some exact solutions for the bending of beams with spatially stochastic stiffness to the best of the knowledge of the authors there are no exact solutions available for the bending of beams with spatially stochastic stiffness investigators are therefore utilizing various approximate techniques in the present study a new method is developed to obtain exact solutions for first and second moments of displacements for statically determinate beams that have spatially random stiffness the method is based on the full probabilistic characterization of the random stiffness so that the solutions are valid for any value of the coefficient of variation of the stiffness the deterministic governing equations and boundary conditions derived for both first and second moments allow apparently for the first time in the literature the exact solutions for the mean and covariance functions of the displacements to be determined two governing equations are uncoupled from each other and can be solved separately several exact solutions for the mean and covariance functions are obtained to illustrate the application of the method it is hoped that the exact solutions determined will serve as benchmark solutions to enable the researchers to check the accuracy of various approximate analytical and numerical techniques on the test solutions presented in this study

a uniform approach to obtain diagonal sets in complexity classes abstract a uniform method for constructing sets which diagonalize over certain complexity classes while preserving other complexity properties is given we obtain some known results as well as some new ones as corollaries of our main theorem the new results concern the complexity classes p np co np pspace apt almost polynomial time r random polynomial time and the polynomial hierarchy

sublattices of the polynomial time degrees we show that any countable distributive lattice can be embedded in any interval of polynomial time degrees furthermore the embeddings can be chosen to preserve the least or the greatest element this holds for both polynomial time bounded many one and turing reducibilities as well as for all of the common intermediate reducibilities

3d inverse scattering abstract we give a self contained presentation of the method developed by r newton for solving the 3d inverse scattering problem and give an analysis of what is proved and what is not proved in this theory

multidimensional inverse problems and completeness of the products of solutions to pde a method is given for proving uniqueness theorems for some inverse problems the method is based on a result on completeness of the products of solutions to pde as an example the following uniqueness theorems are proved 1 the scattering amplitude a k known for all s2 and a fixed k 0 determines the compactly supported q x l2 d uniquely 2 the surface data u x y k known for all x y p x x3 0 and a fixed k 0 determine the compactly supported x l2 d d r 3 x x3 0 is arbitrarily small determine aj x j 1 2 uniquely here 2u k2u k2a1x a2 x u x y in r3 a1 l2 d a2 h2 d the same conclusion holds if the surface data are known at two distinct frequencies 4 the surface data u x y k known for all x y p and all k 0 determine v x and h k uniquely here 2 k2 k2 x u x y h k x l2 d h k is fourier transform of a wavelet of compact support 5 the conductivity x w2 2 d x c 0 is uniquely determined by the measurements of u and un on d here n is the outward normal to d d r3 is a bounded domain with a smooth boundary d x u 0 in d 6 necessary and sufficient conditions are given for a function a k s2 k 0 is fixed to be the scattering amplitude corresponding to a local potential from a certain class

new formulation of fem for deterministic and stochastic beams through generalization of fuchs approach this paper proposes an alternative way of constructing the global stiffness matrix of the finite element method for bending beams it also applies the new formulation to first and second moment analysis of stochastic beams which involve spatially uncertain bending stiffness originating from fuchs idea of decoupling the shear and bending components in the bending beam the element level stiffness matrix is diagonalized the generalized stress strain strain displacement and equilibrium relationships are assembled respectively and then are combined to form the global stiffness matrix the advantage of the new formulation is that the bending stiffness explicitly appears in the global stiffness matrix the mean vector and covariance matrix of the displacement of the beam are then obtained in terms of probabilistic characteristics of the uncertain bending stiffness this is in contrast to the conventional finite element method in stochastic setting which is based on the perturbation technique the example is given to illustrate the efficacy of the new formulation and its application to bending of stochastic beams

explicit optimum design abstract the optimization of a trussed type structure of given geometry and material properties can be formulated as an exact and explicit mathematical programming problem in a mixed space of design variables and behaviour variables three techniques are presented corresponding to the three classical analysis methods of structural theory in the case of a single loading condition without variables linking the proposed method is very efficient since it eliminates the problem of multiple reanalysis without increasing the dimensionality of the problem and the number of constraints in the other cases the numerical efficiency of the technique depends on the specific problem to be solved

