a refined quasi 3d isogeometric analysis for functionally graded microplates based on the modified couple stress theory the isogeometric analysis associated with a novel quasi 3d shear deformation theory is proposed to investigate size dependent behaviours of functionally graded microplates the modified couple stress theory with only one material length scale parameter is employed to effectively capture the size dependent effects within the microplates meanwhile the quasi 3d theory which is constructed from a novel seventh order shear deformation refined plate theory with four unknowns is able to consider both shear deformations and thickness stretching effect without requiring shear correction factors the nurbs based isogeometric analysis is integrated to exactly describe the geometry and approximately calculate the unknown fields with higher order derivative and continuity requirements the proposed approach is successfully applied to study the static bending free vibration and buckling responses of rectangular and circular functionally graded microplates with various types of boundary conditions in which some benchmark numerical examples are presented a number of investigations are also conducted to illustrate the effects of the material length scale material index and aspect ratios on the responses of the microplates

vibration of two dimensional imperfect functionally graded 2d fg porous nano micro beams abstract this study presents analysis on the vibration behavior of the two dimensional functionally graded 2d fg nano and microbeams which are made of two kinds of porous materials for the first time based on timoshenko beam theory the material of the nano and microbeams is modeled as 2d fgms according to the power law the eringen s nonlocal elasticity and the modified couple stress theories are used respectively in case of nano and microbeams the boundary conditions are considered as clamped cc simply supported ss clamped simply supported cs and cantilever cf the governing equations are solved using the generalized differential quadrature method gdqm the effects of fg power indexes along length and thickness are studied along each other and also exclusively the parametric studies are presented for functionally graded along thickness fg functionally graded along length afg and 2d fg porous and perfect micro nanobeams under different boundary conditions

uniqueness for plane crack problems in dipolar gradient elasticity and in couple stress elasticity the present work deals with the uniqueness theorem for plane crack problems in solids characterized by dipolar gradient elasticity the theory of gradient elasticity derives from considerations of microstructure in elastic continua mindlin r d 1964 micro structure in linear elasticity arch ration mech anal 16 51 78 and is appropriate to model materials with periodic structure according to this theory the strain energy density assumes the form of a positive definite function of the strain as in classical elasticity and the second gradient of the displacement additional term specific cases of the general theory employed here are the well known theory of couple stress elasticity and the recently popularized theory of strain gradient elasticity these cases are also treated in the present study we consider an anisotropic material response of the cracked plane body within the linear version of gradient elasticity and conditions of plane strain or anti plane strain it is emphasized that for crack problems in general a uniqueness theorem more extended than the standard kirchhoff theorem is needed because of the singular behavior of the solutions at the crack tips such a theorem will necessarily impose certain restrictions on the behavior of the fields in the vicinity of crack tips in standard elasticity a theorem was indeed established by knowles and pucik knowles j k pucik t a 1973 uniqueness for plane crack problems in linear elastostatics j elast 3 155 160 who showed that the necessary conditions for solution uniqueness are a bounded displacement field and a bounded body force field in our study we show that the additional to the two previous conditions requirement of a bounded displacement gradient field in the vicinity of the crack tips guarantees uniqueness within the general form of the theory of dipolar gradient elasticity in the specific cases of couple stress elasticity and pure strain gradient elasticity the additional requirement is less stringent this only involves a bounded rotation field for the first case and a bounded strain field for the second case

directional tests for one sided alternatives in multivariate models consider one sided testing problems for a multivariate exponential family model through conditioning or other considerations the problem oftentimes reduces to testing a null hypothesis that the natural parameter is a zero vector against the alternative that the natural parameter lies in a closed convex cone b the problems include testing homogeneity of parameters testing independence in contingency tables testing stochastic ordering of distributions and many others a test methodology is developed that directionalizes the usual test procedures such as likelihood ratio chi square fisher and so on the methodology can be applied to families of tests where the family is indexed by a size parameter so as to enable nonrandomized testing by p values for discrete models a refined family of tests provides a refined grid for better testing by p values the tests have essential monotonicity properties that are required for admissibility and for desirable power properties two examples are given

a reformulation of constitutive relations in the strain gradient elasticity theory for isotropic materials abstract the general isotropic strain gradient elasticity theory with five higher order elastic constants is reformulated by introducing two different orthogonal decompositions of the strain gradient tensor just applying the mathematical reformulations no extra conditions needed the constitutive relations equilibrium equation and boundary conditions are reformulated in the reformulated theory the number of independent higher order elastic constants is proved to be three for isotropic materials which indicates that the five higher order elastic constants in the general isotropic strain gradient elasticity theory are dependent with each other therefore the general strain gradient elasticity theory contains only three independent material length scale parameters for isotropic materials in addition to the lame constants the new theory is different from the existed strain gradient elasticity theory with one or three material length scale parameters which introduces extra conditions during deriving process moreover the reformulated theory can be directly reduced to that of incompressible materials by assuming the terms associated with hydrostatic strains to be zero some examples such as torsion of cylindrical bars shearing of fixed end layers and pure bending of thin beams are performed to reveal the necessity of including multi length scale parameters in the strain gradient elasticity theory to predict size effects at micron scale

