inverse avalanches in the abelian sandpile model we define and study the inverse of particle addition process in the abelian sandpile model we show how to obtain the unique recurrent configuration corresponding to a single particle deletion by a sequence of operations called inverse avalanches we study the probability distribution of s 1 the number of untopplings in the first inverse avalanche for a square lattice we determine prob s 1 exactly for s 1 0 1 2 and 3 for large s 1 we show that prob s 1 varies as s 1 11 8 in the direct avalanches this is related to the probability distribution of the number of sites which topple as often as the site where the particle was added these results are verified by numerical simulations

scaling and river networks a landau theory for erosion we propose a coarse grained theory for the formation of a river network in the form of a langevin equation for the erosion of the landscape coupled to a conservation law for the surface water flow we claim that this is the universal form for large scale behavior we show by simulations of a discrete model that represents the same dynamics that the slope area law the basin size distribution law and horton s laws agree with real rivers we discuss the relationship to optimal channel networks and to self organized criticality

sandpiles on a sierpinski gasket abstract we perform extensive simulations of the sandpile model on a sierpinski gasket critical exponents for waves and avalanches are determined we extend the existing theory of waves to the present case this leads to an exact value for the exponent w which describes the distribution of wave sizes w ln 9 5 ln 3 numerically it is found that the number of waves in an avalanche is proportional to the number of distinct sites toppled in the avalanche this leads to a conjecture for the exponent that determines the distribution of avalanche sizes 1 w ln 27 5 ln 3 our predictions are in good agreement with the numerical results

polynomially computable bounds for the probability of the union of events we consider the problem of finding upper and lower bounds for the probability of the union of events when the probabilities of the single events and the probabilities of the intersections of up to m events are given it is known that the best possible bounds can be obtained by solving linear programming problems with a number of variables that is exponential in the number of events because of their size and structure these large linear programs are known to be very hard to solve in the literature simpler polynomially sized aggregations are considered and numerous closed form or polynomially computable bounds are derived from those we present here a new approach that introduces additional constraints to the dual linear programming problems in such a way that those become polynomially solvable by using different sets of additional constraints we introduce three new classes of polynomially computable upper and lower bounds we show that they dominate almost all efficiently computable bounds known in t

lifting and separation procedures for the cut polytope the max cut problem and the associated cut polytope on complete graphs have been extensively studied over the last 25 years however in comparison only little research has been conducted for the cut polytope on arbitrary graphs in particular separation algorithms have received only little attention in this study we describe new separation and lifting procedures for the cut polytope on general graphs these procedures exploit algorithmic and structural results known for the cut polytope on complete graphs to generate valid and sometimes facet defining inequalities for the cut polytope on arbitrary graphs in a cutting plane framework we report computational results on a set of well established benchmark problems

expansion and contraction of avalanches in the two dimensional abelian sandpile we present a detailed analysis of large scale simulations of avalanches in the 2d abelian sandpile model we compare statistical properties of two different decompositions of avalanches into clusters of topplings and waves of topplings auxiliary critical exponents are introduced and the existence of the exponent governing the contraction of avalanches claimed in our previous work priezzhev et al prl 76 2093 1996 is confirmed we also give more elaborated argumentation for the exact values of the exponents characterizing the statistics of waves

critical dynamics of self organizing eulerian walkers the model of self organizing eulerian walkers is numerically investigated on the square lattice the critical exponents for the distribution of a number of steps tau l and visited sites tau s characterizing the process of transformation from one recurrent configuration to another are calculated using the finite size scaling analysis two different kinds of dynamical rules are considered the results of simulations show that both the versions of the model belong to the same class of universality with the critical exponents tau l tau s 1 75 pm 0 1

on nonconvex quadratic programming with box constraints nonconvex quadratic programming with box constraints is a fundamental mathcal np hard global optimization problem recently some authors have studied a certain family of convex sets associated with this problem we prove several fundamental results concerned with these convex sets we determine their dimension characterize their extreme points and vertices show their invariance under certain affine transformations and show that various linear inequalities induce facets we also show that the sets are closely related to the boolean quadric polytope a fundamental polytope in the field of polyhedral combinatorics finally we give a classification of valid inequalities and show that this yields a finite recursive procedure to check the validity of any proposed inequality

