natural frequencies of structures with interval parameters abstract this paper deals with the evaluation of the lower and upper bounds of the natural frequencies of structures with uncertain but bounded parameters the solution of the generalized interval eigenvalue problem is pursued by taking into account the actual variability and dependencies of uncertain structural parameters affecting the mass and stiffness matrices to this aim interval uncertainties are handled by applying the improved interval analysis via extra unitary interval eui recently introduced by the first two authors by associating an eui to each uncertain but bounded parameter the cases of mass and stiffness matrices affected by fully disjoint completely or partially coincident uncertainties are considered then based on sensitivity analysis it is shown that the bounds of the interval eigenvalues can be evaluated as solution of two appropriate deterministic eigenvalue problems without requiring any combinatorial procedure if the eigenvalues are monotonic functions of the uncertain parameters then the exact bounds are obtained the accuracy of the proposed method is demonstrated by numerical results concerning truss and beam structures with material and or geometrical uncertainties

improved parallel construction of wavelet trees and rank select structures abstract existing parallel algorithms for wavelet tree construction have a work complexity of o n log this paper presents parallel algorithms for the problem with improved work complexity our first algorithm is based on parallel integer sorting and has either o n log log n log log n log log n work and polylogarithmic depth or o n log log n work and sub linear depth we also describe another algorithm that has o n log log n work and o log n depth we then show how to use similar ideas to construct variants of wavelet trees arbitrary shaped binary trees and multiary trees as well as wavelet matrices in parallel with lower work complexity than prior algorithms finally we show that the rank and select structures on binary sequences and multiary sequences which are stored on wavelet tree nodes can be constructed in parallel with improved work bounds matching those of the best existing sequential algorithms for constructing rank and select structures

on parallel integer merging the problem of merging two sorted arrays a a1 a2 an1 and b b1 b2 bn2 is considered for input elements that are drawn from a domain of integers 1 s we present an algorithm that runs in o log log log s time using n log log log s crew pram processors optimal speed up and o ns space where n n1 n2 for input elements that are drawn from a domain of integers 1 n we present a second algorithm that runs in o n time where n is the inverse of ackermann s function using n n crew pram processors and linear space this second algorithm is non uniform however it can be made uniform at a price of a certain loss of speed or by using a crcw pram

the generalized eigenvalue problem for tridiagonal symmetric interval matrices an algorithm to compute the eigenintervals of the generalized eigenvalue problem for tridiagonal symmetric matrices a b with interval entries is developed the algorithm is based on the sturm sequences and requires twice as much computational effort as for point matrices the algorithm is applied to the problem of computing the natural frequencies of a vibrating beam where the manufacturing tolerance is taken into consideration

interval analysis of vibrating systems interval calculus is a tool to evaluate a mathematical expression for ranges of values of its parameters the basic mathematical operations are defined in the interval algebra vibrating systems having system parameters initial conditions or forcing functions defined as intervals rather than single valued quantities have been modelled using interval calculus the eigenvalue problem for determining natural frequencies and vibrating modes of a system in the case of system matrix elements given as a range of values cannot be solved by exhaustion due to the prohibitively large number of solutions of the point number eigenvalue problem eigenvalue solution with interval evaluation of the commonly used numerical techniques is not feasible because if they are applied directly the solution intervals diverge an optimization technique was used to obtain the minimum radius intervals of the solution for the eigenvalue sensitivity problem to assure monotonicity and absolute inclusion necessary for convergence to the exact interval a converging interval halving sequence was developed for finite width interval matrices for numerical tests of the method a monte carlo solution was developed the results showed that interval analysis can predict the range of the eigenvalues with sufficient accuracy the response of a linear system to general excitation for interval matrices of the system parameters cannot be found with interval evaluation of the commonly used numerical techniques because if they are applied directly the solution intervals diverge interval modal analysis and the interval solution of the eigenvalue problem were developed an application is presented for the dynamic response of a rotor with interval bearing properties

computing bounds to real eigenvalues of real interval matrices in this study a new method with algorithms for computing bounds to real eigenvalues of real interval matrices is developed the algorithms are based on the properties of continuous functions the method can provide the tightest eigenvalue bounds and improve some former research results numerical examples illustrate the applicability and effectiveness of the new method copyright 2007 john wiley sons ltd

a filtering method for the interval eigenvalue problem we consider the general problem of computing intervals that contain the real eigenvalues of interval matrices given an outer approximation superset of the real eigenvalue set of an interval matrix we propose a filtering method that iteratively improves the approximation even though our method is based on a sufficient regularity condition it is very efficient in practice and our experimental results suggest that it improves in general significantly the initial outer approximation the proposed method works for general as well as for symmetric interval matrices

