a characterization of ideal polyhedra in hyperbolic 3 space the goals of this paper are to provide a characterization of dihedral angles of convex ideal those with all vertices on the sphere at infinity polyhedra in h3 and also of those convex polyhedra with some vertices on the sphere at infinity and some in the finite part of h3 these characterizations are given in respectively theorems 0 1 and 10 5 the first theorem is proved in detail while the proof of the second which is similar is only outlined the results of this paper grow out of the general framework of the author s doctoral dissertation 13 as published in 20 a lot of the language and some of the auxilary results come from there as well so familiarity with the latter reference is very helpful the necessity of the conditions postulated in theorem 0 1 has been shown in 14 hence only the sufficiency direction will be shown in the current paper in 1832 jakob steiner asked for a combinatorial characterization of convex polyhedra inscribed in the sphere this was considered intractable it took almost a hundred years to find a single example of an uninscribable combinatorial type by steinitz 24 however theorem 0 1 gives such a characterization section 11 is dedicated to a very brief historical survey and some purely graph theoretic and computational geometric consequences of theorem 0 1 in order to state this theorem suppose that a convex ideal polyhedron p in h3 is given let p denote the poincar dual of p and assign to each edge e of p a weight w e equal to the exterior dihedral angle at the corresponding edge e of p then the following result holds

magic identities for the conformal four point integrals the minkowski metric case abstract the original magic identities are due to j m drummond j henn v a smirnov and e sokatchev they assert that all n loop box integrals for four scalar massless particles are equal to each other 3 the authors give a proof of the magic identities for the euclidean metric case only and claim that the result is also true in the minkowski metric however the minkowski case is much more subtle and requires specification of the relative positions of cycles of integration to make these identities correct in this article we prove the magic identities in the minkowski metric case and in particular specify the cycles of integration our proof of magic identities relies on previous results from 7 8 where we give a mathematical interpretation of the n loop box integrals in the context of representations of a lie group u 2 2 and quaternionic analysis the main result of 7 8 is a weaker operator version of the magic identities no prior knowledge of physics or feynman diagrams is assumed from the reader we provide a summary of all relevant results from quaternionic analysis to make the article self contained

a vervaat like path transformation for the reflected brownian bridge conditioned on its local time at 0 we describe a vervaat like path transformation for the reflected brownian bridge conditioned on its local time at 0 up to random shifts this process equals the two processes constructed froma brownian bridge and a brownian excursion by adding a drift and then taking the excursions over the current minimum as a consequence these three processes have the same occupation measure which is easily found the three processes arise as limits in three different ways of profiles associated to hashing with linear probing or equivalently to parking functions

an adaptive recursive staggering strategy for simulating multifield coupled processes in microheterogeneous solids in this work an adaptive recursive staggering strategy is developed for the solution of partial differential equations arising from descriptions of multifield coupled processes in microheterogeneous solids in order to illustrate the solution strategy a multifield model problem is studied which describes the diffusion of a detrimental dilute solute into a solid material the coupled equations to be solved are 1 a phenomenological diffusion reaction equation 2 a phenomenological damage evolution law 3 a balance of energy and 4 a balance of momentum the finite element method is used for the spatial discretization and finite differences for the temporal discretization in order to accurately capture the microstructure of the material use of very fine finite element meshes is inescapable therefore in order to reduce computation time one would like to take as large time steps as possible provided that the associated numerical accuracy can be maintained classical staggering approaches solve each field equation in an uncoupled manner by allowing only the primary field variable to be active and momentarily freezing all others after the solution of each field equation the primary field variable is updated and the next field equation is treated in a similar manner in standard approaches after this process has been applied only once to all of the field equations the time step is immediately incremented this non recursive process is highly sensitive to the order in which the staggered field equations are solved furthermore since the staggering error accumulates with each passing time step the process may require very small time steps for sufficient accuracy in the approach developed here in order to reduce the error within a time step the staggering methodology is formulated as a recursive fixed point iteration whereby the system is repeatedly re solved until fixed point type convergence is achieved a sufficient condition for the convergence of such a fixed point scheme is that the spectral radius of the coupled operator which depends on the time step size must be less than unity this observation is used to adaptively maximize the time step sizes while simultaneously controlling the coupled operator s spectral radius in order to deliver solutions below an error tolerance within a prespecified number of desired iterations this recursive staggering error control allows substantial reduction of computational effort by the adaptive use of large time steps three dimensional numerical examples are given to illustrate the approach copyright 2001 john wiley sons ltd