on the cut polytope the cut polytopep c g of a graphg v e is the convex hull of the incidence vectors of all edge sets of cuts ofg we show some classes of facet defining inequalities ofp c g we describe three methods with which new facet defining inequalities ofp c g can be constructed from known ones in particular we show that inequalities associated with chordless cycles define facets of this polytope moreover for these inequalities a polynomial algorithm to solve the separation problem is presented we characterize the facet defining inequalities ofp c g ifg is not contractible tok 5 we give a simple characterization of adjacency inp c g and prove that for complete graphs this polytope has diameter one and thatp c g has the hirsch property a relationship betweenp c g and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied

universality in sandpile models a classification of sandpile models into universality classes is presented on the basis of extensive numerical simulations in which we measure an extended set of exponents the manna two state model s s manna j phys a 24 l363 1991 is found to belong to a universality class of random neighbor models which is distinct from the universality class of the original model of bak tang and wiesenfeld p bak c tang and k wiesenfeld phys rev lett 59 381 1987 directed models are found to belong to a universality class which includes the directed model introduced and solved by dhar and ramaswamy d dhar and r ramaswamy phys rev lett 63 1659 1989

waves of topplings in an abelian sandpile abstract we study the structure of the avalanche process in the 2d abelian sandpile model an avalanche may be represented as a sequence of waves each consisting of sites that toppled only once in that wave it is shown that the waves are in one to one correspondence with the set of two rooted spanning trees and can be described in terms of green functions we argue that the probability distribution of waves of size s varies as 1 s for large s we prove also that the avalanches started at the open boundary consist of only one wave and find the asymptotic distribution of their sizes which varies as s 3 2 we establish the equivalence between waves and inverse avalanches introduced by dhar and manna and reproduce their result 11 8 for the critical exponent of the size distribution of the first inverse avalanche or in our terms the last wave

distribution of sizes of erased loops for loop erased random walks the loop erased random walk lerw is a simpler variant of the well known self avoiding walk saw problem which retains the no self intersection property of saws but is closer to the ordinary random walk problem in this paper we study the distribution of sizes of erased loops for lerws on arbitrary graphs and relate it to the distribution of sizes of loops formed when a bond is added to a random spanning tree on the same graph in the themodynamic limit this distribution has a power law tail and we express the exponent in terms of the fractal dimension of the chemical paths on spanning trees we show also how this relation is modified for deterministic fractals in spite of the fact that the lerw model is somewhat more tractable analytically than the saw problem the number of papers devoted to this problem has remained rather small the model was defined by lawler 1 who called it the loop erased self avoiding walk lesaw this terminology is somewhat inappropriate and

construction of confidence limits about effect measures a general approach it is widely accepted that confidence interval construction has important advantages over significance testing for the presentation of research results as now facilitated by readily available software however for a number of effect measures procedures are either not available or not satisfactory in samples of small to moderate size in this paper we describe a general approach for estimating a difference between effect measures which can also be used to obtain confidence limits for a risk ratio and a lognormal mean numerical evaluation shows that this closed form procedure outperforms existing methods including the bootstrap

conformal field theory correlations in the abelian sandpile model we calculate all multipoint correlation functions of all local bond modifications in the two dimensional abelian sandpile model both at the critical point and in the model with dissipation the set of local bond modifications includes as the most physically interesting case all weakly allowed cluster variables the correlation functions show that all local bond modifications have scaling dimension 2 and can be written as linear combinations of operators in the central charge ensuremath 2 logarithmic conformal field theory in agreement with a form conjectured earlier by mahieu and ruelle in phys rev e 64 066130 2001 we find closed form expressions for the coefficients of the operators and describe methods that allow their rapid calculation we determine the fields associated with adding or removing bonds both in the bulk and along open and closed boundaries some bond defects have scaling dimension 2 while others have scaling dimension 4 we also determine the corrections to bulk probabilities for local bond modifications near open and closed boundaries