improved deterministic parallel integer sorting abstract we consider the problem of deterministic sorting of integers on a parallel ram pram the best previous result t hagerup 1987 inform and comput 75 39 51 states that n integers of size polynomial in n can be sorted in time o log n on a priority crcw pram with o n log log n log n processors we prove that n integers drawn from a set 0 m 1 can be sorted on an arbitrary crcw pram in time o log n log log n log log m with a time processor product of o n log log m in particular if m n log n o 1 the time and number of processors used are o log n log log n and o n log log n 2 log n respectively this improves the previous result in several respects the new algorithm is faster it works on a weaker pram model and it is closer to optimality for input numbers of superpolynomial size if log log m o log n log log n the new algorithm is optimally fast for any polynomial number of processors and if log log m 1 1 log log n and log log m 0 log n it has optimal speedup relative to the fastest known sequential algorithm the space needed is o nme for arbitrary but fixed e 0 the sorting algorithm derives its speed from a fast solution to a special list ranking problem of possible independent interest the monotonic list ranking problem in monotonic list ranking each list element has an associated key and the keys are known to increase monotonically along the list we show that monotonic list ranking problems of size n can be solved optimally in time o log n log log n we also discuss and attempt to solve some of the problems arising in the precise description and implementation of parallel recursive algorithms as part of this effort we introduce a new pram variant the allocated pram

eigenvalue bounds of structures with uncertain but bounded parameters many analysis and design problems in engineering and science involve uncertainty to varying degrees this paper is concerned with the structural vibration problem involving uncertain material or geometric parameters specified as bounds on these parameters this produces interval stiffness and mass matrices and the problem is transformed into a generalized interval eigenvalue problem in interval mathematics however tighter bounds on the eigenvalues may be obtained by using the formulation of the structural dynamic problem often the stiffness and mass matrices can be formed as a non negative decomposition in the uncertain structural parameters in this case the eigenvalue bounds may be obtained from the parameter vertex solutions even more efficiently using interval extension from interval mathematics the generalized interval eigenvalue problem may be divided into two generalized eigenvalue problems for real symmetric matrix pairs the parameter vertex solution algorithm is compared with deif s solution the eigenvalue inclusion principle and the interval perturbation method in numerical examples

the extreme eigenvalues and stability of real symmetric interval matrices a novel khoritonov like algorithm for computing the minimal and maximal eigenvalues of n n dimensional symmetric interval matrices is presented it is proved that the maximal eigenvalue of a given set of interval matrices coincides with the maximal eigenvalue of a special set of 2 sup n 1 symmetric vertex matrices whereas its minimal eigenvalue coincides with the minimal of another special set of 2 sup n 1 symmetric vertex matrices as immediate corollaries of this algorithm weak necessary and sufficient conditions for testing the hurwitz and schur stability of symmetric interval matrices where one has to test the stability of 2 sup n 1 and 2 sup n symmetric vertex matrices respectively are obtained

improved fast integer sorting in linear space we present a fast deterministic algorithm for integer sorting in linear space our algorithm sorts n integers in the range 0 1 2 m 1 in linear space in o n log log n log log log n time when log m log2 n 0 we can further achieve o n log log n time this improves the o n log log n 2 time bound given in m thorup 1998 in proc 1998 acm siam symp on discrete algorithms soda 98 pp 550 555 this result is obtained by combining our new technique with that of thorup s signature sorting a andersson t hagerup s nilsson and r raman 1995 in proc 1995 symposium on theory of computing pp 427 436 a andersson s result 1996 in proc 1996 ieee symp on foundations of computer science pp 135 141 r raman s result 1996 lecture notes in computer science vol 1136 pp 121 137 springer verlag berlin new york and our previous result y han and x shen 1999 in proc 1999 tenth annual acm siam symposium on discrete algorithms soda 99 baltimore md january pp 419 428 are also used for the design of our algorithms we provide an approach and techniques which are totally different from previous approaches and techniques for the problem as a consequence our technique can be extended to apply to nonconservative sorting and parallel sorting our nonconservative sorting algorithm sorts n integers in 0 1 m 1 in time o n log log n 2 log k log log log n using word length k log m n where k log n our erew parallel algorithm sorts n integers in 0 1 m 1 in o log n 2 time and o n log log n 2 log log log n operations provided log m log n 2

optimal bounds for decision problems on the crcw pram optimal o log n log log n lower bounds on the time for crcw prams with polynomially bounded numbers of processors or memory cells to compute parity and a number of related problems are proven a strict time hierarchy of explicit boolean functions of n bits on such machines that holds up to o log n log log n time is also exhibited that is for every time bound t within this range a function is exhibited that can be easily computed using polynomial resources in time t but requires more than polynomial resources to be computed in time t 1 finally it is shown that almost all boolean functions of n bits require log n log log n o 1 time when the number of processors is at most polynomial in n the bounds do not place restrictions on the uniformity of the algorithms nor on the instruction sets of the machines