rigidity of inversive distance circle packings revisited abstract inversive distance circle packing metric was introduced by p bowers and k stephenson 7 as a generalization of thurston s circle packing metric 34 they conjectured that the inversive distance circle packings are rigid for nonnegative inversive distance guo 22 proved the infinitesimal rigidity and then luo 27 proved the global rigidity in this paper based on an observation of zhou 37 we prove this conjecture for inversive distance in 1 by variational principles we also study the global rigidity of a combinatorial curvature introduced in 14 16 19 with respect to the inversive distance circle packing metrics where the inversive distance is in 1

a massive feynman integral and some reduction relations for appell functions new explicit expressions are derived for the one loop two point feynman integral with arbitrary external momentum and masses m12 and m22 in d dimensions the results are given in terms of appell functions manifestly symmetric with respect to the masses mi2 equating our expressions with previously known results in terms of gauss hypergeometric functions yields reduction relations for the involved appell functions that are apparently new mathematical results

the exploration process of inhomogeneous continuum random trees and an extension of jeulin s local time identity we study the inhomogeneous continuum random trees icrt that arise as weak limits of birthday trees we give a description of the exploration process a function defined on 0 1 that encodes the structure of an icrt and also of its width process determining the size of layers in order of height these processes turn out to be transformations of bridges with exchangeable increments which have already appeared in other icrt related topics such as stochastic additive coalescence the results rely on two different constructions of birthday trees from processes with exchangeable increments on weak convergence arguments and on general theory on continuum random trees

a geometrical angle on feynman integrals a direct link between a one loop n point feynman diagram and a geometrical representation based on the n dimensional simplex is established by relating the feynman parametric representations to the integrals over contents of n 1 dimensional simplices in non euclidean geometry of constant curvature in particular the four point function in four dimensions is proportional to the volume of a three dimensional spherical or hyperbolic tetrahedron which can be calculated by splitting into birectangular ones it is also shown that the known formula of reduction of the n point function in n 1 dimensions corresponds to splitting the related n dimensional simplex into n rectangular ones

a variational principle for weighted delaunay triangulations and hyperideal polyhedra we use a variational principle to prove an existence and uniqueness theorem for planar weighted delaunay triangulations with non intersecting site circles with prescribed combinatorial type and circle intersection angles such weighted delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space thus the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles more generally we consider weighted delaunay triangulations in piecewise flat surfaces allowing cone singularities with prescribed cone angles in the vertices the material presented here extends work by rivin on delaunay triangulations and ideal polyhedra

general results for massive n point feynman diagrams with different masses general expressions are obtained for scalar one loop massive feynman diagrams with any number of external lines and with arbitrary momenta of these lines with arbitrary values of masses and powers of denominators of internal lines the space time dimension n also being arbitrary special cases are considered when two or more masses are equal to 0 the results are represented in terms of hypergeometric functions

computation of strongly coupled multifield interaction in particle fluid systems the present work develops a flexible and robust solution strategy to resolve coupled systems comprised of large numbers of flowing particles embedded within a fluid a model problem consisting of particles which may undergo inelastic collisions in the presence of near field forces is considered the particles are surrounded by a continuous interstitial fluid which is assumed to obey the compressible navier stokes equations thermal effects are also considered such particle fluid systems are strongly coupled due to the mechanical forces and heat transfer induced by the fluid onto the particles and vice versa because the coupling of the various particle and fluid fields can dramatically change over the course of a flow process a primary focus of this work is the development of a recursive staggering solution scheme whereby the time steps are adaptively adjusted to control the error associated with the incomplete resolution of the coupled interaction between the various solid particulate and continuum fluid fields a central feature of the approach is the ability to account for the presence of particles within the fluid in a straightforward manner that can be easily incorporated within any standard computational fluid mechanics code based on finite difference finite element or finite volume type discretization a three dimensional example is provided to illustrate the overall approach 2007 elsevier b v all rights reserved