power spectrum of mass and activity fluctuations in a sandpile we consider a directed abelian sandpile on a strip of size 2 ifmmode times else texttimes fi n driven by adding a grain randomly at the left boundary after every t timesteps we establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeros in the ternary base representation of the position of a random walker on a ring of size 3 n we find that while the fluctuations of mass have a power spectrum that varies as 1 f for frequencies in the range 3 ensuremath 2n ensuremath ll f ensuremath ll 1 t the activity fluctuations in the same frequency range have a power spectrum that is linear in f

mean field self organized criticality the authors consider a variant of directed percolation in which the set of nodes is itself dependent on the progress of the percolation process for the mean field model studied the system is driven to its critical point from a broad range of initial conditions an advantage of the model is its explicit and easy solvability

solving quadratic 0 1 problems by semidefinite programs and cutting planes we present computational experiments for solving quadratic 0 1 problems our approach combines a semidefinite relaxation with a cutting plane technique and is applied in a branch and bound setting our experiments indicate that this type of approach is very robust and allows to solve many moderately sized problems having say less than 100 binary variables in a routine manner 1998 the mathematical programming society inc published by elsevier science b v

bayesian modeling of multivariate average bioequivalence bioequivalence trials are usually conducted to compare two or more formulations of a drug simultaneous assessment of bioequivalence on multiple endpoints is called multivariate bioequivalence despite the fact that some tests for multivariate bioequivalence are suggested current practice usually involves univariate bioequivalence assessments ignoring the correlations between the endpoints such as auc and cmax in this paper we develop a semiparametric bayesian test for bioequivalence under multiple endpoints specifically we show how the correlation between the endpoints can be incorporated in the analysis and how this correlation affects the inference resulting estimates and posterior probabilities borrow strength from one another where the amount and the direction of the strength borrowed are determined by the prior correlations the method developed is illustrated using a real data set copyright 2007 john wiley sons ltd

exact solution of a stochastic directed sandpile model we introduce and analytically solve a directed sandpile model with stochastic toppling rules the model clearly belongs to a different universality class from its counterpart with deterministic toppling rules previously solved by dhar and ramaswamy the critical exponents are d 7 4 ensuremath tau 10 7 in two dimensions and d 3 2 ensuremath tau 4 3 in one dimension the upper critical dimension of the model is three at which the exponents apart from logarithmic corrections reach their mean field values d 2 ensuremath tau 3 2

self organized critical state in a directed sandpile automaton on bethe lattices equivalence to site percolation the author considers a directed sandpile automaton on the bethe lattice and shows that the system evolves ergodically to every configuration with equal probability irrespective of the initial state the author makes it clear that the system is driven into the self organized critical state which is in equivalence to site percolation at the critical percolation threshold on the bethe lattice

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

unified view of scaling laws for river networks scaling laws that describe the structure of river networks are shown to follow from three simple assumptions these assumptions are 1 river networks are structurally self similar 2 single channels are self affine and 3 overland flow into channels occurs over a characteristic distance drainage density is uniform we obtain a complete set of scaling relations connecting the exponents of these scaling laws and find that only two of these exponents are independent we further demonstrate that the two predominant descriptions of network structure tokunaga s law and horton s laws are equivalent in the case of landscapes with uniform drainage density the results are tested with data from both real landscapes and a special class of random networks

self organized critical pinball machine abstract the nature of self organized criticality soc is pin pointed with a simple mechanical model a pinball machine its phase space is fully parameterized by two integer variables one describing the state of an on going game the other describing the state of the machine this is the simplest possible soc system having only two degrees of freedom and no spatial correlations yet is not solvable by analytical means

area distribution for directed random walks we study the probability distribution for the area under a directed random walk in the plane the walk can serve as a simple model for avalanches based on the idea that the front of an avalanche can be described by a random walk and the size is given by the area enclosed this model captures some of the qualitative features of earthquakes avalanches and other self organized critical phenomena in one dimension by finding nonlinear functional relations for the generating functions we calculate directly the exponent in the size distribution law and find it to be 4 3