parallel depth first search in general directed graphs a directed cycle separator of an n vertex directed graph is a vertex simple directed cycle such that when the vertices of the cycle are deleted the resulting graph has no strongly connected component with more than n 2 vertices it is shown that the problem of finding a directed cycle separator is in randomized nc it is also proved that computing cycle separators and conducting depth first search in directed graphs are deterministically nc equivalent these two results together yield the first randomized nc algorithm for depth first search in general directed graphs

sufficient conditions on stability of interval matrices connections and new results the stability properties of interval matrices are discussed sufficient conditions are given for these properties based upon gershgorin s theorem and its extension and they are shown to improve previous results the derivation helps clarify relations between certain previously available results and shows they can be derived using gershgorin s theorem furthermore the problem of determining the stability of interval matrices is related to that of characterizing certain nonsingular m matrices

fuzzy finite element method for frequency response function analysis of uncertain structures a concept is presented for incorporating fuzzy uncertainties in dynamic e nite element analyses of uncertain structures the objective is twofold the e rst goal is to clarify and extend the classical fuzzy e nite element ffe method as it was introduced for static analyses the shortcomings of the classical approach are described and an extension to a generalized approach is proposed this generalized approach is proven to be a more realistic and therefore more reliable concept for taking uncertainty into account the second goal is to illustrate the applicability of the method for dynamic analyses the classical and the generalized approach are compared using an eigenvalue analysis of a simple numerical example the ffe method is also applied to the calculation of the total envelope frequency response function frf using the modal superposition principle this method requires safe approximations of the individual mode envelope frequency response functions for this purpose a number of safe approximate optimization strategies are introduced the numerical example shows that useful results are obtained using this ffe approach for frf calculations

hybrid uncertain natural frequency analysis for structures with random and interval fields abstract this paper presents a robust non deterministic free vibration analysis for engineering structures involving hybrid yet spatially dependent uncertain system parameters distinguished from the conventional hybrid uncertain eigenvalue problem the concept of interval field is enclosed with random field model such that both the stochastic and non stochastic representations of the spatial dependency of the uncertainties are simultaneously incorporated within a unified non deterministic free vibration analysis in order to determine the probabilistic characteristics i e means and standard deviations of the extremities of structural natural frequencies an extended unified interval stochastic sampling x uiss method is implemented for the purpose of effective hybrid uncertain free vibration analysis by meticulously blending sharpness promised interval eigenvalue analysis with stochastic sampling techniques the stochastic profiles i e probability density functions pdfs and the cumulative distribution functions cdfs of the extreme bounds of the structural natural frequencies can be rigorously established by utilizing the adequate statistical inference methods the applicability and effectiveness of the proposed computational framework are evidently demonstrated through the numerical investigations on various practically motivated engineering structures

polylog time and near linear work approximation scheme for undirected shortest paths shortest paths computations constitute one of the most fundamental network problems nonetheless known parallel shortest paths algorithms are generally inefficient they perform significantly more work product of time and processors than their sequential counterparts this gap known in the literature as the transitive closure bottleneck poses a long standing open problem our main result is an o mn e 0 s m n1 e 0 work polylog time randomized algorithm that computes paths within 1 o 1 polylog n of shortest from s source nodes to all other nodes in weighted undirected networks with n nodes and m edges for any fixed e 0 0 this work bound nearly matches the o d sm sequential time in contrast previous polylog time algorithms required nearly min o d n3 o d m2 work even when s 1 and previous near linear work algorithms required near o n time we also present faster sequential algorithms that provide good approximate distances only between distant vertices we obtain an o m sn n e 0 time algorithm that computes paths of weight 1 o 1 polylog n dist o w max polylog n where dist is the corresponding distance and w max is the maximum edge weight our chief instrument which is of independent interest are efficient constructions of sparse hop sets a d e hop set of a network g v e is a set e of new weighted edges such that mimimum weight d edge paths in v e e have weight within 1 e of the respective distances in g we construct hop sets of size o n1 e 0 where e o 1 polylog n and d o polylog n

optimal parallel algorithms on planar graphs abstract few existing parallel graph algorithms achieve optimality when applied to very sparse graphs such as planar graphs we describe optimal pram algorithms for the connected components spanning tree biconnected components and strong orientation problems that work on classes of undirected graphs including planar graphs and graphs of bounded genus the running times achieved for n vertex input graphs are o log n on the crcw pram and o log n log n on the erew pram we also give non optimal randomized erew pram algorithms using o log n time and n processors and non uniform deterministic erew pram algorithms using o log n time and o n 2 processors