chi squared tests for and against an order restriction on multinomial parameters abstract chi squared tests are considered for testing homogeneity against an order restriction and for testing an order restriction against all alternatives on multinomial parameters closed form expressions are obtained for the minimum chi squared estimator and the neyman modified chi squared estimator the asymptotic equivalence between these chi squared tests and the likelihood ratio test is established the latter has been studied by robertson 1978 their asymptotic null distribution is the chi bar squared distribution these chi squared tests are applied to a numerical example in horse racing also included are the results of some simulation studies conducted to investigate the rate of convergence to the limiting null distributions and to compare the powers of these tests chi squared tests are widely used statistical tools for making inferences about a collection p 1 p 2 pk of multinomial parameters many such collections encountered in practice exhibit a trend for instance pi may represent

comparison of several treatments with a control using multiple contrasts abstract a problem frequently encountered in the practice of statistics is the comparison of several treatments with a control or standard we consider an experimental situation where prior knowledge indicates that all of the treatments are at least as effective as the control and the problem is to determine if any are significantly better than the control a number of statistical procedures have been proposed for this situation of which the best known is dunnett s 1955 multiple comparison procedure dunnett s test rejects equality of the treatments and the control for a large value of the maximum contrast of the data vector with several vectors that are located symmetrically within the alternative region we study a large class of such tests which includes dunnett s test as a particular case one of these tests which is based on the maximum contrast of the data vector with several orthogonal vectors is very easy to implement and has an uncomplicated good and fairly uniform power function over t

significance levels in exploratory projection pursuit summary in exploratory data analysis projection pursuit methods explore the nonlinear structure of high dimensional data it is useful to have a significance test to help us decide whether apparent structure is real or just the effect of noise monte carlo methods can be helpful to achieve this but unfortunately in this case they are computationally expensive in this paper under a suitable null hypothesis we derive a theoretical approximation for the p value associated with friedman s projection pursuit index friedman 1987 the result of monte carlo simulations is compared with our analytical result some practical aspects are discussed

stochastic estimation and testing on introduit plusieurs procedures stochastiques pour lesquelles on developpe une theorie asymptotique estimations et tests de distance minimum stochastique estimation du maximum de vraisemblance stochastique ensembles de confiance pour une probabilite inconnue etc

on measuring the conformity of a parameter set to a trend with applications abstract consider the hypothesis h 1 theta 1 or theta 2 or or theta k regarding a collection theta 1 theta 2 theta k of unknown parameters it is clear that this trend is reflected in certain possible parameter sets more than in others a quantification of this notion of conformity to a trend which was suggested by barlow and brunk 1972 is studied applications of the resulting theory to several order restricted hypothesis tests are presented author

nonlinear size dependent dynamics of an imperfect shear deformable microplate abstract the nonlinear size dependent dynamics of a geometrically imperfect shear deformable microplate is investigated based on the modified couple stress theory an initial imperfection in the out of plane direction is taken into account the third order shear deformation theory is employed to model the system by retaining in plane displacements and inertia the governing equations of motion of the system are obtained employing an energy method through use of the lagrange equations which upon employing an assumed mode technique yields a set of second order nonlinear ordinary differential equations with coupled terms a high dimensional discretised system is considered and made double dimensional via a change of variables in order to obtain a set of first order nonlinear ordinary differential equations the resulting equations are solved using a direct time integration technique resulting in time dependent generalised coordinates for the in plane and out of plane displacements and two rotations from these generalised coordinates phase plane portraits and fast fourier transforms ffts are also obtained moreover the frequency response and force response curves of the system are obtained using a continuation technique stability analysis is conducted via the floquet theory the effect of the initial imperfection as well as the length scale parameter on the system response is also examined

free vibration of size dependent mindlin microplates based on the modified couple stress theory this paper develops a mindlin microplate model based on the modified couple stress theory for the free vibration analysis of microplates this non classical plate model contains an internal material length scale parameter related to the material microstructures and is capable of interpreting the size effect that the classical mindlin plate model is unable to describe the higher order governing equations of motion and boundary conditions are derived using the hamilton principle the p version ritz method is employed to determine the natural frequencies of the microplate with different boundary conditions a detailed parametric study is conducted to study the influences of the length scale parameter side to thickness ratio and aspect ratio on the free vibration characteristics of the microplate it is found that the size effect is significant when the thickness of microplate is close to the material length scale parameter

size dependent nonlinear analysis and damping responses of fg cntrc micro plates abstract this paper presents a nonlinear numerical model which is based on the modified couple stress theory mcst and trigonometric shear deformation theory coupled with isogeometric analysis the present approach captures the small scale effects on the geometrically nonlinear behaviors of functionally graded carbon nanotube reinforced composite fg cntrc micro plate with four patterns distribution the equations of motion are established based on a galerkin weak form associated with von karman nonlinear strains the mcst utilizes only one material length scale parameter to predict the size effect in fg cntrc micro plate for which its material properties are derived from an extended rule of mixture the solutions of nonlinear static equation are obtained by using the newton raphson technique and the newmark time iteration procedure in association with picard method is assigned to get responses of the nonlinear dynamic problems in addition the rayleigh damping is applied to consider the influence of damping characteristic on the oscillation of fg cntrc micro plates comparisons are performed to verify the proposed approach afterward the numerical examples are used to show the effects of the distribution of carbon nanotubes cnt their volume fraction the material length scale parameter and the boundary conditions on the nonlinear static and dynamic behaviors of fg cntrc micro plates