semiparametric bayesian modeling of multivariate average bioequivalence bioequivalence trials are usually conducted to compare two or more formulations of a drug simultaneous assessment of bioequivalence on multiple endpoints is called multivariate bioequivalence despite the fact that some tests for multivariate bioequivalence are suggested current practice usually involves univariate bioequivalence assessments ignoring the correlations between the endpoints such as auc and cmax in this paper we develop a semiparametric bayesian test for bioequivalence under multiple endpoints specifically we show how the correlation between the endpoints can be incorporated in the analysis and how this correlation affects the inference resulting estimates and posterior probabilities borrow strength from one another where the amount and direction of the strength borrowed are deter department of mathematics and statistics georgia state university atlanta ga 30303 3083 usa email pghosh mathstat gsu edu department of epidemiology and biostatistics memorial sloan kettering cancer center 1275 york avenue new york ny 10021

the upper critical dimension of the abelian sandpile model the existing estimation of the upper critical dimension of the abelian sandpile model is based on a qualitative consideration of avalanches as self avoiding branching processes we find an exact representation of an avalanche as a sequence of spanning subtrees of two component spanning trees using equivalence between chemical paths on the spanning tree and loop erased random walks we reduce the problem to determination of the fractal dimension of spanning subtrees then the upper critical dimension du 4 follows from lawler s theorems for intersection probabilities of random walks and loop erased random walks

in vitro bioequivalence testing a statistical test is proposed for in vitro bioequivalence testing between drug products such as nasal aerosols and nasal sprays the proposed test generalizes the one recommended in the fda 1999 guidance to the situation where replicated observations obtained from each sampled canister or bottle of the drug product are available the technique developed by hyslop hsuan and holder is used so that the proposed test is asymptotically accurate the type i error probability and power of the proposed test are investigated through a simulation study a method for determining the required sample size to achieve a desired power is also proposed a numerical example is given for illustration

a lagrangian relaxation approach to the edge weighted clique problem the b clique polytope cp n b is the convex hull of the node and edge incidence vectors of all subcliques of size at most b of a complete graph on n nodes including the boolean quadric polytope qp n as a special case and being closely related to the quadratic knapsack polytope it has received considerable attention in the literature in particular the max cut problem is equivalent with optimizing a linear function over qp n n the problem of optimizing linear functions over cp n b has so far been approached via heuristic combinatorial algorithms and cutting plane methods we study the structure of cp n b in further detail and present a new computational approach to the linear optimization problem based on lucena s suggestion of integrating cutting planes into a lagrangian relaxation of an integer programming problem in particular we show that the separation problem for tree inequalities becomes polynomial in our lagrangian framework finally computational results are presented

the boolean quadric polytope some characteristics facets and relatives we study unconstrained quadratic zero one programming problems havingn variables from a polyhedral point of view by considering the boolean quadric polytope qpn inn n 1 2 dimensions that results from the linearization of the quadratic form we show that qpn has a diameter of one descriptively identify three families of facets of qpn and show that qpn is symmetric in the sense that all facets of qpn can be obtained from those that contain the origin by way of a mapping the naive linear programming relaxation qpnlp of qpn is shown to possess the trubin property and its extreme points are shown to be 0 1 2 1 valued furthermore o n3 facet defining inequalities of qpn suffice to cut off all fractional vertices of qpnlp whereas the family of facets described by us has at least o 3n members the problem is also studied for sparse quadratic forms i e when several or many coefficients are zero and conditions are given for the previous results to carry over to this case polynomially solvable problem instances are discussed and it is shown that the known polynomiality result for the maximization of nonnegative quadratic forms can be obtained by simply rounding the solution to the linear programming relaxation in the case that the graph induced by the nonzero coefficients of the quadratic form is series parallel a complete linear description of the associated boolean quadric polytope is given the relationship of the boolean quadric polytope associated to sparse quadratic forms with the vertex packing polytope is discussed as well

