role of hypergeometric functions in function spaces and to prove some new results for these functions the first part of this paper proves results such as monotone convexity and concavity properties of sums of products of hypergeometric functions the second part of our results deals with the space a of all normalized analytic functions f f 0 0 f 0 1 in the unit disk and the subspace r f a a n r such that re ei f z 0 z for f a we consider integral transforms of the type v f 01 t f tz t dt where t is a real valued nonnegative weight function normalized so that 01 t 1 we obtain conditions on and the function such that v f takes each member of r into a starlike function of order 0 1 2 these results extend and improve the earlier known results in these directions we end the paper with an open problem

on the average number of steps of the simplex method of linear programming the goal is to give some theoretical explanation for the efficiency of the simplex method of george dantzig fixing the number of constraints and using dantzig s self dual parametric algorithm we show that the number of pivots required to solve a linear programming problem grows in proportion to the number of variables on the average

some basic information on information based complexity theory abstract a branch of complexity theory called information based complexity theory ibct produces an abundance of impressive results about the quest for approximate solutions to mathematical problems why then do most numerical analysts turn a cold shoulder to ibct close analysis of two papers representative of ibct s best efforts reveals a mixture of nice new observations misdirected examples and misleading theorems some elements in the framework of ibct erected to support a rigorous yet flexible theory make it difficult to judge whether a model is off target or reasonably realistic for instance a sharp distinction is made between information and algorithms restricted to this information yet the information itself usually comes from an algorithm and so the distinction clouds the issues and can lead to true but misleading inferences another troublesome aspect of ibct is a free parameter f the class of admissible problem instances whose membership fee is completely ignored in ascertaining the cost of solving the worst case in f sometimes this leads to unrealistic models we conclude that one s satisfaction with each result of ibct must be inversely proportional to what one knows about the problem the surprising results known to us pertain only to unnatural situations and ibct s genuinely new insights might serve us better if expressed in the conventional mode of error bounds and approximation theory

can adaption help on the average we study adaptive information for approximation of linear problems in a separable hilbert space equipped with a probability measure it is known that adaption does not help in the worst case for linear problems we prove that adaption also doesnot help on the average that is there exists nonadaptive information which is as powerful as adaptive information this result holds for orthogonally invariant measures we provide necessary and sufficient conditions for a measure to be orthogonally invariant examples of orthogonally invariant measures include gaussian measures and in the finite dimensional case weighted lebesgue measures

the distribution of the sample minimum variance frontier in this paper we present a finite sample analysis of the sample minimum variance frontier under the assumption that the returns are independent and multivariate normally distributed we show that the sample minimum variance frontier is a highly biased estimator of the population frontier and we propose an improved estimator of the population frontier in addition we provide the exact distribution of the out of sample mean and variance of sample minimum variance portfolios this allows us to understand the impact of estimation error on the performance of in sample optimal portfolios

computational complexity on the geometry of polynomials and a theory of cost ii this paper deals with traditional algorithms newton s method and a higher order generalization due to euler these iterations schemes and their modifications have had a great success in solving nonlinear systems of equations we give some understanding of this phenomenon by giving estimates of efficiency the problem we focus on is that of finding a zero of a complex polynomial

evaluating rational functions infinite precision is finite cost and tractable on average if one is interested in the computational complexity of problems whose natural domain of discourse is the reals then one is led to ask what is the cost of obtaining solutions to within a prescribed absolute accuracy varepsilon 1 2 s or precision s log 2 varepsilon the loss of precision intrinsic to solving a problem independent of method of solution gives a lower bound on the cost it also indicates how realistic it is to assume that basic arithmetic operations are exact and or take one step for complexity analyses for the relative case the analogous notion is the loss of significance let p x q x be a rational function of degree d dimension n and real coefficients of absolute value bounded by rho the loss of precision in evaluating p q will depend on the input x and in general can be arbitrarily large we show that w r t normalized lebesgue measure on b r the ball of radius r about the origin in r n the average loss is small loglinear

commutation with codes the centralizer of a set of words x is the largest set of words c x commuting with x xc x c x x it has been a long standing open question due to j h conway regular algebra and finite machines chapman hall london 1971 whether the centralizer of any rational set is rational while the answer turned out to be negative in general see m kunc proc of icalp 2004 lecture notes in computer science vol 3142 springer berlin 2004 pp 870 881 we prove here that the situation is different for codes the centralizer of any rational code is rational and if the code is finite then the centralizer is finitely generated this result has been previously proved only for binary and ternary sets of words in a series of papers by the authors and for prefix codes in an ingenious paper by b ratoandromanana rairo inform theor 23 4 1989 425 444 many of the techniques we use in this paper follow her ideas we also give in this paper an elementary proof for the prefix case