lattice models for conditional independence in a multivariate normal distribution the lattice conditional independence model n x is defined to be the set of all normal distributions on wri such that for every pair l m e x xl and xm are conditionally independent given xl n m here x is a ring of subsets of the finite index set i and for k e x xk is the coordinate projection of x e hii onto rk statistical properties of n y may be studied for example maximum likelihood inference invariance and the problem of testing ho n x vs h n 1 when w is a subring of x the set j x of join irreducible elements of xy plays a central role in the analysis of n x this class of statistical models occurs in the analysis of nonnested multivariate missing data patterns and nonnested dependent linear regression models

tensor products of semilattices and fuzzy ideals we study complete fuzzy ideals on semilattices from the point of view of tensor products of semilattices we show that the lattice of all complete fuzzy ideals on a semilattice is an extension of tensor product we define the notion of semicomplete bi ideals and show that the lattice of all fuzzy ideals on a distributive lattice is isomorphic to the lattice of all semicomplete bi ideals moreover we show that the lattice of all complete fuzzy ideals on a distributive algebraic lattice is distributive

properties of sufficiency and statistical tests 1 in a previous paper dealing with the importance of properties of sufficiency in the statistical theory of small samples attention was mainly confined to the theory of estimation in the present paper the structure of small sample tests whether these are related to problems of estimation and fiducial distributions or are of the nature of tests of goodness of fit is considered further

the intrinsic bayes factor for model selection and prediction abstract in the bayesian approach to model selection or hypothesis testing with models or hypotheses of differing dimensions it is typically not possible to utilize standard noninformative or default prior distributions this has led bayesians to use conventional proper prior distributions or crude approximations to bayes factors in this article we introduce a new criterion called the intrinsic bayes factor which is fully automatic in the sense of requiring only standard noninformative priors for its computation and yet seems to correspond to very reasonable actual bayes factors the criterion can be used for nested or nonnested models and for multiple model comparison and prediction from another perspective the development suggests a general definition of a reference prior for model comparison

sign tests in multidimension inference based on the geometry of the data cloud abstract multivariate sign tests attracted several statisticians in the past and it is evident from recent nonparametric literature that they still continue to draw attention one of the most important features of the univariate sign test is that it does not involve much technical assumptions or complicacy and this makes it quite popular among statistics users in this article we have come up with a new method for constructing multivariate sign tests that have reasonable statistical properties and can be used conveniently to solve one sample location problems our principal strategy here is to make a wise utilization of certain geometric structures in the constellation of data points for making inference about the location of their distribution as we proceed with the development of a fairly broad and general methodology we indicate its relationship with previous work done by others and sometimes attempt to unify some of the earlier ideas in particular we pick up some well known tests for uniform dis

a non classical mindlin plate model incorporating microstructure surface energy and foundation effects a non classical model for a mindlin plate resting on an elastic foundation is developed in a general form using a modified couple stress theory a surface elasticity theory and a two parameter winkler pasternak foundation model it includes all five kinematic variables possible for a mindlin plate the equations of motion and the complete boundary conditions are obtained simultaneously through a variational formulation based on hamilton s principle and the microstructure surface energy and foundation effects are treated in a unified manner the newly developed model contains one material length scale parameter to describe the microstructure effect three surface elastic constants to account for the surface energy effect and two foundation parameters to capture the foundation effect the current non classical plate model reduces to its classical elasticity based counterpart when the microstructure surface energy and foundation effects are all suppressed in addition the new model includes the mindlin plate models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases recovers the kirchhoff plate model incorporating the microstructure surface energy and foundation effects and degenerates to the timoshenko beam model including the microstructure effect to illustrate the new mindlin plate model the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulae derived

subgrid interaction and micro randomness novel invariance requirements in infinitesimal gradient elasticity abstract we present a micromechanically motivated form of the curvature energy in infinitesimal isotropic gradient elasticity the basis is a homogenization averaging scheme using a micro randomness assumption imposed on a directional higher gradient interaction term these directional interaction terms are matrix valued allowing to apply the standard orthogonal cartan lie algebra decomposition averaging over all subgrid directions leads to three quadratic curvature terms which are conformally invariant when neglecting volumetric effects restricted to rotational inhomogeneities we motivate thereby a symmetric couple stress tensor in the infinitesimal indeterminate couple stress model of koiter mindlin toupin type relations are established to a novel conformally invariant linear cosserat model

couple stress theory for solids by relying on the definition of admissible boundary conditions the principle of virtual work and some kinematical considerations we establish the skew symmetric character of the couple stress tensor in size dependent continuum representations of matter this fundamental result which is independent of the material behavior resolves all difficulties in developing a consistent couple stress theory we then develop the corresponding size dependent theory of small deformations in elastic bodies including the energy and constitutive relations displacement formulations the uniqueness theorem for the corresponding boundary value problem and the reciprocal theorem for linear elasticity theory next we consider the more restrictive case of isotropic materials and present general solutions for two dimensional problems based on stress functions and for problems of anti plane deformation finally we examine several boundary value problems within this consistent size dependent theory of elasticity