applications of cut polyhedra ii abstract in this paper and in its continuation part ii this issue we group within a unified framework many applications of the following polyhedra cut boolean quadric hypermetric and metric polyhedra we treat in particular the following applications l1 and l1 metrics in functional analysis the max cut problem the boole problem and multicommodity flow problems in combinatorial optimization lattice holes in geometry of numbers density matrices of many fermions systems in quantum mechanics we present some other applications in probability theory statistical data analysis and design theory in this first part after introducing the main definitions and operations for cut polyhedra we describe the connections with l1 metrics and with other metric properties and we consider in particular the applications to some classes of metrics arising from graphs normed spaces and lattices

rare events and breakdown of simple scaling in the abelian sandpile model due to intermittency and conservation the abelian sandpile in two dimensions obeys multifractal rather than finite size scaling in the thermodynamic limit a vanishingly small fraction of large avalanches dominates the statistics and a constant gap scaling is recovered in higher moments of the toppling distribution thus rare events shape most of the scaling pattern and preserve a meaning for effective exponents which can be determined on the basis of numerical and exact results

self organizing height arrow model numerical and analytical results the recently introduced self organizing height arrow ha model is studied numerically on the square lattice and analytically on the bethe lattice the concentration of occupied sites and critical exponents of distributions of avalanches are evaluated for two slightly different versions of the model the exponents found for distributions of avalanches by mass area duration and appropriate fractal dimensions are close to those for the original sandpile model which suggests that the ha model belongs to the same universality class for comparison the concentration of occupied sites in the ha model is calculated exactly on the bethe lattice with coordination number q 4

critical exponents for boundary avalanches in two dimensional abelian sandpile we investigate the properties of boundary avalanches in 2d abelian sandpile model asm we construct the one to one correspondence of boundary avalanches and two rooted spanning trees using the connection between the obtained graph representation and lattice green functions we calculate the exact values of critical exponents for size and lifetime distributions of avalanches starting at the open boundary that forms an angle alpha we find that the probability of a boundary avalanche of the size s varies as s 1 pi 2 alpha for large s and the probability of an avalanche of lifetime t varies as t 1 4 pi 5 alpha for large t the obtained values are verified by numerical simulations

self organized criticality in a sandpile model with threshold dissipation we study a nonconservative sandpile model in one dimension in which if the height at any site exceeds a threshold value the site topples by transferring one particle along each bond connecting it to its neighbors its height is then set to 1 irrespective of the initial value the model shows nontrivial critical behavior we solve this model analytically in one dimension for all driving rates we calculate all the two point correlation functions in this model and find that the average local height decreases as the inverse of the distance from the nearest boundary and the power spectrum of fluctuations of the total mass varies as 1 f

toppling distributions in one dimensional abelian sandpiles we consider toppling distributions for the abelian sandpile model in one dimension we study the avalanche mass and duration distributions in the thermodynamic limit we also investigate their dependence on the seeding distribution which describes how sand is dropped none of the models shows criticality

the noise spectrum in the model of self organised criticality the explanation of 1 f noise with the help of models of self organised criticality makes use of combining scaling and the distributed time constant picture limits of applicability of this type of arguments are presented the numerical study of the two dimensional model of bak tang and wiesenfeld 1987 shows that although non trivial scaling is present the power spectrum of the total process has a approximately 1 f2 behaviour

facets of the bipartite subgraph polytope the bipartite subgraph polytope pbg of a graph g v e is the convex hull of the incidence vectors of all edge sets of bipartite subgraphs of g we show that all complete subgraphs of g of odd order and all so called odd bicycle wheels contained in g induce facets of pbg moreover we describe several methods with which new facet defining inequalities of pbg can be constructed from known ones examples of these methods are contraction of node sets in odd complete subgraphs odd subdivision of edges certain splittings of nodes and subdivision of all edges of a cut using these methods we can construct facet defining inequalities of pbg having coefficients of order v 2