necessary and sufficient condition for l to have property c abstract let l m j 0aj j be a formal differential operator with constant coefficients in r n n 2 j is a multi index we say that l has property c iff for any bounded domain d r n the set of products uw u w nd l u l u 0 in d u hm d is complete in l2 d here hm d is the sobolev space a necessary and sufficient condition is given for l to have property c

unimodal beam elements abstract a uniform prismatic elastic element in small displacements theory can be modeled as consisting of extension torsion and flexural elements acting in parallel the extension and torsion elements are unimodal whereas the flexural elements are bimodal due to the combined action of bending moments and shear forces this dissimilarity between the elements is the source of inconsistency between the analysis of a truss and a framed structure especially in a force approach it is shown that by an appropriate change of basis one can uncouple the flexural element into a unimodal moment element which carries the average moment of the element and a unimodal shear element which carries the shear force and related moment as a result any framed structure can be viewed as a generalized truss and analyzed accordingly in a standard form it is further shown that it is possible to physically construct a moment element and a shear element both exhibiting unimodal deformation patterns consequently one can replace a classical beam element by a moment element and a shear element assembled in parallel the approach is tested numerically in the case of beams of constant height subjected to several loading conditions preliminary results indicate that in theory substantial weight reductions can be obtained when designing structures composed of parallel unimodal elements

structural properties of bounded relations with an application to np optimization problems we introduce the notion bounded relation which comprises most resource bounded reducibilities which can be found in the literature including non uniform bounded reducibilities such as tp poly we state conditions on bounded relations which are again satisfied for most bounded reducibilities and which imply that every countable partial ordering can be embedded into every proper interval of the recursive degrees as corollaries we obtain that every countable partial ordering can be embedded into every proper interval of rec tp poly as well as into every proper interval between either two maximization or two minimization problems in the structures npo e and npo l for the two latter structures we show further that the result about embeddings of partial orderings extends to embeddings of arbitrary countable distributive lattices where in addition the least or the greatest element of the lattice can be preserved among other corollaries we obtain that for both structures every non trivial np optimization problem bounds a minimal pair in connection with embeddings into npo we introduce a representation of maximization or minimization problems by polynomial time computable functions we consider decidability issues w r t this representation and show for example that there is no effective procedure which decides for a subrecursive index of a polynomial time computable function whether the corresponding maximization problem is approximable within a constant factor

stochastic reduced basis methods stochastic reduced basis methods for solving large scale linear random algebraic systems of equations such as those obtained by discretizing linear stochastic partial differential equations in space time and the random dimension are introduced the fundamental idea employed is to represent the system response using a linear combination of stochastic basis vectors with undetermined deterministic coefficients or random functions we present a theoretical justification for employing basis vectors spanning the preconditioned stochastic krylov subspace to approximate the response process subsequently variants of the bubnov galerkin scheme are employed to compute the undetermined coefficients which allow explicit expressions for the response quantities to be derived we also examine some theoretical properties of the projection scheme and procedures for computing the response statistics numerical studies are presented for static and dynamic analysis of stochastic structural systems we demonstrate that significant improvements over the neumann expansion scheme as well as other relevant techniques in the literature can be achieved

formula for the radius of the support of the potential in terms of scattering data let q r r x be a real valued square integrable compactly supported function and be the smallest interval containing the support of q r let be the corresponding scattering amplitude at a fixed positive energy let be the phase shifts at k 1 it is proved that provided that q r does not change sign in some arbitrary small neighbourhood of a

canonical disjoint np pairs of propositional proof systems we prove that every disjoint np pair is polynomial time many one equivalent to the canonical disjoint np pair of some propositional proof system therefore the degree structure of the class of disjoint np pairs and of all canonical pairs is identical we show that this degree structure is not superficial assuming there exist p inseparable disjoint np pairs every countable distributive lattice can be embedded into every interval of polynomial np degrees of disjoint pairs by maps that preserve the least and greatest element respectively as one consequence of this embedding under the same assumption there exist intermediate disjoint np pairs that is if a b is a p separable disjoint np pair and c d is a p inseparable disjoint np pair then there exist p inseparable incomparable np pairs e f and g h whose degrees lie strictly between a b and c d furthermore between any two disjoint np pairs that are comparable and inequivalent such a diamond exists

