astala s conjecture on distortion of hausdorff measures under quasiconformal maps in the plane let e subset mathbb c be a compact set g mathbb c to mathbb c be a k quasiconformal map and let 0 t 2 let mathcal h t denote t dimensional hausdorff measure then mathcal h t e 0 quad rightarrow quad mathcal h t left ge right 0 quad t frac 2kt 2 left k 1 right t

a theory of semiprimitive groups abstract a transitive permutation group is semiprimitive if each of its normal subgroups is transitive or semiregular interest in this class of groups is motivated by two sources problems arising in universal algebra related to collapsing monoids and the graph restrictive problem for permutation groups here we develop a theory of semiprimitive groups which encompasses their structure their quotient actions and a method by which all finite semiprimitive groups are constructed we also extend some results from the theory of primitive groups to semiprimitive groups and conclude with open problems of a similar nature

exact closed form solution for the vibration modes of the euler bernoulli beam with multiple open cracks abstract in this study exact closed form expressions for the vibration modes of the euler bernoulli beam in the presence of multiple concentrated cracks are presented the proposed expressions are provided explicitly as functions of four integration constants only to be determined by the standard boundary conditions the enforcement of the boundary conditions leads to explicit expressions of the natural frequency equations besides the evaluation of the natural frequencies neither computational work nor recurrence expressions for the vibration modes are required the cracks that are not subjected to the closing phenomenon are modelled as a sequence of dirac s delta generalised functions in the flexural stiffness the eigen mode governing equation is formulated over the entire domain of the beam without enforcement of any continuity conditions which are already accounted for in the adopted flexural stiffness model the vibration modes of beams with different numbers of cracks under different boundary conditions have been analysed by means of the proposed closed form expressions in order to show their efficiency

a two dimensional version of the goldschmidt sims conjecture abstract the goldschmidt sims conjecture asserts that there is a finite number of conjugacy classes of edge transitive lattices in the automorphism group of a regular tree with prime valence we prove a similar theorem for irreducible lattices transitive on the 2 cells of the product of two regular trees of prime valences

amalgams of prime index this serves as a motivation for our main theorem from the theory of the classification of finite simple groups indeed our result classifies all such configurations and in particular it says that in all the interesting cases we must have ip n p i d 2 and ip p n p i 3 or 5 the basic definitions we use are the same as in gs which we quickly repeat for the convenience of the reader an umalgatn consists of two monomorphisms 4 and dz of groups such that

on locally semiprimitive graphs and a theorem of weiss abstract in this paper we investigate graphs that admit a group acting arc transitively such that the local action is semiprimitive with a regular normal nilpotent subgroup this type of semiprimitive group is a generalisation of an affine group we show that if the graph has valency coprime to six then there is a bound on the order of the vertex stabilisers depending on the valency alone we also prove a detailed structure theorem for the vertex stabilisers in the remaining case this is a contribution to an ongoing project to investigate the validity of the potocnik spiga verret conjecture

on the order of borel subgroups of group amalgams and an application to locally transitive graphs abstract a permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular given nontrivial finite transitive permutation groups l 1 and l 2 with l 1 not semiprimitive we construct an infinite family of rank two amalgams of permutation type l 1 l 2 and borel subgroups of strictly increasing order as an application we show that there is no bound on the order of edge stabilisers in locally l 1 l 2 graphs we also consider the corresponding question for amalgams of rank k 3 we completely resolve this by showing that the order of the borel subgroup is bounded by the permutation type l 1 l k only in the trivial case where each of l 1 l k is regular

closed form solutions of euler bernoulli beams with singularities abstract the problem of the integration of the static governing equations of the uniform euler bernoulli beam with discontinuities is studied in particular two types of discontinuities have been considered flexural stiffness and slope discontinuities both the above mentioned discontinuities have been modeled as singularities of the flexural stiffness by means of superimposition of suitable distributions generalized functions to a uniform one dimensional field closed form solutions of governing differential equation requiring the knowledge of the boundary conditions only are proposed and no continuity conditions are enforced at intermediate cross sections where discontinuities are shown the continuity conditions are in fact embedded in the flexural stiffness model and are automatically accounted for by the proposed integration procedure finally the proposed closed form solution for the cases of slope discontinuity is compared with the solution of a beam having an internal hinge with rotational spring reproducing the slope discontinuity