a polyhedral study of binary polynomial programs we study the polyhedral convex hull of a mixed integer set x1d4ae defined by a collection of multilinear equations over the unit hypercube such sets appear frequently in the factorable reformulation of mixed integer nonlinear optimization problems in particular the set x1d4ae represents the feasible region of a linearized unconstrained binary polynomial optimization problem we define an equivalent hypergraph representation of the mixed integer set x1d4ae which enables us to derive several families of facet defining inequalities structural properties and lifting operations for its convex hull in the space of the original variables our theoretical developments extend several well known results from the boolean quadric polytope and the cut polytope literature paving a way for devising novel optimization algorithms for nonconvex problems containing multilinear sub expressions

an extended formulation approach to the edge weighted maximal clique problem we consider an extended formulation approach to the edge weighted maximal clique problem the problem is formulated by using additional variables for the set of nodes with the natural variables for the set of edges we show that the proposed formulation is superior to the natural formulation both theoretically and practically by using the projection technique we can also derive new classes of facet defining inequalities for the lower dimensional polytope of the natural variables computational results are reported

sandpile models with and without an underlying spatial structure we present a simple mean field model for the sandpile model introduced by bak tang and wiesenfeld btw phys rev lett 59 381 1987 in the mean field model we are able to pinpoint the process of self organization as well as the emerging scale invariance displayed as a power law distribution of avalanche sizes we discuss the btw sandpile model on a lattice and show that the dynamical behavior can be expressed as a transport problem this implies that the average avalanche size scales with the system size and additional heuristic arguments related to the transport properties more than indicate the origin of the power law behavior we review recent work in which scaling relations and additional constraints between the various critical exponents are addressed we demonstrate that some of the proposed relations are inconsistent we present a coherent theory in which the scaling relations along with additional constraints leave only one exponent unknown

binary positive semidefinite matrices and associated integer polytopes we consider the positive semidefinite psd matrices with binary entries along with the corresponding integer polytopes we begin by establishing some basic properties of these matrices and polytopes then we show that several families of integer polytopes in the literature the cut boolean quadric multicut and clique partitioning polytopes are faces of binary psd polytopes finally we present some implications of these polyhedral relationships in particular we answer an open question in the literature on the max cut problem by showing that the rounded psd inequalities define a polytope

equivalence between the abelian sandpile model and the q 0 limit of the potts model abstract we establish an equivalence between the undirected abelian sandpile model and the q 0 limit of the q state potts model the equivalence is valid for arbitrary finite graphs two dimensional abelian sandpile models thus correspond to a conformal field theory with central charge c 2 the equivalence also gives a monte carlo algorithm to generate random spanning trees we study the growth process of the spread of fire under the burning algorithm in the background of a random recurrent configuration of the abelian sandpile model the average number of sites burnt upto time t varies at ta in two dimensions our numerically determined value of a agrees with the theoretical prediction a 8 5 we relate this exponent to the conventional exponents characterizing the distributions of avalanche sizes

patterns formed by addition of grains to only one site of an abelian sandpile abstract we study the patterns formed by adding n sand grains at a single site in the abelian sandpile model on an infinite square lattice when in the initial configuration all sites have same height when this height is 2 stable heights being 0 3 we show that the perturbed region is a square whose length increases as n for large n if all lengths are rescaled by n the pattern tends to a limiting pattern in which the locally averaged height takes piece wise constant rational values we study the structure of these patterns and also other initial configurations we introduce a toppling function that fully describes these we show that it is piecewise quadratic and determine its form as well as the relation with the height configurations of the pattern

asymmetric abeiian sandpile models in the abelian sandpile models introduced by dhar long time behavior is determined by an invariant measure supported uniformly on a set of implicitly defined recurrent configurations of the system dhar proposed a simple procedure theburning algorithm as a possible test of whether a configuration is recurrent and later with majumdar verified the correctness of this test when the toppling rules of the sandpile are symmetric we observe that the test is not valid in general and give a new algorithm which yields a test correct for all sandpiles we also obtain necessary and sufficient conditions for the validity of the original test the results are applied to a family of deterministic one dimensional sandpile models originally studied by lee liang and tzeng