an efficient parallel algorithm for the single function coarsest partition problem we describe an efficient parallel algorithm to solve the single function coarsest partition problem the algorithm runs in o log n time using o n log log n operations on the arbitrary crcw pram the previous best known algorithms run in o log2 n time using o n log2n operations on the crew pram and o log n time using o n log n operations on the arbitrary crcw pram our solution is based on efficient algorithms for solving several subproblems that are of independent interest in particular we present efficient parallel algorithms to find a minimal starting point of a circular string with respect to lexicographic ordering and to sort lexicographically a list of strings of different lengths

factorisation properties of group scheme actions let h be an algebraic group scheme over a field k acting on a commutative k algebra a which is a unique factorisation domain we show that under certain mild assumptions the monoid of nonzero h stable principal ideals in a is free commutative from this we deduce in certain special cases results about the monoid of nonzero semi invariants and the algebra of invariants we use an infinitesimal method which allows us to work over an arbitrary base field

an evolution strategy method for computing eigenvalue bounds of interval matrices in this paper a modified method by means of an evolution strategy es is introduced for computing eigenvalue bounds of interval matrices then a sufficient condition theorem of the evolutionary strategy is presented which guarantees the convergence in probability of the method the numerical examples show the method can effectively yield accurate bounds for interval eigenvalues

ellipsoidal bounds for uncertain linear equations and dynamical systems abstract in this paper we discuss semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations the proposed technique is based on the combination of a quadratic embedding of the uncertainty and the s procedure this formulation leads to convex optimization problems that can be essentially solved in o n3 n being the size of unknown vector by means of suitable interior point barrier methods as well as to closed form results in some particular cases we further show that the uncertain linear equations paradigm can be directly applied to various state bounding problems for dynamical systems subject to set valued noise and model uncertainty

stability analysis of dynamic interval systems abstract the stability analysis of interval matrices with certain stability margins is considered by means of similarity transformation and gershgorin s theorem new criteria are developed and illustrated by examples we also demonstrate a counter example to a recent result by argoun 1986

novel estimations for the eigenvalue bounds of complex interval matrices our work proposes two methods that estimate the eigenvalue bounds left right for real and imaginary parts of complex interval matrices the first method expresses each bound as an algebraic sum of weighted matrix measures where the measure corresponds to the spectral norm and the weighting matrix is diagonal and positive definite with unknown entries the optimization with respect to the entries of the weighting matrix yields the best value of the bound the computational approach is ensured as the minimization of a linear objective function subject to bilinear matrix inequality constraints and interval constraints the bounds are proved to be better than those provided by other estimation techniques the second method constructs four real matrices so that each of them can be exploited as a comparison matrix for the complex interval matrix and allows the estimation of one of the eigenvalue bounds the two methods we propose rely on different mathematical backgrounds and between the resulting bounds no firm inequality can be stated therefore they are equally useful in applications as reflected by the numerical case studies presented in the paper

an interval algorithm for uncertain dynamic stability analysis abstract analysis on the stability of dynamic systems is a very important field in structural elasticity theory for possible engineering situations where the system parameters are uncertain but bounded the propagation from the initial uncertainties of parameters to the consequent uncertainty of dynamic stability of structure is elaborately investigated by interval analysis for the first time the interval dynamic stability issue is studied through the column structure as a generalized interval eigenvalue problem and an analytic theorem is rigorously derived and numerically verified for solving the generalized interval eigenvalue problem moreover to avoid the exaggeration on actual bounds of solution an effective approach is established to limit the dependency phenomenon in the interval dynamic stability analysis the interval relationships of the excitation parameter and the critical load frequency with five types of system parameters are further bridged according to the interval arithmetic to propose a three stage interval scheme for evaluating the effects of interval system parameters on the dynamic stability of structures numerical studies demonstrate that the uncertainty of the constant part of load influences the boundaries of principal instability region much more than that of the variable part of load within all structural parameters the uncertainty of column length has the most effect on the boundaries which could make the critical frequency of load be magnified about ten times the uncertainty of parameter in particular if all the system parameters are interval with an identical uncertainty degree the uncertainty of parameter would be propagated in the system of dynamic stability and enlarged as high as 20 times the uncertainty of parameter consequently the impacts of parameter uncertainties on the dynamic stability of structures are fairly significant

extreme structural response analysis of truss structures under material uncertainty via linear mixed 0 1 programming this paper concerns the extreme structural response analysis of truss structures taking the possible uncertainties of young s moduli of the bars into consideration the uncertainty of young s modulus is described by a non probability but bounded interval model with the use of this uncertain model it is shown that the structural responses such as nodal displacement and bar stress will attain their extreme values when young s moduli of the bars take either their upper or lower bound value based on this conclusion and simultaneous analysis and design formulation a linear mixed 0 1 programming problem can be constructed to find the extreme values of the structural response by the well established branch and bound approach with global optimality the proposed approach is applied to several test problems for the demonstration of its validity