mathematical model of a three layer micro and nano beams based on the hypotheses of the grigolyuk chulkov and the modified couple stress theory abstract the mathematical model of three layered beams developed based on the hypothesis of the grigolyuk chulkov and the modified couple stress theory and the size depended equations governing the layers motions on the micro and nano scales is constructed the hamilton s principle yields the novel equations of motion as well as the boundary initial conditions regarding beams displacement the latter ones clearly exhibit the size dependent dynamics of the studied micro and nano beams and the introduced theory overlaps with the classical beam equations for large enough layer thickness in particular a three layer beam with the micro layer thickness has been investigated with respect to the classical theory of grigolyuk chulkov the derived boundary problem is of sixth order and can be solved analytically in the case of statics the carried out numerical experiments allowed to detect and explain size dependent effects exhibited by the micro beams the beam deflections and stress yielded by the employed couple stress model are less than those predicted by the classical grigolyuk chulkov theory while the estimated eigen frequencies are higher respectively it has been shown that the proposed model can be reduced to the classical three layer grigolyuk chulkov beam through increase of the layers thickness which validates our approach

mathematical modeling of vibration problem of nano sized annular sector plates using the nonlocal continuum theory via eight node discrete singular convolution transformation abstract in the present study free vibration analysis of nano sized annular sector plate is analyzed using the nonlocal continuum theory the method of discrete singular convolution dsc is used for numerical computations firstly equation of motion of thin plates is formulated via nonlocal elasticity then irregular physical domain is transformed into a rectangular domain by using geometric coordinate transformation the dsc procedures are then applied to discretization of the transformed set of governing equations and related boundary conditions the effects of nonlocal parameter mode numbers sector angle and radius ratio on the vibration frequencies are investigated in detail it is seen that the size effects are significant in vibration analysis of nano scaled annular sector plates and need to be included in the mechanical model

a distribution free multivariate sign test based on interdirections abstract distribution free tests are investigated for the one sample multivariate location problem counts called interdirections which measure the angular distance between two observation vectors relative to the positions of the other observations are introduced these counts are invariant under nonsingular linear transformations and have a small sample distribution free property over a broad class of population models called distributions with elliptical directions which includes all elliptically symmetric populations and many skewed populations a sign test based on interdirections is described including as special cases the two sided univariate sign test and blumen s bivariate sign test the statistic is shown to have a limiting 2 p null distribution and because it is based on interdirections it is also seen to be invariant and to have a small sample distribution free property pitman asymptotic relative efficiencies and a monte carlo study show the test to perform well compared with hotell

on the admissibility and consistency of tests for homogeneity of variances 1 introduction and summary the problem of testing homogeneity of variances in a one way fixed analysis of variance model has been studied for the past 50 years if we let s2 denote the sample variance for the ith population i 1 2 p then the s2 s are independently distributed and each estimates its corresponding variance a under the assumption of normality of each of the p populations and the further assumption that each s2 is based on n 1 the same number of observations laue 1965 studied a two parameter family of tests t x q which reject the homogeneity hypothesis when r x rq d where

an approach to multivariate rank tests in multivariate analysis of variance abstract a class of multivariate rank like quantities is defined and used to develop multivariate tests to mimic popular one dimensional rank tests such as the mann whitney wilcoxon two sample test the jonckheere terpstra test for trend and the kruskal wallis one way analysis of variance test tests in one way analysis of variance are developed based on qualitative orthogonal contrasts allowing decomposition of an overall statistic into asymptotically independent components based on the contrasts the class of tests includes the usual normal theory tests and the componentwise rank tests but the main focus is on the tests based on a particular definition of multivariate rank a study of the pitman efficiency of the latter tests to those based on multivariate medians shows them to be superior at the normal slightly heavy tailed and light tailed distributions whereas the median based tests are superior for heavy tails these results are analogous to the univariate case

fully nonparametric hypotheses for factorial designs i multivariate repeated measures designs abstract we introduce nonparametric versions for many of the hypotheses tested in analysis of variance and repeated measures models such as the hypotheses of no main effects no interaction effects and no factor effects these natural extensions of the nonparametric hypothesis of equality of the k distributions in the k sample problem have appealing practical interpretations we concentrate on multivariate repeated measures designs and obtain simple rank statistics for testing these hypotheses these statistics are the rank transform rt versions of the classical statistics for testing hypotheses in repeated measures designs we emphasize that even though recent research has demonstrated the inappropriateness of the rt method for many parametric hypotheses the rt procedure is always valid for testing our nonparametric hypotheses we show that the rank statistics converge in distribution to central chi squared distributions under their respective nonparametric null hypotheses the noncentrality paramet

