rings of invariants for modular representations of the klein four group we study the rings of invariants for the indecomposable modular representations of the klein four group for each such representation we compute the noether number and give minimal generating sets for the hilbert ideal and the field of fractions we observe that with the exception of the regular representation the hilbert ideal for each of these representations is a complete intersection

on the castelnuovo mumford regularity of rings of polynomial invariants we show that when a group acts on a polynomial ring over a eld the ring of invariants has castelnuovo mumford regularity at most zero as a consequence we prove a well known conjecture that the invariants are always generated in degrees at most n jgj 1 where n 1 is the number of polynomial generators andjgj 1 is the order of the group we also prove some other related conjectures in invariant theory

invariants of the diagonal cp action on v3 abstract let c p denote the cyclic group of order p where p 3 is prime we denote by v 3 the indecomposable three dimensional representation of c p over a field f of characteristic p we compute a set of generators in fact a sagbi basis for the ring of invariants f v 3 v 3 c p our main result confirms the conjecture of shank r j shank classical covariants and modular invariants in h e a campbell d l wehlau eds invariant theory in all characteristics crm proc lecture notes vol 35 amer math soc 2004 pp 241 249 for this example that all modular rings of invariants of c p are generated by rational invariants norms and transfers

an arbitrary polynomial chaos expansion approach for response analysis of acoustic systems with epistemic uncertainty abstract by introducing the arbitrary polynomial chaos theory the evidence theory based arbitrary polynomial chaos expansion method etapcem is proposed to improve the computational accuracy of polynomial chaos expansion methods for the evidence theory based analysis of acoustic systems with epistemic uncertainty in etapcem the epistemic uncertainty of acoustic systems is treated with evidence theory the response of acoustic systems in the range of variation of evidence variables is approximated by the arbitrary polynomial chaos expansion through which the lower and upper bounds of the response over all focal elements can be efficiently calculated by a number of numerical solvers inspired by the application of polynomial chaos theory in the interval and random analysis the weight function of the optimal polynomial basis of etapcem for evidence theory based uncertainty analysis is derived from the uniformity approach compared with the conventional evidence theory based polynomial chaos expansion methods including the recently proposed evidence theory based jacobi expansion method the main advantage of etapcem is that the polynomial basis orthogonalized with arbitrary weight functions can be obtained to construct the polynomial chaos expansion thereby the optimal polynomial basis of polynomial chaos expansion for arbitrary types of the evidence variable can be established by using etapcem the effectiveness of the proposed method for acoustic problems has been fully demonstrated by comparing it with the conventional evidence theory based polynomial chaos expansionmethods

vector invariants for the two dimensional modular representation of a cyclic group of prime order in this paper we study the vector invariants of the 2 dimensional indecomposable representation v2 of the cyclic group cp of order p over a field f of characteristic p f mv2 cp this ring of invariants was first studied by david richman 1990 20 who showed that the ring required a generator of degree m p 1 thus demonstrating that the result of noether in characteristic 0 that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group does not extend to the modular case he also conjectured that a certain set of invariants was a generating set with a proof in the case p 2 this conjecture was proved by campbell and hughes 1997 in 3 later shank and wehlau 2002 in 24 determined which elements in richman s generating set were redundant thereby producing a minimal generating set we give a new proof of the result of campbell and hughes shank and wehlau giving a minimal algebra generating set for the ring of invariants f mv2 cp in fact our proof does much more we show that our minimal generating set is also a sagbi basis for f mv2 cp further our results provide a procedure for finding an explicit decomposition of f mv2 into a direct sum of indecomposable cp modules finally noting that our representation of cp on v2 is as the p sylow subgroup of sl2 fp we describe a generating set for the ring of invariants f mv2 sl2 fp and show that p m 2 p 1 is an upper bound for the noether number for m 2

reliability analysis of structures with interval uncertainties under stationary stochastic excitations reliability analysis of linear discretized structural systems with interval uncertainties subjected to stationary gaussian random excitation is addressed under the assumption of independent up crossings of a specified threshold an efficient procedure for the evaluation of the bounds of the interval reliability function of the generic response process is presented the first step of the proposed approach is to derive approximate expressions of the interval mean value and spectral moments of the response along with the associated bounds to this aim the improved interval analysis via extra unitaryinterval is applied in conjunction with a novel series expansion of the inverse of an interval matrix with modifications called interval rational series expansion irse then the lower bound and upper bounds of the interval reliability function are readily evaluated by properly combining the bounds of the interval mean value and spectral moments of the response two numerical examples are provided to demonstrate the accuracy of the proposed procedure and its usefulness in view of decision making in engineering practice