upper bounds for sorting integers on random access machines two models of random access machines suitable for sorting integers are presented our main results show that i a ram with addition subtraction multiplication and integer division can sort n integers in the range 0 2 cn in o n log c n steps ii a ram with addition subtraction and left and right shifts can sort any n integers in linear time iii a ram with addition subtraction and left and right shifts can sort n integers in the range 0 n c in o n log c n steps where all intermediate results are bounded in value by the largest input

dynamic perfect hashing upper and lower bounds the dynamic dictionary problem is considered provide an algorithm for storing a dynamic set allowing the operations insert delete and lookup a dynamic perfect hashing strategy is given a randomized algorithm for the dynamic dictionary problem that takes o 1 worst case time for lookups and o 1 amortized expected time for insertions and deletions it uses space proportional to the size of the set stored furthermore lower bounds for the time complexity of a class of deterministic algorithms for the dictionary problem are proved this class encompasses realistic hashing based schemes that use linear space such algorithms have amortized worst case time complexity omega log n for a sequence of n insertions and lookups if the worst case lookup time is restricted to k then the lower bound becomes omega k cdot n 1 k

a survey of non probabilistic uncertainty treatment in finite element analysis the objective of this paper is to critically review the emerging non probabilistic approaches for uncertainty treatment in finite element analysis the paper discusses general theoretical and practical aspects of both the interval and fuzzy finite element analysis first the applicability of the non probabilistic concepts for numerical uncertainty analysis is discussed from a theoretical viewpoint the necessary conditions for a useful application of the non probabilistic concepts are determined and are proven to be complementary rather than competitive to the classical probabilistic approach the second part of the paper focuses on numerical aspects of the interval finite element method it describes two principal strategies for the implementation i e the anti optimisation and the interval arithmetic approach and gives a state of the art of the interval finite element algorithms available from literature it is shown how the application of the interval arithmetic approach to the classical finite element procedure can result in a severe overestimation of the uncertainty on the output and the main sources of this conservatism are identified a numerical example in the final part of the paper illustrates the capabilities of the different strategies on an eigenfrequency analysis of a built up benchmark structure

the complexity of parallel prefix problems on small domains abstract we establish non trivial lower bounds for several prefix problems in the crcw pram model the chaining problem is given a binary input for each 1 in the input to find the index of the nearest 1 to its left our main result is that for an input of n bits solving the chaining problem using o n processors requires inverse ackerman time this matches the previously known upper bound we also give a reduction to show that the same lower bound applies to a parenthesis matching problem again matching the previously known upper bound we also give reductions to show that similar lower bounds hold for the prefix maxima and the range maxima problem

sorting strings and constructing digital search trees in parallel we describe two simple optimal work parallel algorithms for sorting a list l x1 x2 xm of m strings over an arbitrary alphabet where i 1m xi n and two elements of can be compared in unit time using a single processor the first algorithm is a deterministic algorithm that runs in o log2m log log m time and the second is a randomized algorithm that runs in o logm time both algorithms use o m log m n operations compared to the best known parallel algorithms for sorting strings our algorithms offer the following improvements 1 1 the total number of operations used by our algorithms is optimal while all previous parallel algorithms use a nonoptimal number of operations 2 2 we make no assumption about the alphabet while the previous algorithms assume that the alphabet is restricted to 1 2 no 1 3 3 the computation model assumed by our algorithms is the common crcw pram unlike the known algorithms that assume the arbitrary crcw pram 4 4 our algorithms use o m log m n space while the previous parallel algorithms use o n1 e space where e is a positive constant we also present optimal work parallel algorithms to construct a digital search tree for a given set of strings and to search for a string in a sorted list of strings we use our parallel sorting algorithms to solve the problem of determining a minimal starting point of a circular string with respect to lexicographic ordering our solution improves upon the previous best known result to solve this problem

improved parallel integer sorting without concurrent writing we show thatnintegers in the range 1 ncan be sorted stably on an erew pram usingo t time ando n lognloglogn logn 2 t operations for arbitrary givent lognloglogn and on a crew pram usingo t time ando n logn logn 2t logn operations for arbitrary givent logn in addition we are able to sortnarbitrary integers on a randomized crew pram within the same resource bounds with high probability in each case our algorithm is a factor of almost logn closer to optimality than all previous algorithms for the stated problem in the stated model and our third result matches the operation count of the best previous sequential algorithm we also show thatnintegers in the range 1 mcan be sorted ino logn 2 time witho n operations on an erew pram using a nonstandard word length ofo lognloglognlogm bits thereby greatly improving the upper bound on the word length necessary to sort integers with a linear time processor product even sequentially our algorithms were inspired by and in one case directly use the fusion trees of fredman and willard