a new kirchhoff plate model based on a modified couple stress theory in this paper a new kirchhoff plate model is developed for the static analysis of isotropic micro plates with arbitrary shape based on a modified couple stress theory containing only one material length scale parameter which can capture the size effect the proposed model is capable of handling plates with complex geometries and boundary conditions from a detailed variational procedure the governing equilibrium equation of the micro plate and the most general boundary conditions are derived in terms of the deflection using the principle of minimum potential energy the resulting boundary value problem is of the fourth order instead of existing gradient theories which is of the sixth order and it is solved using the method of fundamental solutions mfs which is a boundary type meshless method several plates of various shapes aspect and poisson s ratios are analyzed to illustrate the applicability of the developed micro plate model and to reveal the differences between the current model and the classical plate model moreover useful conclusions are drawn from the micron scale response of this new kirchhoff plate model

an efficient computational approach for size dependent analysis of functionally graded nanoplates abstract in this paper an efficient computational approach based on refined plate theory rpt including the thickness stretching effect namely quasi 3d theory in conjunction with isogeometric formulation iga is proposed for the size dependent bending free vibration and buckling analysis of functionally graded nanoplate structures the present novel quasi 3d theory not only possesses 4 variables as refined plate theory but also accounts for both shear deformation and stretching effect without any requirement of shear correction factors scfs the size dependent effect is taken into account by nonlocal elasticity theory isogeometric analysis shows a great advantage in dealing with the high continuity and high order derivative requirements of the displacement fields used in quasi 3d and nonlocal theory the reliability and accuracy of the present method are ascertained by comparing the obtained results with other published ones numerical examples are also performed to show the significance of nonlocal effect material distribution profile aspect ratios and boundary conditions on the behaviour of fgm nanoplates

geometric material analogy for multiscale modelling of twisted plates abstract it is well known that the macroscopic behaviour of many engineering materials is strongly affected by the role of underlying microstructure currently though mathematical expressions linking behaviour of large scale structures to the geometry of their microscopic structure are largely lacking in this respect establishing quantitative links across different material lengthscales may offer new pathways for engineering design in the present work an analogy between cross sectional geometrical properties representing macrostructure and a material length parameter representing microstructure is presented the analogy is established through the study of a thin plate subject to axial loading undergoing finite displacements from two alternative perspectives first we consider a thin elastic plate with a pretwist about the loading axis where a warping term is introduced accounting for the out of plane deformation of the cross section the coupled governing differential equations and the corresponding coupled boundary conditions are explicitly derived employing a classical structural mechanics approach utilising an energy variational statement secondly an axially loaded thin flat plate i e with no pretwist is studied with strain gradient elasticity theory incorporating only one material length parameter representing the microstructure in addition to the two classical lame stiffness constants the ensuing analogy emerges by comparison of the governing equations of the two formulations which shows a mathematical expression can be identified which incorporates both geometric and material length variables that formalises the link between microscale and macroscale this mathematical expression which constitutes the kernel of the proposed multiscale approach admits a twofold interpretation depending on the assumed independent variable on the one hand the proposed multiscale modelling approach suggests that a plate with complex global geometry can be substituted by a structurally equivalent flat plate with constitutive relations given by a non local strain gradient theory on the other hand the material length parameter can be interpreted on a physical basis because for the first time it has been identified as a known function of geometrical features of the structure through simple algebraic relationships for various cross sectional profiles

algebraic fuzzy systems abstract a general theory of algebraic fuzzy systems is developed by introducing the notion of a fuzzy l subset corresponding to a set l of subsets of a nonempty set x assuming truth values in a complete brouwerian lattice it is observed that the set fl of all fuzzy l subsets of x is a closure algebraic closure fuzzy set system if and only if l is a closure algebraic closure set system on x a fuzzy l subset generated by a fuzzy subset is described modularity and distributivity of the complete lattices l and fl are discussed a method to construct fuzzy l subsets from certain family of members of l is established certain applications of the general theory are mentioned

testing a point null hypothesis the irreconcilability of p values and evidence abstract the problem of testing a point null hypothesis or a small interval null hypothesis is considered of interest is the relationship between the p value or observed significance level and conditional and bayesian measures of evidence against the null hypothesis although one might presume that a small p value indicates the presence of strong evidence against the null such is not necessarily the case expanding on earlier work especially edwards lindman and savage 1963 and dickey 1977 it is shown that actual evidence against a null as measured say by posterior probability or comparative likelihood can differ by an order of magnitude from the p value for instance data that yield a p value of 05 when testing a normal mean result in a posterior probability of the null of at least 30 for any objective prior distribution objective here means that equal prior weight is given the two hypotheses and that the prior is symmetric and nonincreasing away from the null other definiti

reconciling bayesian and frequentist evidence in the one sided testing problem abstract for the one sided hypothesis testing problem it is shown that it is possible to reconcile bayesian evidence against h 0 expressed in terms of the posterior probability that h 0 is true with frequentist evidence against h 0 expressed in terms of the p value in fact for many classes of prior distributions it is shown that the infimum of the bayesian posterior probability of h 0 is equal to the p value in other cases the infimum is less than the p value the results are in contrast to recent work of berger and sellke 1987 in the two sided point null case where it was found that the p value is much smaller than the bayesian infimum some comments on the point null problem are also given