a novel interval finite element method based on the improved interval analysis abstract static analysis of linear elastic structures with uncertain parameters subjected to deterministic loads is addressed the uncertain structural properties are modeled as interval variables with assigned lower bound and upper bound a novel interval finite element method is formulated in the framework of the improved interval analysis via extra unitary interval recently proposed to limit the conservatism affecting the classical interval analysis the key idea of the novel method is to associate an extra unitary interval to each uncertain parameter in order to keep physical properties linked to the finite elements in both the assembly and solution phases this allows one to reduce overestimation and perform standard assembly of the interval element matrices the lower bound and upper bound of interval displacements and stresses are evaluated by applying two different strategies both based on the so called interval rational series expansion for deriving the approximate explicit inverse of the interval global stiffness matrix numerical examples concerning 2d and 3d structures with uncertain young s modulus are presented to demonstrate the accuracy and efficiency of the proposed procedure

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 noether bound in invariant theory of finite groups abstract let r be a commutative ring v a finitely generated free r module and g glr v a finite group acting naturally on the graded symmetric algebra a sym v let ag denote the minimal number m such that the ring ag of invariants can be generated by finitely many elements of degree at most m furthermore let h g be a normal subgroup such that the index g h is invertible in r in this paper we prove the inequality a g a h g h for h 1 and g invertible in r we obtain noether s bound ag g which so far had been shown for arbitrary groups only under the assumption that the factorial of the group order g is invertible in r

coinvariants and the regular representation of a cyclic p group we consider an indecomposable representation of a cyclic p group z p r over a field of characteristic p we show that the top degree of the corresponding ring of coinvariants is less than frac r 2 3r p r 2 this bound also applies to the degrees of the generators for the invariant ring of the regular representation

the encounter of interval and probabilistic approaches to structural reliability at the design point abstract the consideration of uncertainties in structural analysis has traditionally been made in the field of structural reliability within the framework of classical probability theory in many cases however there is no sufficient probabilistic information to substantiate such an approach in recent years several non probabilistic methods have been proposed as alternatives one of them is the interval analysis oriented to estimate the range of variation of output structural variables given the ranges of the input ones the method however requires the use of special arithmetics which renders its application unpractical for structural analysis based on finite elements or other numerical techniques in this paper a monte carlo method that can be applied to probabilistic as well as to interval approaches to reliability analysis is proposed it is based on the optimal properties of the classical design point of probabilistic structural reliability in particular it is shown that the order statistics of the output variable is concealed in a plot defined by the design point vector this facilitates the selection of the relevant samples for either interval or probability based analysis the examples show that the desired interval of the response or the failure probability can be accurately estimated on such a basis the paper also discusses the use of monte carlo methods for both reliability and interval analysis from the point of view of the proposed representation

probabilistic interval reliability of structural systems the probabilistic reliability approach is the most widely used method for reliability analysis the recent research shows that the reliabilities of structural systems strongly depend on the parameters of the probability model it is possible that the little error in the estimation of the parameters may lead to the remarkable error of the resulting probability in this study we introduce the interval approach into the conventional reliability theory we present a novel approach which allows us to obtain the system failure probability interval from the statistical parameter intervals of the basic variables this approach is a combination of the two techniques namely the classical reliability theory and the interval analysis in the end of this paper we show the feasibility of the proposed approach through two examples of the truss systems

an exact solution for the static and dynamic analysis of fe discretized uncertain structures the probabilistic analysis of fe discretized uncertain linear structures in the static and dynamic domain is the goal of this paper in particular a procedure able to give the exact relationship between the response and the random variables representing the structural uncertainties is presented under the assumption that a point discretization method is used for the representation of the uncertain random field this procedure is based on the properties of the structural deformation modes and in particular on the number of principal deformation modes of the fe type used for the structural discretization the use of these relationships leads to the exact probabilistic characterization of the response once that the probabilistic description of the uncertainty is fixed moreover they give the approximate relationship when only some of the random variables chosen by a sensitivity procedure are included in the analysis this allows to obtain an optimum level of accuracy through a reduced computational effort