second order cone programming second order cone programming socp problems are convex optimization problems in which a linear function is minimized over the intersection of an affine linear manifold with the cartesian product of second order lorentz cones linear programs convex quadratic programs and quadratically constrained convex quadratic programs can all be formulated as socp problems as can many other problems that do not fall into these three categories these latter problems model applications from a broad range of fields from engineering control and finance to robust optimization and combinatorial optimization on the other hand semidefinite programming sdp that is the optimization problem over the intersection of an affine set and the cone of positive semidefinite matrices includes socp as a special case therefore socp falls between linear lp and quadratic qp programming and sdp like lp qp and sdp problems socp problems can be solved in polynomial time by interior point methods the computational effort per iteration required by these methods to solve socp problems is greater than that required to solve lp and qp problems but less than that required to solve sdp s of similar size and structure because the set of feasible solutions for an socp problem is not polyhedral as it is for lp and qp problems it is not readily apparent how to develop a simplex or simplex like method for socp while socp problems can be solved as sdp problems doing so is not advisable both on numerical grounds and computational complexity concerns for instance many of the problems presented in the survey paper of vandenberghe and boyd vb96 as examples of sdps can in fact be formulated as socps and should be solved as such in 2 3 below we give socp formulations for four of these examples the convex quadratically constrained quadratic programming qcqp problem problems involving fractional quadratic functions rutcor rutgers university e mail alizadeh rutcor rutgers edu research supported in part by the u s national science foundation grant ccr 9901991 ieor columbia university e mail gold ieor columbia edu research supported in part by the department of energy grant de fg02 92er25126 national science foundation grants dms 94 14438 cda 97 26385 and dms 01 04282

modal analysis of structures with uncertain but bounded parameters via interval analysis in this paper the modal interval analysis method to estimate modal parameters frequency response function frf and mode shapes of structures with uncertain but bounded is presented although the system parameters or properties are uncertain in many engineering problems but their probable range of values i e upper and lower bounds can be provided from practical experience and engineering knowledge moreover to avoid the resonance of a structure and to consider the dynamic response of an uncertain one and also for reliability and stability analysis we often need the bounds of the ranges of structural characteristic parameters such as natural frequency and normal mode by using modal analysis and interval calculus we investigate the method of computing upper and lower bounds of paramenters such as natural frequencies modal shapes and frfs theoretically it is possible to analyze the uncertain but bounded of a structure by using modal analysis and interval calculus they can be estimated by modal interval analysis on the basis of the estimated intervals the engineering structure parameters can be applied into engineering design a numerical example is presented for a tower structure and the results illustrate that the proposed method is effective a comparison of the modal interval method with the results of monte carlo simulation serves to validate the solutions and to identify the bounded ranges of parameters

the zassenhaus variety of a reductive lie algebra in positive characteristic let g be the lie algebra of a connected reductive group g over an algebraically closed field k of characteristic p 0 let z be the centre of the universal enveloping algebra u u g of g its maximal spectrum is called the zassenhaus variety of g we show that under certain mild assumptions on g the field of fractions frac z of z is g equivariantly isomorphic to the function field of the dual space g with twisted g action in particular frac z is rational this confirms a conjecture of j alev furthermore we show that z is a unique factorisation domain confirming a conjecture of a braun and c hajarnavis recently a premet used the above result about frac z a result of colliot thelene kunyavskii popov and reichstein and reduction mod p arguments to show that the gelfand kirillov conjecture cannot hold for simple complex lie algebras that are not of type a c or g2

solution theorems for the standard eigenvalue problem of structures with uncertain but bounded parameters generalized eigenvalue problems from the modal analysis are often converted to the standard eigenvalue problems in this paper it evaluates the upper and lower bounds on the eigenvalues of the standard eigenvalue problem of structures subject to severely deficient information about the structural parameters here we focus on non probabilistic interval analysis models of uncertainty which are adapted to the case of severe lack of information on uncertainty non probabilistic interval analysis method in which uncertainties are defined by interval numbers appears as an alternative to the classical probabilistic models for the standard eigenvalue problem of structures with uncertain but bounded parameters the vertex solution theorem the positive semi definite solution theorem and the parameter decomposition solution theorem for the standard eigenvalue problem are presented and compared with deif s solution theorem in numerical examples it is shown that for the upper and lower bounds on the eigenvalues of the standard eigenvalue problem with uncertain but bounded parameters the presented vertex solution theorem is unconditional and the positive semi definite solution theorem and the parameter decomposition solution theorem have less limitary conditions compared with deif s solution theorem the effectiveness of the vertex solution theorem the positive semi definite solution theorem and the parameter decomposition solution theorem are illustrated by numerical examples