nonlinear thermo elastic bending behavior of graphene sheets embedded in an elastic medium based on nonlocal elasticity theory this paper investigates the large deflection behavior of orthotropic single layered graphene sheet slgs embedded in a winkler pasternak elastic medium under a uniform transverse load in thermal environments using the nonlocal differential constitutive relations of eringen the slgs is modeled as a nonlocal orthotropic plate using the principle of virtual work the coupled nonlinear equilibrium equations are obtained based on first order shear deformation theory fsdt and the von karman geometrical model the differential quadrature dq discretized form of the governing equations with clamped and simply supported boundary conditions is derived the newton raphson iterative scheme is used to solve the resulting system of nonlinear algebraic equations effects of small scale parameter thermal environment width to length elasticity ratio aspect ratio thickness elastic foundation load value and boundary conditions are considered in detail the results show that unlike the simply supported boundary conditions increase of small scale parameter plays a decreasing role in effect of thermal environment on the deflections of nanoplates with clamped edges it is also observed that increasing the plate thickness thermal load effects decline noticeably and depending on the small scale value the thermal loads do not have any significant effect on the results beyond a specified thickness

free transverse vibrations of nano to micron scale beams in the present work the integral equation approach and the non local elasticity theory are employed to investigate the free transverse vibrations of nano to micron scale beams the frequency equation is analytically formulated into an eigenvalue problem of a matrix with an infinite order the numerical calculation is implemented by truncating this matrix to a finite order one it is found that the impact of the non local effect on the natural frequencies and vibrating modes is negligible for the beams with micrometre scale length but when the length of beams reaches the nanoscale the non local effect becomes important especially for the high order natural frequencies and vibrating modes

variational analysis of gradient elastic flexural plates under static loading gradient elastic flexural kirchhoff plates under static loading are considered their governing equation of equilibrium in terms of their lateral deflection is a sixth order partial differential equation instead of the fourth order one for the classical case a variational formulation of the problem is established with the aid of the principle of virtual work and used to determine all possible boundary conditions classical and non classical ones two circular gradient elastic plates clamped or simply supported at their boundaries are analyzed analytically and the gradient effect on their static response is assessed in detail a rectangular gradient elastic plate simply supported at its boundaries is also analyzed analytically and its rationally obtained boundary conditions are compared with the heuristically obtained ones in a previous publication of the authors finally a plate with two opposite sides clamped experiencing cylindrical bending is also analyzed and its response compared against that for the cases of micropolar and couple stress elasticity theories

an analysis of means type test for variances from normal populations after a brief review of the literature a test for homogeneity of variances is presented the test which is the variances analog of the analysis of means anom can be presented in a graphical form this graphical presentation makes it easy for practitioners to assess both practical and statistical significance tables of critical values are presented for balanced designs a large sample version of the test which uses anom critical values is also presented as well as a version of the test for unbalanced designs for which approximate critical values can be calculated monte carlo results indicate that this test has power comparable to well known normal based tests for homogeneity of variances

on some fundamental misunderstandings in the indeterminate couple stress model a comment on recent papers of a r hadjesfandiari and g f dargush abstract in a series of papers which are either published hadjesfandiari a dargush g f 2011a couple stress theory for solids int j solids struct 48 18 2496 2510 hadjesfandiari a dargush g f 2013 fundamental solutions for isotropic size dependent couple stress elasticity int j solids struct 50 9 1253 1265 or available as preprints hadjesfandiari a dargush g f 2010 polar continuum mechanics preprint arxiv 1009 3252 hadjesfandiari a r dargush g f 2011b couple stress theory for solids int j solids struct 48 2496 2510 hadjesfandiari a r 2013 on the skew symmetric character of the couple stress tensor preprint arxiv 1303 3569 hadjesfandiari a r dargush g f 2015a evolution of generalized couple stress continuum theories a critical analysis preprint arxiv 1501 03112 hadjesfandiari a r dargush g f 2015b foundations of consistent couple stress theory preprint arxiv 1509 06299 hadjesfandiari and dargush have reconsidered the linear indeterminate couple stress model they are postulating a certain physically plausible split in the virtual work principle based on this postulate they claim that the second order couple stress tensor must always be skew symmetric since they do not consider that the set of boundary conditions intervening in the virtual work principle is not unique their statement is not tenable and leads to some misunderstandings in the indeterminate couple stress model this is shown by specifying their development to the isotropic case however their choice of constitutive parameters is mathematically possible and we show that it still yields a well posed boundary value problem

couple stress based strain gradient theory for elasticity the deformation behavior of materials in the micron scale has been experimentally shown to be size dependent in the absence of stretch and dilatation gradients the size dependence can be explained using classical couple stress theory in which the full curvature tensor is used as deformation measures in addition to the conventional strain measures in the couple stress theory formulation only conventional equilibrium relations of forces and moments of forces are used the couple s association with position is arbitrary in this paper an additional equilibrium relation is developed to govern the behavior of the couples the relation constrained the couple stress tensor to be symmetric and the symmetric curvature tensor became the only properly conjugated high order strain measures in the theory to have a real contribution to the total strain energy of the system on the basis of this modification a linear elastic model for isotropic materials is developed the torsion of a cylindrical bar and the pure bending of a flat plate of infinite width are analyzed to illustrate the effect of the modification