self organized critical forest fire model a forest fire model is introduced which contains a lightning probability f this leads to a self organized critical state in the limit f ensuremath rightarrow 0 provided that the time scales of tree growth and burning down of forest clusters are separated we derive scaling laws and calculate all critical exponents the values of the critical exponents are confirmed by computer simulations for a two dimensional system we show that the forest density in the critical state assumes its minimum possible value i e that energy dissipation is maximum

solving the maximum edge weight clique problem via unconstrained quadratic programming the unconstrained quadratic binary program uqp is proving to be a successful modeling and solution framework for a variety of combinatorial optimization problems experience reported in the literature with several problem classes has demonstrated that this approach works surprisingly well in terms of solution quality and computational times often rivaling and sometimes surpassing more traditional methods in this paper we report on the application of uqp to the maximum edge weighted clique problem computational experience is reported illustrating the attractiveness of the approach

unbounded convex sets for non convex mixed integer quadratic programming this paper introduces a fundamental family of unbounded convex sets that arises in the context of non convex mixed integer quadratic programming it is shown that any mixed integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a face of a set in the family some fundamental properties of the convex sets are derived along with connections to some other well studied convex sets several classes of valid and facet inducing inequalities are also derived

a multivariate test for population bioequivalence in this article we propose a multivariate generalization of the criteria for testing univariate population bioequivalence recently a number of approaches for testing multivariate equivalence have appeared in the literature most of them consider a multivariate equivalence region which implies simultaneous comparison of means in each dimension in contrast our proposal combines a comparison of means and a comparison of variances into a single aggregate criterion using the trace of the covariance matrix as a scalar measure of the total variability we use a confidence interval approach to multivariate population bioequivalence testing similar to the univariate case two versions of the modified large sample confidence interval for the linearized multivariate criterion are constructed in a simulation study we evaluate the empirical coverage of these confidence intervals and rejection rates of the corresponding tests in finite samples the proposed methodology is illustrated with an example of testing equivalence of the spray pattern of nasal sprays

cut polytopes boolean quadratic polytopes and nonnegative quadratic pseudo boolean functions in this paper we present a class of valid inequalities as well as a class of facets for the cut polytope of the complete graph it is shown that many of the known classes of valid inequalities e g triangle hypermetric and cycle inequalities belong to this class by specializing some of the parameters large classes of facets can be defined it is shown that each of these classes contains at least n 3n 3 facets where n 10 denotes the number of vertices and that the corresponding separation problems are polynomially solvable these results are based on a one to one correspondence established between the set of valid inequalities facets for the cut polytope of kn 1 and the set of nonnegative extremal quadratic pseudo boolean functions in n variables

scaling optimality and landscape evolution a nonlinear model is studied which describes the evolution of a landscape under the effects of erosion and regeneration by geologic uplift by mean of a simple differential equation the equation already in wide use among geomorphologists and in that context obtained phenomenologically is here derived by reparametrization invariance arguments and exactly solved in dimension d 1 results of numerical simulations in d 2 show that the model is able to reproduce the critical scaling characterizing landscapes associated with natural river basins we show that configurations minimizing the rate of energy dissipation optimal channel networks are stationary solutions of the equation describing the landscape evolution numerical simulations show that a careful annealing of the equation in the presence of additive noise leads to configurations very close to the global minimum of the dissipated energy characterized by mean field exponents we further show that if one considers generalized river network configurations in which splitting of the flow i e braiding and loops are allowed the minimization of the dissipated energy results in spanning loopless configurations under the constraints imposed by the continuity equations this is stated in the form of a general theorem applicable to generic networks suggesting that other branching structures occurring in nature may possibly arise as optimal structures minimizing a cost function

exactly solved model of self organized critical phenomena we define a variant of the model of bak tang and wiesenfeld of self organized critial behavior by introducing a preferred direction we characterize the critical state and by establishing equivalence to a voter model determine the critical exponents exactly in arbitrary dimension d the upper critical dimension for this model is three in two dimensions the model is equivalent to an earlier solved special case of directed percolation