unified polynomial expansion for interval and random response analysis of uncertain structure acoustic system with arbitrary probability distribution abstract for structure acousticsystem with uncertainties the interval model the random model and the hybrid uncertain model have been introduced in the interval model and the random model the uncertain parameters are described as either the random variable with well defined probability density function pdf or the interval variable without any probability information whereas in the hybrid uncertain model both interval variable and random variable exist simultaneously for response analysis of these three uncertain models of structure acoustic problem involving arbitrary pdfs a unified polynomial expansion method named as the interval and random arbitrary polynomial chaos method irapcm is proposed in irapcm the response of the structure acoustic system is approximated by apc expansion in a unified form particularly only the weight function of polynomial basis is required to be changed to construct the apc expansion for the response of different uncertain models through the unified apc expansion the uncertain properties of the response of three uncertain models can be efficiently obtained as the apc expansion can provide a free choice of the polynomial basis the optimal polynomial basis for the random variable with arbitrary pdfs can be obtained by using the proposed irapcm the irapcm has been employed to solve a mathematical problem and a structure acoustic problem and the effectiveness of the unified irapcm for response analysis of three uncertain models is demonstrated by fully comparing it with the hybrid first order perturbation method and several existing polynomial chaos methods

inflation centres of the cut and project quasicrystals a rather general form of the conventional cut and project scheme is used to define quasicrystals as point sets in real n dimensional euclidean space the inflation or equivalently the self similarity properties of such quasicrystals are studied here assuming only the convexity of the acceptance window our result is a description of inflation centres of all types in a quasicrystal and a proof that our description is complete there are no other inflation centres for any chosen quasicrystal point internal inflation centre u its inflation properties are given as a set of scaling factors it turns out that the scaling factors form a one dimensional quasicrystal with a u dependent acceptance window scaling window the intersection of the scaling windows associated with all points of a quasicrystal is the one dimensional quasicrystal of universal internal scaling symmetries its acceptance window is the interval external inflation centres of a cut and project quasicrystal are those which are not among quasicrystal points their complete description is given analogically to the description of the internal ones imposing some additional requirements on the scaling factors between any two adjacent quasicrystal points one finds a countable infinity of external inflation centres the scaling factors belonging to any such centre u form an infinite u dependent subset of points of the quasicrystal with acceptance window containing

non probabilistic convex model process a new method of time variant uncertainty analysis and its application to structural dynamic reliability problems in this paper we propose a method for time variant uncertainty analysis namely the non probabilistic convex model process which provides an effective mathematical tool for the analysis of structural dynamic uncertainty when lacking relevant information in the convex model process we express the variables at any time with intervals and establish the corresponding auto covariance function and correlation coefficient function to depict the correlation between variables at different times we also define several important characteristic parameters for the uni and bi dimensional convex model processes including the mid value function variance function auto covariance function and cross covariance function we provide the definition for the stationary convex model process and its ergodicity then by combining the convex model process with the first passage failure mechanism we propose a non probabilistic analysis model of structural dynamic reliability and formulate the solving algorithm based on monte carlo simulation finally through the analysis of numerical examples we verify the effectiveness of the convex model process and the model of dynamic reliability analysis proposed in this paper

dynamics of multiple cracked prismatic beams with uncertain but bounded depths under deterministic and stochastic loads abstract dynamic response of cracked beams changes with respect to undamaged ones due to the change in mechanical properties moreover usually the crack characteristics crack size crack position applied stress etc are assumed to be known however in practice they possess considerable scatter of uncertainty the aim of this paper is to evaluate the time varying upper and lower bounds of the response of damaged beams with multiple cracks subjected to deterministic or stochastic excitations assuming uncertain crack depths as interval variables in this study the analysis is performed adopting a finite element model for the damaged beam extended to the case where multiple non propagating cracks are presented in order to provide the bounds of the response the authors propose an approach developed in the time domain starting from the derivation of the bounds of the interval eigenvalues evaluated as solution of two appropriate deterministic eigenvalue problems to assess the efficiency and the accuracy of the presented method a damaged cantilever prismatic aluminum beam with multiple uncertain but bounded cracks is examined the bounds of the free end displacement and variance are calculated and compared with the exact solution via combinatorial procedure the proposed results show a high level of accuracy for both increasing amplitude value of interval deviation and increasing number of uncertain parameters