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

the helicoidal modeling in computational finite elasticity part i variational formulation the finite elasticity mechanics of continua capable of a polar description is formulated by an alternative modeling to keeping position and orientation as uncoupled fields the rototranslation between two material particles can be described by a single complex tensorial quantity which is recognized to be orthogonal its linearization gives the characteristic curvature and differential vectors underlying the helicoidal modeling in both the sense of the body geometric description and the evolution of a deforming body after due introduction to dual tensors and rototranslations the polar description of the continuum is addressed with particular care to mixed differentiations of the rototranslation field then a thorough variational framework is established for the most general polar continuum under hyperelasticity hypothesis and the three field two field and one field principles are drawn and linearized the proposed modeling is expected to be profitably exploited in non linear finite element analyses of solids undergoing finite displacements rotations and strains

projected tests for order restricted alternatives consider the model where x ij i 1 2 k j 1 2 n are independent random variables distributed according to a one parameter exponential family with natural parameter i we test h 0 1 k versus h 1 c h 0 where 1 k and c is a cone determined by a 0 where the rows of a are contrasts with two nonzero elements we offer a method of generating good tests for h 0 versus h 1 the method is to take a good test for h 0 versus h 2 not h 0 and apply the test to projected sample points where the projection is onto c good tests for h 0 versus h 1 are tests that are schur convex good tests for h 0 versus h 1 are tests which are monotone with respect to a cone order we demonstrate that if the test function for h 0 versus h 2 is a constant size schur convex test then the resulting projected test is monotone

inclusions and inhomogeneities in strain gradient elasticity with couple stresses and related problems abstract at small length scales and or in presence of large field gradients the implicit long wavelength assumption of classical elasticity breaks down postulating a form of second gradient elasticity with couple stresses as a suitable phenomenological model for small scale elastic phenomena we herein extend eshelby s classical formulation for inclusions and inhomogeneities while the modified size dependent eshelby s tensor and hence the complete elastic state of inclusions containing transformation strains or eigenstrains is explicitly derived the corresponding inhomogeneity problem leads to integrals equations which do not appear to have closed form solutions to that end eshelby s equivalent inclusion method is extended to the present framework in form of a perturbation series that then can be used to approximate the elastic state of inhomogeneities the approximate scheme for inhomogeneities also serves as the basis for establishing expressions for the effective properties of composites in second gradient elasticity with couple stresses the present work is expected to find application towards nano inclusions and certain types of composites in addition to being the basis for subsequent non linear homogenization schemes

affine invariant multivariate one sample signed rank tests abstract brown and hettmansperger introduced affine invariant bivariate analogs of the sign rank and signed rank tests based on the oja median in this article affine invariant k variate extensions of the one sample signed rank test and the hodges lehmann estimate are considered the necessary distribution theory is developed and asymptotic pitman efficiencies with respect to hotelling s t 2 test under multivariate t distributions are tabulated an application of the signed rank tests to a repeated measurement setting is presented

bayesian hypothesis testing in two arm trials with dichotomous outcomes summary this article is motivated by an interest in comparing inferences made when using a bayesian or frequentist statistical approach the article addresses the study of one sided superiority and noninferiority bayesian tests these tests are stated in terms of the posterior probability that the null hypothesis is true for the binomial distribution and in terms of one sided credible limits we restrict our considerations to conjugate beta priors with integer parameters under this assumption the posterior probabilities of tested hypotheses can be transformed into the frequentist probabilities of bernoulli trials with an adjusted number of events and population sizes the method resembles a standard frequentist problem formulation by using an appropriate choice of prior parameters the posterior probabilities of the null hypothesis can be made smaller or larger than the p values of frequentist tests

a refined nonconforming quadrilateral element for couple stress strain gradient elasticity c0 1 patch test int j numer meth engng 2004 61 433 454 proposed by soh and chen is a reliable method to ensure convergence of nonconforming finite element for the couple stress strain gradient elasticity the c0 1 patch test function is a complete quadratic polynomial that satisfies the equilibrium equations to pass the c0 1 patch test the element displacement functions used to calculate strains must satisfy c0 continuity or weak c0 continuity and quadratic completeness in this paper a 24 dof degrees of freedom quadrilateral element cq12 rdkq for the couple stress strain gradient elasticity is developed by combining the refined thin plate element rdkq and the nonconforming element cq12 the element rdkq which satisfies weak c1 continuity is used to calculate strain gradients whereas strains are computed by the element cq12 which is established based on an extended variational functional and satisfies weak c0 continuity and quadratic completeness numerical examples show that the element cq12 rdkq passes the c0 1 patch test and it is also more efficient than the existing available triangular and quadrilateral elements in stress concentration problems with size effects copyright 2010 john wiley sons ltd

non local elastic plate theories a non local plate model is proposed based on eringen s theory of non local continuum mechanics the basic equations for the non local kirchhoff and the mindlin plate theories are derived these non local plate theories allow for the small scale effect which becomes significant when dealing with micro nanoscale plate like structures as illustrative examples the bending and free vibration problems of a rectangular plate with simply supported edges are solved and the exact non local solutions are discussed in relation to their corresponding local solutions