numerical determination of the avalanche exponents of the bak tang wiesenfeld model we consider the bak tang wiesenfeld sandpile model on a two dimensional square lattice of lattice sizes up to l 4096 a detailed analysis of the probability distribution of the size area duration and radius of the avalanches will be given to increase the accuracy of the determination of the avalanche exponents we introduce a new method for analyzing the data which reduces the finite size effects of the measurements the exponents of the avalanche distributions differ slightly from previous measurements and estimates obtained from a renormalization group approach

self organized formulation of standard percolation phenomena abstract we consider standard percolation processes such as epidemic processes with or without immunization we show that their dynamics can be formulated so that they mimic self organized critical phenomena the wetting probability p needs not to be fine tuned to its critical value p c in order to arrive at criticality but it rather emerges as a singularity in some time dependent distribution on the one hand this casts doubts on the significance of self organized as opposed to ordinary criticality on the other hand it suggests very efficient algorithms where percolation problems are studied at several values of p in a single run as an example we apply such an algorithm to directed percolation in 2 1 dimensions where it allows a very precise determination of critical behavior

introduction to the sandpile model abstract this article is based on a talk given by one of us evi at the conference statphys taipei 1997 it overviews the exact results in the theory of the sandpile model and discusses briefly the yet unsolved problem of calculation of avalanche distribution exponents the key ingredients include the analogy with the critical reaction diffusion system the spanning tree representation of height configurations and the decomposition of the avalanche process into waves of topplings

theoretical studies of self organized criticality these notes are intended to provide a pedagogical introduction to the abelian sandpile model of self organized criticality and its related models the abelian group the algebra of particle addition operators the burning test for recurrent states equivalence to the spanning trees problem are described the exact solution of the directed version of the model in any dimension is explained the model s equivalence to scheidegger s model of river basins takayasu s aggregation model and the voter model is discussed for the undirected case the solution for one dimensional lattices and the bethe lattice is briefly described known results about the two dimensional case are summarized generalization to the abelian distributed processors model is discussed time dependent properties and the universality of critical behavior in sandpiles are briefly discussed i conclude by listing some still unsolved problems

structure of two dimensional sandpile i height probabilities the height probabilities of the two dimensional abelian sandpile model are the fractionial numbers of lattice sites having heights 1 2 3 4 a combinatorial method for evaluation of these quantities is proposed the method is based on mapping the set of allowed sandpile configurations onto the set of spanning trees covering a given lattice exact analytical expressions for all probabilities are obtained

algebraic aspects of abelian sandpile models the abelian sandpile models feature a finite abelian group g generated by the operators corresponding to particle addition at various sites we study the canonical decomposition of g as a product of cyclic groups g z d1 z d2 z d3 z dg where g is the least number of generators of g and di is a multiple of di 1 the structure of g is determined in terms of the toppling matrix delta we construct scalar functions linear in the height variables of the pile that are invariant under toppling at any site these invariants provide convenient coordinates to label the recurrent configurations of the sandpile for an l l square lattice we show that g l in this case we observe that the system has non trivial symmetries transcending the obvious symmetries of the square namely those coming from the action of the cyclotomic galois group gall of the 2 l 1 th roots of unity which operates on the set of eigenvalues of h these eigenvalues are algebraic integers the product of which is the order mod g mod with the help of gall we are able to group the eigenvalues into certain subsets the products of which are separately integers and thus obtain an explicit factorization of mod g mod we also use gall to define other simpler sets of toppling invariants

logarithmic corrections of the avalanche distributions of sandpile models at the upper critical dimension we study numerically the dynamical properties of the btw model on a square lattice for various dimensions the aim of this investigation is to determine the value of the upper critical dimension where the avalanche distributions are characterized by the mean field exponents our results are consistent with the assumption that the scaling behavior of the four dimensional btw model is characterized by the mean field exponents with additional logarithmic corrections we benefit in our analysis from the exact solution of the directed btw model at the upper critical dimension which allows to derive how logarithmic corrections affect the scaling behavior at the upper critical dimension similar logarithmic corrections forms fit the numerical data for the four dimensional btw model strongly suggesting that the value of the upper critical dimension is four