duality structures and discrete conformal variations of piecewise constant curvature surfaces abstract a piecewise constant curvature manifold is a triangulated manifold that is assigned a geometry by specifying lengths of edges and stipulating the simplex has an isometric embedding into a constant curvature background geometry euclidean hyperbolic or spherical with the specified edge lengths additional geometric structure leads to a notion of discrete conformal structure generalizing circle packings and their generalizations as studied by thurston and others we analyze discrete conformal variations of piecewise constant curvature 2 manifolds giving particular attention to the variation of angles formulas are derived for the derivatives of angles in each background geometry which yield formulas for the derivatives of curvatures and to curvature functionals finally we provide a complete classification of possible definitions of discrete conformal structures in each of the background geometries

eternal additive coalescents and certain bridges with exchangeable increments aldous and pitman have studied the asymptotic behavior of the additive coalescent processes using a nested family random forests derived by logging certain inhomogeneous continuum random trees here we propose a different approach based on partitions of the unit interval induced by certain bridges with exchangeable increments the analysis is made simple by an interpretation in terms of an aggregating server system

a fragmentation process connected to brownian motion let b s s 0 be a standard brownian motion and t 1 its first passage time at level 1 for every t 0 we consider ladder time set t of the brownian motion with drift t b t s b s ts and the decreasing sequence f t f 1 t f 2 t of lengths of the intervals of the random partition of 0 t 1 induced by t the main result of this work is that f t t 0 is a fragmentation process in the sense that for 0 t t f t is obtained from f t by breaking randomly into pieces each component of f t according to a law that only depends on the length of this component and independently of the others we identify the fragmentation law with the one that appears in the construction of the standard additive coalescent by aldous and pitman 3

fluid structure interaction with large structural displacements the calculation of viscous flows inside deformable structures faces major difficulties when enforcing the kinematic compatibility at the fluid structure interface and updating the geometry of the domain this paper proposes to overcome these problems by considering the fluid and the structure as a unique continuous medium to be studied in a fixed reference configuration the resulting problem is then split into a fluid and a structural part through an additive decomposition of the space of kinematically admissible test functions this approach treats the structure in a fully lagrangian way and uses an associated arbitrary lagrangian eulerian ale formulation for the fluid such a strategy has three advantages which are detailed in the present paper it can be discretised in time by implicit stable energy conserving time integration schemes and solved by simple iterative uncoupled algorithms the fluid and the structural problems can be approximated by independent finite element spaces the most recent geometrically exact nonlinear shell models can be used for the structures the full strategy has been implemented into an industrial cfd code results obtained in the simulation of industrial hydraulic shock absorbers are presented at the end of the paper

combinatorial ricci flows on surfaces we show that the analogue of hamilton s ricci flow in the combinatorial setting produces solutions which converge exponentially fast to thurston s circle packing on surfaces as a consequence a new proof of thurston s existence of circle packing theorem is obtained as another consequence ricci flow suggests a new algorithm to find circle packings

universality in one dimensional hierarchical coalescence processes motivated by several models introduced in the physics literature to study the nonequilibrium coarsening dynamics of one dimensional systems we consider a large class of hierarchical coalescence processes hcp an hcp consists of an infinite sequence of coalescence processes n n 1 each process occurs in a different epoch indexed by n and evolves for an infinite time while the evolution in subsequent epochs are linked in such a way that the initial distribution of n 1 coincides with the final distribution of n inside each epoch the process described by a suitable simple point process representing the boundaries between adjacent intervals domains evolves as follows only intervals whose length belongs to a certain epoch dependent finite range are active that is they can incorporate their left or right neighboring interval with quite general rates inactive intervals cannot incorporate their neighbors and can increase their length only if they are incorporated by active neighbors the activity ranges are such that after a merging step the newly produced interval always becomes inactive for that epoch but active for some future epoch without making any mean field assumption we show that i if the initial distribution describes a renewal process then such a property is preserved at all later times and all future epochs ii the distribution of certain rescaled variables for example the domain length has a well defined and universal limiting behavior as n independent of the details of the process merging rates activity ranges this last result explains the universality in the limiting behavior of several very different physical systems e g the east model of glassy dynamics or the paste all model which was observed in several simulations and analyzed in many physics papers the main idea to obtain the asymptotic result is to first write down a recursive set of nonlinear identities for the laplace transforms of the relevant quantities on different epochs and then to solve it by means of a transformation which in some sense linearizes the system