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

towards optimal parallel bucket sorting we present a simple deterministic parallel algorithm that runs on a crcw pram and sorts n integers of size polynomial in n in time o log n using o n log log nlog n processors it is closer to optimality than any previously known deterministic algorithm that solves the stated restricted sorting problem in polylog time

computation of bounds for eigenvalues of structures with interval parameters in this paper the computation of eigenvalue bounds for generalized interval eigenvalue problem is considered two algorithms based on the properties of continuous functions are developed for evaluating upper and lower eigenvalue bounds of structures with interval parameters the method can provide the tightest bounds within a given precision numerical examples illustrate the effectiveness of the proposed method

time work tradeoffs for parallel algorithms some parallel algorithms have the property that as they are allowed to take more time the total work that they do is reduced this paper describes several algorithms with this property these algorithms solve important problems on directed graphs including breadth first search topological sort strong connectivity and and the single source shorest path problem all of the algorithms run on the erew pram model of parallel computer except the algorithm for strong connectivity which runs on the probabilistic erew pram

zassenhaus varieties of general linear lie algebras abstract let g be a lie algebra over an algebraically closed field of characteristic p 0 and let u g be the universal enveloping algebra of g we prove in this paper for g gl n and g sl n that the centre of u g is a unique factorisation domain and its field of fractions is rational for g sl n our argument requires the assumption that p n while for g gl n it works for any p it turned out that our two main results are closely related to each other the first one confirms in type a a recent conjecture of a braun and c hajarnavis while the second answers a question of j alev

assessing the stability of linear time invariant continuous interval dynamic systems it is known that stability analysis of linear time invariant dynamic systems under parameter uncertainties can be equated to estimating the range of the eigenvalues of matrices whose elements are intervals in this note first the problem of finding tight outer bounds on the eigenvalue ranges is considered a method for computing such bounds is suggested which consists essentially of setting up and solving a system of n mildly nonlinear algebraic equations n being the size of the interval matrix investigated the main result of the note however is a method for determining the right end point of the exact eigenvalue ranges the latter makes use of the outer bounds it is applicable if certain computationally verifiable monotonicity conditions are fulfilled the methods suggested can be applied for robust stability analysis of both continuous and discrete time systems numerical examples illustrating the applicability of the new methods are also provided

optimal and sublogarithmic time randomized parallel sorting algorithms this paper assumes a parallel ram random access machine model which allows both concurrent reads and concurrent writes of a global memory the main result is an optimal randomized parallel algorithm for integer sort i e for sorting n integers in the range 1 n this algorithm costs only logarithmic time and is the first known that is optimal the product of its time and processor bounds is upper bounded by a linear function of the input size also given is a deterministic sublogarithmic time algorithm for prefix sum in addition this paper presents a sublogarithmic time algorithm for obtaining a random permutation of n elements in parallel and finally sublogarithmic time algorithms for general sort and integer sort are presented our sub logarithmic general sort algorithm is also optimal

real eigenvalue bounds of standard and generalized real interval eigenvalue problems abstract in this paper we study the real eigenvalue bounds of standard and generalized real interval eigenvalue problems based on some known sufficient conditions for the regularity of interval matrices a new algorithm for computing real eigenvalue bounds for real interval matrices is proposed it can easily be extended for applying to generalized real interval eigenvalue problems one advantage of the method is that the computation procedure takes less time than former methods we have proposed therefore it can be applied for solving larger interval eigenvalue problems finally some numerical examples are presented which show the effectiveness of the method

eigenvalues of tridiagonal symmetric interval matrices an algorithm for the determination of the eigenvalues of tridiagonal symmetric interval matrices is presented the intervals of the eigenvalues are not overestimated but exactly calculated the algorithm is an efficient one because it only needs twice as many operations as the sturm algorithm which is used for real matrices

robust optimization methodology and applications robust optimization ro is a modeling methodology combined with computational tools to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set the paper surveys the main results of ro as applied to uncertain linear conic quadratic and semidefinite programming for these cases computationally tractable robust counterparts of uncertain problems are explicitly obtained or good approximations of these counterparts are proposed making ro a useful tool for real world applications we discuss some of these applications specifically antenna design truss topology design and stability analysis synthesis in uncertain dynamic systems we also describe a case study of 90 lps from the netlib collection the study reveals that the feasibility properties of the usual solutions of real world lps can be severely affected by small perturbations of the data and that the ro methodology can be successfully used to overcome this phenomenon