the invariants of modular indecomposable representations of mathbb z _ p 2 we consider the invariant ring for an indecomposable representation of a cyclic group of order p 2 over a field mathbb f of characteristic p we describe a set of mathbb f algebra generators of this ring of invariants and thus derive an upper bound for the largest degree of an element in a minimal generating set for the ring of invariants this bound as a polynomial in p is of degree two

hybrid uncertain static analysis with random and interval fields abstract uncertain static analysis of an engineering structure with diverse type of non deterministic system parameter is investigated in this study unlike the traditional hybrid uncertain static analysis involving random and interval variables the concept of random and interval fields has been implemented to model the spatially dependent uncertainties associated with the system inputs a novel computational approach namely the extended unified interval stochastic sampling x uiss method is proposed to calculate the statistical characteristics i e mean and standard deviation of the extreme bounds i e lower and upper bounds of the concerned responses e g displacement and stress of engineering structure involving hybrid spatially dependent uncertainties subsequently by utilizing either parametric or nonparametric statistical analysis the probability density functions pdfs as well as the cumulative distribution functions cdfs of the extreme bounds of the concerned structural responses can be effectively established consequently the upper and lower bounds of either the concerned responses of the engineering structure at any particular percentile of probability or the structural reliability against any specified capacities can be effectively secured the applicability and effectiveness of the proposed computational analysis framework are illustrated through the numerical investigations on various examples

interval analysis of structures with uncertain but bounded axial stiffness abstract a method to evaluate the static response of structures with interval axial stiffnesses is proposed the procedure overcomes the drawbacks arising in traditional interval analysis due to dependency phenomenon the inversion of the interval stiffness matrix is pursued by means of a novel approximate relationship based on an interval valued sherman morrison woodbury formula numerical results on truss and frame structures evidence the accuracy and the effectiveness of the proposed approach to evaluate bounds of displacements and internal forces

the noether numbers for cyclic groups of prime order abstract the noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants we compute the noether number for an arbitrary representation of a cyclic group of prime order and as a consequence prove the 2 p 3 conjecture

structural reliability analysis using non probabilistic convex model abstract by using the convex model approach the bounds of the uncertain parameters are only required rather than their precise probability distributions based on which it can be made possible to conduct the reliability analysis for many complex engineering problems with limited information in this paper a non probabilistic reliability model is given for structures with convex model uncertainty which is defined as a ratio of the multidimensional volume falling into the reliability domain to the one of the whole convex model a monte carlo simulation is then formulated for the reliability model to obtain a reference solution a first order approximation method foam is proposed to solve the reliability model based on a linear approximation of the failure surface a second order approximation method soam is further formulated to improve the reliability analysis precision for limit state functions with relatively strong nonlinearity by introducing the suggested reliability model a convex model based system reliability method is also formulated eight numerical examples are provided to demonstrate the effectiveness of the present methods

invariants of finite groups over fields of characteristicp abstract let v denote a finite dimensional k vector space and let g denote a finite group of k linear automorphisms of v let v m denote the direct sum of m copies of v and let g act on the symmetric algebra k v m of v m by the diagonal action on v m a result of noether implies that if char k 0 then k v m g can be generated as a k algebra by polynomials whose degrees are g no matter how large m is this paper proves that this result no longer holds when the characteristic of k divides g more precisely it is proved in this case that there is a positive number depending only on g and char k such that every set of k algebra generators of k v m g contains a generator whose degree is m

a new approach for the stochastic analysis of finite element modelled structures with uncertain parameters a method for evaluating the static response of uncertain finite element fe discretised structures is presented the method is comparable with the perturbation procedures from a computational point of view but it overcomes the drawbacks related to these procedures in fact the present method gives excellent level of accuracy even for high amount of uncertainties moreover it produces the exact solution for statically determinate structures the accuracy is remarkable even in the evaluation of high order statistics and probability density function of the response so that reliability consideration can be drawn

vector invariants ofu2 fp a proof of a conjecture of richman abstract in this paper we prove a conjecture of d r richman concerning the vector invariants of the group u 2 f p let v be a vector space of dimension 2 with basis x y over the field f p and let f p x y be the symmetric algebra of v over f p if denotes a generator of u 2 f p then we may assume x x and y x y let a n be the symmetric algebra of v n we obtain an automorphism of a n of order p by using the diagonal action of extended to the whole of a n the subalgebra of polynomials left invariant by this action is called the ring of vector invariants of u 2 f p richman conjectured that these rings of invariants have certain sets of generators and gave a proof in the case p 2 we prove his conjecture for all primes