the lattice of l ideals of a ring is modular in this paper we extend the notion of a tip extended pair of fuzzy subgroups to l ideals of a ring we prove that the sum of two tip extended l ideals of an arbitrary pair of l ideals of a ring is the least l ideal containing the union of the given l ideals using this construction of join of l ideals we prove that the lattice of all l ideals of a given ring is modular

post processing posterior predictive p values this article addresses issues of model criticism and model comparison in bayesian contexts and focuses on the use of the so called posterior predictive p values ppp these involve a general discrepancy or conflict measure and depend on the prior the model and the data they are used in statistical practice to quantify the degree of surprise or conflict in data and to compare different combinations of prior and model the distribution of such ppp values is far from uniform however as we demonstrate for different models making their interpretation and comparison a difficult matter we propose a natural calibration of the ppp values where the resulting cppp values are uniform on the unit interval under model conditions the cppp values which in general rely on a double simulation scheme for their computation may then be used to assess and compare different priors and models our methods also make it possible to compare parametric and nonparametric model specifications in that genuine measures of

nonlocal third order shear deformation plate theory with application to bending and vibration of plates abstract the third order shear deformation plate theory of reddy a simple higher order theory for laminated composite plates j appl mech 51 1984 745 752 is reformulated using the nonlocal linear elasticity theory of eringen this theory has ability to capture the both small scale effects and quadratic variation of shear strain and consequently shear stress through the plate thickness analytical solutions of bending and free vibration of a simply supported rectangular plate are presented using this theory to illustrate the effect of nonlocal theory on deflection and natural frequency of the plates finally the relations between nonlocal third order first order and classical theories are discussed by numerical results

notions of limiting p values based on data depth and bootstrap abstract in this article we propose some new notions of limiting p values for hypothesis testing the limiting p value lp here not only provides the usual attractive interpretation of a p value as the strength in support of the null hypothesis coming from the observed data but also has several advantages first it allows us to resample directly from the empirical distribution in the bootstrap implementations rather than from the estimated population distribution satisfying the null constraints second it serves as a test statistic and as a p value simultaneously and thus enables us to obtain test results directly without having to construct an explicit test statistic and then establish or approximate its sampling distribution these are the two steps generally required in a standard testing procedure using bootstrap and the concept of data depth we have provided lp s for a broad class of testing problems where the parameters of interest can be either finite or infinite dimensional some compute

a quality index based on data depth and multivariate rank tests let f and g be the distribution functions of two given populations on rp p 1 we introduce and study a parameter q q f g which measures the overall outlyingness of population g relative to population f the parameter q can be defined using any concept of data depth its value ranges from 0 to 1 and is 5 when f and g are identical we show that within the dass of elliptical distributions when g departs from f in location or g has a larger spread or both the value of q dwindles down from 5 hence q can be used to detect the loss of accuracy or precision of a manufacturing process and thus it should serve as an important measure in quality assurance this in fact is the reason why we refer to q as a quality index in this article in addition to studying the properties of q we provide an exact rank test for testing q 5 vs q 5 this can be viewed as a multivariate analog of wilcoxon s rank sum test the tests proposed here have power against location change and scale increase simultaneo

lattice of fuzzy subalgebras and closure systems in i x abstract let x be a non empty set and i be the unit interval of real numbers the elements of i x are called fuzzy subsets of x i x is a lattice with the pointwise ordering of i now if a x f is an universal algebra with a specified set f of finitary operations on x then each operation can be extended to a finitary operation on i x an element of i x is defined to be a fuzzy subalgebra of a using these extended operations we also define algebraic closure system in i x we then show that the lattice of fuzzy subalgebras of a is an algebraic closure system in i x

the lattice of fuzzy left right ideals of a ring is modular we use the technique of nested set to study the structure of the lattice of all fuzzy left right ideals of a given ring we give a way to depict the supremum of fuzzy left right ideals by nested set and prove that the lattice of all fuzzy left right ideals of a given ring is modular

triangular mindlin microplate element in this article based on the most general form of strain gradient theory mgsgt a novel extended triangular mindlin plate element is proposed to accomplish this aim first the quadratic form of energy functional is obtained by vectorizing the higher order tensors of energy pairs from which the stiffness and mass matrices of the element are readily derived in comparison with the standard mindlin plate element the new element needs three additional nodal degrees of freedom dof including derivatives of lateral deflection and rotations which means a total of nine dofs per node also as compared to the standard mindlin plate element which requires only c0 shape functions the present one requires c1 continuous smooth shape functions due to second derivatives of deflection and rotations hence cubic polynomials are used to interpolate the displacement components the new element can be reduced to that based on the modified strain gradient theory msgt and the modified couple stress theory mcst moreover the standard mindlin plate element is recovered when the gradient based material parameters tend to zero the mindlin microplates with different boundary conditions are considered as the problem under study whose free vibration and bending are analyzed the results are compared with the exact solutions and excellent agreement is achieved

affine invariant multivariate multisample sign tests summary invariant sign type tests for the multivariate multisample location models are developed the definition of sign is based on the oja criterion function emphasis is on the twosample location model and the associated equivariant estimate of shift asymptotic theory is developed to provide large sample approximations for the null distribution of the test and to provide a framework for a discussion of efficiency tests are then developed for the one way and two way lay outs