necessary and sufficient conditions for stability of a class of interval matrices in this paper some necessary and sufficient conditions for stability complete instability and composite stability type of a class of interval matrices are given

semidefinite programming for dynamic steady state analysis of structures under uncertain harmonic loads this paper presents semidefinite programming approaches for computing confidential bounds for the dynamic steady state responses of a damped structure subjected to uncertain driving loads we assume that amplitudes of harmonic driving loads obey a non probabilistic uncertainty model the semidefinite programming problems are formulated for finding confidential bounds for various characteristic amounts of dynamic steady state response including the modulus and phase angle of the complex amplitude of the displacement and stress numerical examples demonstrate that sufficiently tight bounds can be obtained by solving the presented semidefinite programming problems

simple fast parallel hashing by oblivious execution a hash table is a representation of a set in a linear size data structure that supports constant time membership queries we show how to construct a hash table for any given set of n keys in o lg lg n parallel time with high probability using n processors on a weak version of a concurrent read concurrent write parallel random access machine crcw pram our algorithm uses a novel approach of hashing by oblivious execution based on probabilistic analysis the algorithm is simple and has the following structure partition the input set into buckets by a random polynomial of constant degree for t 1 to o lg lg n do allocate mt memory blocks each of size kt let each bucket select a block at random and try to injectively map its keys into the block using a random linear function buckets that fail carry on to the next iteration the crux of the algorithm is a careful a priori selection of the parameters mt and kt the algorithm uses only o lg lg n random words and can be implemented in a work efficient manner

an algorithm to determine linear independence of a set of interval vectors this paper presents an algorithm to determine the linear dependence and independence of a set of interval vectors the algorithm is developed on the basis of a necessary and sufficient condition for linear independence of an arbitrary interval vector relative to other linearly independent interval vectors the necessary and sufficient condition can be checked by a linear programming method then the developed necessary and sufficient condition will be employed to check the linear independence of a set of interval vectors through numerical examples the validity of the developed algorithm is illustrated

on sufficient conditions for the stability of interval matrices using gershgorin s diagonal dominance theorem new sufficient conditions for the stability of interval matrices are obtained the new conditions remove a restriction in heinen s 1983 analysis and are applicable to a wider class of problems than previously possible

equivalence between priority queues and sorting we present a general deterministic linear space reduction from priority queues to sorting implying that if we can sort up to n keys in s n time per key then there is a priority queue supporting delete and insert in o s n time and find min in constant time conversely a priority queue can trivially be used for sorting first insert all keys to be sorted then extract them in sorted order by repeatedly deleting the minimum asymptotically this settles the complexity of priority queues in terms of that of sorting previously at soda 96 such a result was presented by the author for the special case of monotone priority queues where the minimum is not allowed to decrease besides nailing down the complexity of priority queues to that of sorting and vice versa our result yields several improved bounds for linear space integer priority queues with find min in constant time deterministically we get o log log n update time using a sorting algorithm of han from stoc 02 this improves the o log log n log log log n update time of han from soda 01 randomized we get o slog log n expected update time using a randomized sorting algorithm of han and thorup from focs 02 this improves the o log log n expected update time of thorup from soda 96 deterministically in ac0 without multiplication for any e 0 we get o log log n 1pe update time using an ac0 sorting algorithm of han and thorup from focs 02 this improves the o log log n 2 update time of thorup from soda 98 randomized in ac0 we get o log log n expected update time using a randomized ac0 sorting algorithm of thorup from soda 97 this improves the o log log n 1pe expected update time of thorup also from soda 97 the above bounds assume that each integer is stored in a single word and that word operations take unit time as in the word ram

dynamic ordered sets with exponential search trees we introduce exponential search trees as a novel technique for converting static polynomial space search structures for ordered sets into fully dynamic linear space data structures this leads to an optimal bound of o slog n log log n for searching and updating a dynamic set x of n integer keys in linear space searching x for an integer y means finding the maximum key in x which is smaller than or equal to y this problem is equivalent to the standard text book problem of maintaining an ordered set the best previous deterministic linear space bound was o log n log log n due to fredman and willard from stoc 1990 no better deterministic search bound was known using polynomial space we also get the following worst case linear space trade offs between the number n the word length w and the maximal key u our results are generalized to finger searching and string searching providing optimal results for both in terms of n

eigenvalue bounds for symmetric matrices with entries in one interval in this paper we consider the eigenvalue bounds for real symmetric matrices whose entries are in a given interval based on some matrix computation theories a new algorithm and some theorems are presented which provide ways for solving the left problems in former papers and obtain satisfying results furthermore the ideas are spread to general symmetric interval matrices finally numerical examples are presented which show the validity of the proposed methods