laser induced heating of dynamic particulate depositions in additive manufacturing abstract many applications in additive manufacturing involve the dynamic deposition of powders and the in flight heating of such material by a laser in order to characterize such systems the discrete element method dem is employed specifically this paper focuses on the initial stages of this process by developing a modular discrete element type multiphysics simulation method for the particle dynamics and heating by a laser and detailed thermal behavior the objective is to provide researchers with a framework to construct computational tools for this growing industry in order to achieve this from a simulation standpoint the overall particle mixture system is constructed by coupling submodels for each primary physical process dynamics and heating together an iterative staggering scheme is developed whereby within every time step each individual particle is solved freezing the state of the remaining multiparticle system the state of the particle is then updated and the algorithm moves to the next particle in the system and the process is repeated the overall process sweeps through the entire system repeatedly until convergence is achieved in an appropriate norm as the system evolves an error estimate dictates the time step size that is needed to induce convergence to below an appropriate error level thus the process can be considered as an implicit time stepping scheme which is combined with an internal iterative staggering process in order to control rates of convergence within a time step the algorithm adjusts the time step size if the iterative process does not converge within a desired number of iterations below an error tolerance the time step is reduced the degree of time step reduction is determined by utilizing an estimate of the spectral radius of the coupled system since the construction of model and solution process is modular one can easily replace physical submodels with other choices making it easy to numerically experiment with a variety of models qualitative and quantitative analyses are provided as well as three dimensional numerical examples this paper addresses the first of two overall macrostages of a complex manufacturing process with dem the first stage addressed in this paper concentrates on dry powder materials which are of a discrete particulate character using dem to characterize the deposition and in flight particle heating other papers of the author zohdi 2010 2013 2014 2015 81 82 2008 have addressed stage two namely phase transformations curing cooling and stress analysis using more appropriate continuum and hybrid dem continuum approaches

a representation of exchangeable hierarchies by sampling from random real trees a hierarchy on a set s also called a total partition of s is a collection mathcal h of subsets of s such that s in mathcal h each singleton subset of s belongs to mathcal h and if a b in mathcal h then a cap b equals either a or b or varnothing every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy mathcal h associated as follows with a random real tree mathcal t equipped with root element 0 and a random probability distribution p on the borel subsets of mathcal t given mathcal t p let t 1 t 2 ldots be independent and identically distributed according to p and let mathcal h comprise all singleton subsets of mathbb n and every subset of the form j t j in f x as x ranges over mathcal t where f x is the fringe subtree of mathcal t rooted at x there is also the alternative characterization every exchangeable random hierarchy of positive integers has the same distribution as a random hierarchy mathcal h derived as follows from a random hierarchy mathscr h on 0 1 and a family u j of i i d uniform 0 1 random variables independent of mathscr h let mathcal h comprise all sets of the form j u j in b as b ranges over the members of mathscr h

a computational framework for agglomeration in thermochemically reacting granular flows a computational framework is developed which couples a series of models each describing vastly different physical events in order to characterize particle growth agglomeration in thermochemically reacting granular flows the modelling is purposely simplified to expose the dominant mechanisms which control agglomeration the overall system is comprised of relatively simple coupled submodels describing impact heat production bonding and fragmentation each of which can be replaced by more elaborate descriptions if and when they are available inverse problems solved with a genetic algorithm are then constructed to ascertain system parameters which maximize agglomeration likelihood within a range of admissible data

the standard additive coalescent regard an element of the set x 1 x 2 x 1 x 2 0 i x i 1 as a fragmentation of unit mass into clusters of masses x i the additive coalescent of evans and pitman is the valued markov process in which pairs of clusters of masses x i x j merge into a cluster of mass x i x j at rate x i x j they showed that a version x t t of this process arises as a n weak limit of the process started at time 1 2 log n with n clusters of mass 1 n we show this standard additive coalescent may be constructed from the continuum random tree of aldous by poisson splitting along the skeleton of the tree we describe the distribution of x t on at a fixed time t we show that the size of the cluster containing a given atom as a process in t has a simple representation in terms of the stable subordinator of index 1 2 as t we establish a gaussian limit for centered and normalized cluster sizes and study the size of the largest cluster

explicitly symmetrical treatment of three body phase space we derive expressions for three body phase space that are explicitly symmetrical in the masses of the three particles we study geometrical properties of the variables involved in elliptic integrals and demonstrate that it is convenient to use the jacobian zeta function to express the results in four and six dimensions