approximate reanalysis based on the exact analytic expressions the first author has recently given the exact analytic expressions of the inverse of the stiffness matrix the nodal displacements and the stress resultants in linear elastic structures composed of prismatic elements for structures of constant geometry the expressions are explicit in terms of the unimodal stiffnesses of the components of the structures however the expressions are intractable in their exact form due to their inordinate length it all has to do with the number of statically determinate substructures embedded in common engineering structures this paper describes some preliminary results obtained from approximate analysis models for the internal forces using truncated expressions that are similar in form to the exact analytic ones the approach is illustrated with numerical examples he classical methods of structural analysis methods are numerical techniques that compute the structural response to external loads for a given configuration of the structure and for given stiffnesses of its elements very often engineers are faced with the need to reanalyze structures for the same geo metrical configuration and for the same applied loads but with modified stiffnesses such instances occur in structural design and in particular in optimal structural design where one usually tests many candidate stiffness distributions before an acceptable structure is found 1 structural reanalysis is also present in the analysis of framed structures when the p d effect is accounted for 2 since the axial load in a member modifies its flexural rigidity such structures are solved itera tively at a given iteration the flexural stiffnesses of the members are computed based on the axial loads of the previ ous iteration the procedure is repeated until convergence when the stiffnesses of the individual members change the analysis procedure must in principle be repeated with the modified parameters there is no formula for the structural response in terms of the rigidities of its members to alleviate the computational cost of multiple reanalysis engineers have over the years developed reanalysis techniques that use results of previous analysis to predict the structural response when some or all of the rigidities are modified 3 4 noteworthy is the reciprocal approximation 1 which expresses the nodal dis placements as a linear taylor polynomial in terms of the reciprocals of the member stiffnesses recently the first author has written a series of papers 5 7 describing the analytic solution of the structural analysis prob lem for structures composed of uniform prismatic members of given geometry and assuming linear elastic behavior the analytic expressions are exact equations for the inverse of the stiffness matrix for the nodal displacements and for the internal forces in the members explicit in terms of the cross sectional structural characteristics of the components of the structure however what seems to be eldorado for the struc tural designer an exact formula for the structural response

minimal pairs for p abstract we prove a general minimal pair theorem which yields as corollaries many results about minimal pairs for p obtained previously by various authors furthermore it yields the new counterintuitive result that there exist arbitrarily complex minimal pairs we also investigate the question of whether minimal pairs in np are low

uniqueness theorems for multidimensional inverse problems with unbounded coefficients abstract in a g ramm inverse problems 3 1987 l77 l82 a method to prove uniqueness theorems is given the method uses an estimate of the solution to a partial differential equation with l d coefficients here the estimate is established for l2 d coefficients and the resulting uniqueness theorems are formulated

gap languages and log time complexity classes abstract this paper shows that classical results about complexity classes involving delayed diagonalization and gap languages such as ladner s theorem and schoning s theorem and independence results of a kind noted by schoming and hartmanis apply at very low levels of complexity indeed all the way down in sipser s log time hierarchy this paper also investigates refinements of sipser s classes and notions of log time reductions following on from recent work by cai chen and others

an asymptotic behavior of the s matrix and the inverse scattering problem an asymptotic formula of the s matrix as k will be shown for the general short range scattering in rn n 2 the main tool for the proof is the spectral decomposition theorem for the schrodinger operator by the use of the asymptotic formula the uniqueness of the inverse scattering problem will be shown and a reconstruction formula for the potential will be presented for the general short range scattering