on the existence of prime decompositions we investigate factorizations of regular languages in terms of prime languages a language is said to be strongly prime decomposable if any way of factorizing it yields a prime decomposition in a finite number of steps we give a characterization of the strongly prime decomposable regular languages and using the characterization we show that every regular language over a unary alphabet has a prime decomposition we show that there exist non regular unary languages that do not have prime decompositions we also consider infinite factorizations of unary languages

conjugacy relations of prefix codes it is shown that if x and y are prefix codes and z is a non empty language satisfying the condition x z z y then z is the union of a non empty family p n i i of pairwise disjoint prefix sets such that x p i p i y for all i i consequently the conjugacy relations of prefix codes are explored and under the restriction that both of x and y are prefix codes the solutions of the conjugacy equation x z z y for languages are determined also the decidability of the conjugacy problem for finite prefix codes is confirmed

the commutation of finite sets a challenging problem we prove that given a set x of two nonempty words a set y of nonempty words commutes with x if and only if either y is a union of powers of x or x y subseteq t for some primitive word t we also show that the same holds for certain special types of codes but does not hold in general for sets of cardinality at least four

average case optimality for linear problems abstract we introduce an average case model and define general notions of optimal algorithm and optimal information we prove that the same algorithm and information are optimal in the worst and average cases and that adaptive information is not more powerful than non adaptive information

do linear problems have linear optimal algorithms a selective introduction to the field of information based complexity is presented in the context of the question raised in the title after introducing some of the basic ideas of this relatively new area that interfaces mathematics and theoretical computer science the article surveys results on the existence of linear optimal algorithms for linear problems several illustrative examples are given along with an indication of the variety of disciplines in which techniques of information based complexity are being applied known under aliases such as the general theory of optimal algorithms and the information centered approach to algorithms information based complexity seeks to create a general theory about problems with partial or approximate information and to apply the results to specific problems some of the disciplines where problems with such incomplete information have arisen include computer science economics control theory signal processing and geophysics the goal of this article is to present a selective introduction to the field of information based complexity and to describe the seesaw nature of results in the last decade on the extent to which linear problems have linear optimal algorithms the approach we take will certainly not do justice to the breadth of ideas techniques and applications that are coming into play in the general theory instead we will take a direct route to the results on linear optimal algorithms pausing occasionally to survey the pleasing way in which ideas from linear algebra and functional analysis are used in the final section we mention some other theoretical and applied topics currently under investigation in information based complexity a more general expository overview can be found in traub and wozniakowski 15 and more technical surveys appear in packel and wozniakowski 10 and wozniakowski 21 2 two preliminary examples we set the stage for an exposition of the general linear theory by imposing partial information on two problems which are known to virtually every student of mathematics the problem of integrating a continuous function needs no introduction the fourier series approach to approximating func tions by linear combinations of sines and cosines is almost as familiar we do not develop these ideas as a testimony to the power of the general theory more about that later but rather to have a comfortable base to refer to when the generalizations and abstractions are introduced integration example let f be a function from the space w 0 1 of real valued continuous functions on the interval 0 1 we would like to obtain a value for the

optimal error properties of finite element methods for second order elliptic dirichlet problems we use the informational approach of traub and wozniakowski 9 to study the variational form of the second order elliptic dirichlet problem lu f on q c rn for f g hw q where r 1 a quasi uniform finite element method using n linear functionals fs f4 has h q norm error j n r 1 n we prove that it is asymptotically optimal among all methods using any information consisting of any n linear functionals an analogous result holds if l is of order 2m if f g hr q where r m then there is a finite element method whose ha q norm error is n 2m r a n for 0 n 2 we show that when n 1 there is no method using n function evaluations whose error is better than q n r thus for n 1 the finite element method with quadrature is asymptotically optimal among all methods using n evaluations of f

on effective construction of the greatest solution of language inequality xa bx in this paper we consider effective constructions of the greatest solution of the language inequality xa bx it has been proved by kunc in 2005 that the greatest solution of xa bx is regular provided that b is regular no matter what a is however this proof is based on kruskal s tree theorem and does not provide any effective way to construct the greatest solution we focus on this gap in this paper we give an effective construction of the greatest solution for the following two cases i a b are regular and there exists k 1 such that pref b a k b kpref b a where pref b is the set of prefixes of words in b ii a b are regular and b is a code with finite decoding delay our construction takes the point of view of games as shown by kunc in his regularity proof the construction of the greatest solution can be reduced to the construction of the winning region of a two player game our contribution is to show that the winning regions of the two player game for the two cases can be constructed effectively the main ingredient of the construction for the first case is a shrinking lemma for the words on which one of the players has a winning strategy while the construction for the second case is based on the observation that the two player game can be reduced to a two player reachability game played on the transition graph of a one counter machine