quasiconformal distortion of riesz capacities and hausdorff measures in the plane in this paper we prove the sharp distortion estimates for the quasi conformal mappings in the plane both in terms of the riesz capacities from non linear potential theory and in terms of the hausdorff measures

quasiconformal planes with bi lipschitz pieces and extensions of almost affine maps abstract a quasiplane f v is the image of an n dimensional euclidean subspace v of r n 1 n n 1 under a quasiconformal map f r n r n we give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi lipschitz n manifold and for a quasiplane to have big pieces of bi lipschitz images of r n one main novelty of these results is that we analyze quasiplanes in arbitrary codimension n n to establish the big pieces criterion we prove new extension theorems for almost affine maps which are of independent interest this work is related to investigations by tukia and vaisala on extensions of quasisymmetric maps with small distortion

exact solution of the multi cracked euler bernoulli column abstract the use of distributions generalized functions is a powerful tool to treat singularities in structural mechanics and besides providing a mathematical modelling their capability of leading to closed form exact solutions is shown in this paper in particular the problem of stability of the uniform euler bernoulli column in presence of multiple concentrated cracks subjected to an axial compression load under general boundary conditions is tackled concentrated cracks are modelled by means of dirac s delta distributions an integration procedure of the fourth order differential governing equation which is not allowed by the classical distribution theory is proposed the exact buckling mode solution of the column as functions of four integration constants and the corresponding exact buckling load equation for any number position and intensity of the cracks are presented as an example a parametric study of the multi cracked simply supported and clamped clamped euler bernoulli columns is presented

multiplications of distributions in one dimension and a first application to quantum field theory in a previous paper we introduced a class of multiplications of distributions in one dimension here we furnish different generalizations of the original definition and we discuss some applications of these procedures to the multiplication of delta functions and to quantum field theory

multi cracked euler bernoulli beams mathematical modeling and exact solutions localized flexibility models of cracks enable one for simple and effective representation of the behavior of damaged beams and frames important applications such as the determination of closed form solutions and the development of diagnostic methods of analysis have attracted the interest of many researchers in recent years nevertheless certain fundamental questions have not been completely clarified yet one of these issues concerns with the mechanical justification of the macroscopic model of rotational elastic spring commonly used to describe the presence of an open crack in a beam under bending deformation two main analytical formulations have been recently proposed to take into account the singularity generated by the crack the crack is represented by suitable dirac s delta functions either in the beam s flexural rigidity or in the beam s flexural flexibility both approaches require some caution due to mathematical subtleties of the analysis motivated by these considerations in this paper we propose a justification of the rotational elastic spring model of an open crack in a beam in bending deformation we show that this localized flexibility model of a crack is the variational limit of a family of one dimensional beams when the flexural stiffness of these beams tends to zero in an interval centered at the cracked cross section and simultaneously the length of the interval vanishes in a suitable way we also show that the static and dynamic problem for the flexibility model of cracked beam can be formulated within the classical context of the theory of distributions avoiding the hindrances encountered in previous approaches to the problem in addition the proposed treatment leads to a simple and efficient determination of exact closed form solutions of both static and dynamic problems for beams with multiple cracks

multiplication of distributions in one dimension possible approaches and applications to δ function and its derivatives we introduce a new class of multiplications of distributions in one dimension merging two different regularizations of distributions some of the features of these multiplications are discussed in detail we use our theory to study a number of examples involving products between dirac delta functions and its successive derivatives

several products of distributions on rm the method of the sequential completion has been used efficiently to define products of distributions some results for products of distributions on r 1 have been given by j mikusinski b fisher and cheng lin zhi et al the main aim of this paper is to extend the method to the case for several variables a number of products such as delta r o delta s x r o delta s and x r o delta s etc are considered

multiplication of distributions in any dimension applications to δ function and its derivatives in two previous papers the author introduced a multiplication of distributions in one dimension and he proved that two one dimensional dirac delta functions and their derivatives can be multiplied at least under certain conditions here mainly motivated by some engineering applications in the analysis of the structures we propose a different definition of multiplication of distributions which can be easily extended to any spatial dimension in particular we prove that with this new definition delta functions and their derivatives can still be multiplied

area distortion of quasiconformal mappings in recent years quasiconformal mappings have been an efficient tool in the study of dynamical systems of the complex plane we show here that in turn methods or ideas from dynamical systems can be used to solve a number of open questions in the theory of planar quasiconformal mappings it has been known since the work of ahlfors a and mori mo that k quasiconformal mappings are locally h51der continuous with exponent 1 k the function