lattice embeddings for abstract bounded reducibilities we give an abstract account of resource bounded reducibilities as exemplified by the polynomially time or logarithmically space bounded reducibilities of turing truth table and many one type we introduce a small set of axioms that are satisfied for most of the specific resource bounded reducibilities appearing in the literature some of the axioms are of a more algebraic nature such as the requirement that the reducibility under consideration is a reflexive relation while others are formulated in terms of recursion theory and for example are related to delayed computations of arbitrary recursive sets the main technical result shown is that for any reducibility that satisfies these axioms every countable distributive lattice can be embedded into any proper interval of the structure induced on the recursive sets this extends a corresponding result for polynomially time bounded reducibilities due to ambos spies inform and control 65 1985 pp 63 84 as well as a result on embeddings of partial orderings for axiomatically described reducibilities due to mehlhorn j comput system sci 12 1976 pp 147 178

the recursion theoretic structure of complexity classes abstract we prove easy recursion theoretic results which have as corollaries generalizations of existing diagonalization theorems on complexity classes roughly speaking almost no reasonable time space or even abstract complexity class can be expressed as the non trivial union of two recursively presentable classes which are closed under finite variations e g unless np p np p np complete languages and consequently the non trivial complement of one complexity class in another e g npp provided np p is almost never recursively presentable

symmetry properties of scattering amplitudes and applications to inverse problems abstract for the transformations of the potential q q l where l is one of the following transformations rotation scaling or translation the corresponding transformation of the scattering amplitude is given a necessary and sufficient condition on the scattering amplitude for the scatterer to be spherically symmetric is derived as an application of the above results

a method to derive approximate explicit solutions for structural mechanics problems the availability of explicit solutions i e analytical relationships between the structural response and the design variables allows a more direct and plain treatment of several structural problems this paper is devoted to derive approximate explicit solutions in the framework of linear static analysis of finite element modeled structures with a given layout fixed node positions the proposed procedure is based on a factorization of the element stiffness matrix following the unimodal components concept which allows a non conventional assembly of the global stiffness matrix the exact inversion of that matrix is a trivial task for the case of statically determinate structures structures with few redundancies or few design variables an approximate inverse of the stiffness matrix is herein derived for more general structural problems by resorting to the sherman morrison woodbury formula 2006 elsevier ltd all rights reserved

response variability due to randomness in poisson s ratio for plane strain and plane stress states summary in the ordinary structural materials one of the parameters that can be assumed to have spatial uncertainty is poisson s ratio therefore the independent evaluation of the effects of this parameter on the response variability is of importance the difficulties in obtaining the response variability due to randomness in poisson s ratio lie in the fact that the poisson s ratio enters the stiffness matrix as a non linear parameter in this paper a formulation to determine the response variability in plane strain and plane stress states due to the randomness in the poisson s ratio is given the formulation is accomplished by means of the stochastic decomposition of the constitutive matrix into several sub matrices taking into consideration of the polynomial expansion on the coefficients of constitutive relation to demonstrate the validity of the proposed formulation some example structures are chosen and the results are compared with those obtained by means of monte carlo simulation through the formulation proposed in this study it becomes possible for the weighted integral stochastic finite element analysis to consider all the uncertain material parameters in its application

analytic representation of member forces in linear elastic redundant trusses abstract the process of designing a truss of given geometry and material properties where the design variables are the cross sectional areas of the bars is hampered by the need to reanalyse the structure many times until an acceptable design is obtained currently approximate explicit analysis models based on truncated linear taylor series expansions are used to evaluate the structural response at the various candidate design points due to the approximate nature of the analysis model the structure is designed iteratively until convergence of both the analysis equations and the design process this paper presents for the first time the exact analytic expressions of the internal loads in a truss which is subjected to static loads the stress resultants are the ratio of two multilinear polynomials in the element stillness the number of terms of the polynomials is equal to the number of combinations of statically determinate stable structures which can be derived from the original structures the coefficients of the polynomial expansions can be obtained from equilibrium considerations and from enforcing global compatibility of deformations the expressions are explicit in both the external loads and the element stiffness the applicability of the analytic equations hinges on the number of combinations of statically determinate stable substructures in the case of small size structures the present explicit equations circumvent the need for approximate reanalysis in common engineering structures the number of stable subsets is prohibitively large which renders the analytic expressions intractable the exact analytic expressions may however constitute a starting point for constructing approximate explicit analysis equations of improved quality