variational principles for circle patterns and koebe s theorem we prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces our method is based on two new functionals one for the euclidean and one for the hyperbolic case we show how colin de verdiere s bragger s and rivin s functionals can be derived from ours

one loop derivation of the wilson polygon mhv amplitude duality we discuss the origin of the wilson polygon mhv amplitude duality at a perturbative level it is shown that the duality for the mhv amplitudes at the one loop level can be proven upon a particular change of variables in feynman parametrization and with the use of the relation between feynman integrals at different space time dimensions some generalization of the duality which implies the insertion of a particular vertex operator at the wilson triangle is found for the 3 point function we discuss the analytical structure of wilson loop diagrams and present the corresponding landau equations the geometrical interpretation of the loop diagram in terms of the hyperbolic geometry is discussed

on the volume of hyperbolic polyhedra where v1k is the volume vol 2 sj of the apex sjk sj c s to wjk schl ifli proved this formula for spherical simplices in 1936 h kneser gave a second very skilful proof of l see i l l and 4 sect 5 1 for both the spherical and hyperbolic cases up to a change of sign in the latter case but even for a three dimensional non euclidean simplex the integration of this schl li differential is practically impossible in fact the most basic objects in polyhedral geometry are orthoschemes or orthogonal simplices first introduced by schl fli an n orthoscheme r is an n simplex with vertices po pn such that

quaternionic analysis representation theory and physics we develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group we first review the cauchy fueter and poisson formulas and explain their representation theoretic meaning the requirement of unitarity of representations leads us to the extensions of these formulas in the minkowski space which can be viewed as another real form of quaternions representation theory also suggests a quaternionic version of the cauchy formula for the second order pole remarkably the derivative appearing in the complex case is replaced by the maxwell equations in the quaternionic counterpart we also uncover the connection between quaternionic analysis and various structures in quantum mechanics and quantum field theory such as the spectrum of the hydrogen atom polarization of vacuum one loop feynman integrals we also make some further conjectures the main goal of this and our subsequent paper is to revive quaternionic analysis and to show profound relations between quaternionic analysis representation theory and four dimensional physics

brownian excursions critical random graphs and the multiplicative coalescent let b t s 0 s be reflecting inhomogeneous brownian motion with drift t s at time s started with b t 0 0 consider the random graph script g sign n n 1 tn 4 3 whose largest components have size of order n 2 3 normalizing by n 2 3 the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of b t corollary 2 the dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process the states of this process are vectors x of nonnegative real cluster sizes x i and clusters with sizes x i and x j merge at rate x i x j the multiplicative coalescent is shown to be a feller process on l 2 the random graph limit specifies the standard multiplicative coalescent which starts from infinitesimally small clusters at time the existence of such a process is not obvious

the conformal four point integrals magic identities and representations of u 2 2 abstract in 3 5 we found mathematical interpretations of the one loop conformal four point feynman integral as well as the vacuum polarization feynman integral in the context of representations of a lie group u 2 2 and quaternionic analysis then we raised a natural question of finding mathematical interpretation of other feynman diagrams in the same setting in this article we describe this interpretation for all conformal four point integrals using this interpretation we give a representation theoretic proof of an operator version of the magic identities for the conformal four point integrals described by the box diagrams the original magic identities are due to j m drummond j henn v a smirnov and e sokatchev they assert that all n loop box integrals for four scalar massless particles are equal to each other 2 the authors give a proof of the magic identities for the euclidean metric case only and claim that the result is also true for the minkowski metric case however the minkowski case is much more subtle in this article we prove an operator version of the magic identities in the minkowski metric case and in particular specify the relative positions of cycles of integration that make these identities correct no prior knowledge of physics or feynman diagrams is assumed from the reader we provide a summary of all relevant results from quaternionic analysis to make the article self contained

multidimensional phase space and sunset diagrams we derive expressions for the phase space of a particle of momentum p decaying into n particles that are valid for any number of dimensions these are the imaginary parts of so called sunset diagrams which we also obtain the results are given as a series of hypergeometric functions which terminate for odd dimensions and are also well suited for deriving the threshold behavior

recursive algorithm for evaluating vertex type feynman integrals an algorithm for evaluating vertex type loop integrals is considered it is based on applying the integration by parts technique as an example a class of massless integrals corresponding to triangle diagrams is considered the presented method can also be applied to loop diagrams with a larger number of external lines as well as to integrals with massive denominators