ternary composition algebras and hurwitz s theorem axioms are proposed for a certain alternative kind of ternary composition algebra termed a 3cn algebra the axioms are shown to be for n 2 in a simple correspondence with the axioms for a ternary vector cross product algebra the axioms imply that n 1 2 4 or 8 from which the usual hurwitz theorem is deduced the existence of 3c8 algebras is demonstrated by an explicit construction in four dimensional hilbert space without appeal to the properties of the algebra of octonions

a new view of d 7 clifford algebra a new view of d 7 clifford algebra is described in which starting from a septet of mutually anticommuting clifford operators one introduces a septet of mutually commuting involutions in so 8 these involutions can be seen to arise naturally from a study of ternary vector cross products in eight dimensions

sur les dimensions des algebres absolument valuees abstract let a denote an arbitrary absolute valued real algebra in proc amer math soc 11 1960 861 866 urbanik and wright showed that if a is commutative then a has dimension 1 or 2 and is isomorphic to r or c in this paper we show that if in the previous result we replace the commutativity by the weaker assumption that it is flexible then a is still finite dimensional but in this case a has dimension 1 2 4 or 8 and is isotopic to r c h or c moreover we consider various conditions on a which imply that it is finite dimensional necessarily of dimension 1 2 4 or 8 in fact we prove that it is the case when one of the following holds i x x x 0 for all x a and there exists in a a non zero element which commutes with all elements of a ii there exists a non zero element a in a such that a and a2 commute with all elements of a and x a x 0 for all x a

brief paper guaranteed cost control of stochastic uncertain systems with slope bounded nonlinearities via the use of dynamic multipliers this paper presents a new approach to constructive output feedback robust nonlinear guaranteed cost controller design the approach involves a class of controllers which include copies of the slope bounded nonlinearities occurring in the plant dynamic multipliers are introduced to exploit these repeated nonlinearities the linear part of the controller is synthesized using minimax lqg control theory

ternary vector cross products ternary vector cross products are studied in their own right results include a new proof of hurwitz s theorem and a principle of duplicity upon breaking the symmetry in eight dimensions by choosing a preferred axis this last principle implies the well known triality principle for octonions and so 8 transformations in displaying canonical forms it helps to put the eight basis vectors in correspondence with the eight points of the three dimensional affine geometry over f2

robust mean field linear quadratic gaussian games with unknown l 2 disturbance this paper considers a class of mean field linear quadratic gaussian games with model uncertainty the drift term in the dynamics of the agents contains a common unknown function we take a robust

on absolute valued algebras with involution abstract let a be an absolute valued algebra with involution in the sense of urbanik k urbanik absolute valued algebras with an involution fund math 49 1961 247 258 we prove that a is finite dimensional if and only if the algebra obtained by symmetrizing the product of a is simple if and only if ea s a s where e denotes the unique nonzero self adjoint idempotent of a and a s stands for the set of all skew elements of a we determine the idempotents of a and show that a is the linear hull of the set of its idempotents if and only if a is equal to either mcclay s algebra a a albert a note of correction bull amer math soc 55 1949 1191 the para quaternion algebra or the para octonion algebra we also prove that if a is infinite dimensional then it can be enlarged to an absolute valued algebra with involution having a nonzero idempotent different from the unique nonzero self adjoint idempotent

sedenions algebra and analysis a 16 dimensional cayley dickson algebra is presented its unique algebraic properties its zero divisors and the solutions to a general linear equation are found a theory of function is developed in terms of the regularity monogenicity conditions and some such functions are constructed

robust control of nonlinear jump parameter systems governed by uncertain chains we consider an infinite horizon minimax optimal control problem for stochastic uncertain systems governed by a discrete state uncertain continuous time chain using existing risk sensitive control results a robust suboptimal absolutely stabilizing guaranteed cost controller is constructed conditions are presented under which this suboptimal controller is minimax optimal we then present a numeric algorithm for calculating a robust sub optimal controller using a markov chain approximation technique

dynamic simulation of the centripetal packing of mono sized spheres this paper presents a study of the centripetal packing of mono sized spherical particles simulated by means of the granular dynamic or discrete element method a packing is formed by imposing an assumed centripetal force on particles randomly generated in a spherical space different from the conventional simulation techniques dynamic information of individual particles including transient forces and trajectory is traced in the present simulation structural properties such as packing density radial distribution function coordination number distribution and homogeneity are analyzed with particular reference to the eects of the magnitude of the centripetal force and the number of particles comparison with the literature results suggests that such a dynamic model can satisfactorily simulate the dynamics of forming a packing and produce more realistic structural information in particular it is conrmed that a centripetal packing is not homogeneous in structure becoming looser as its size or the number of particles increases the packing has a limit packing density of 0 637 0 645 an overall mean coordination number of around 6 0 and a radial distribution function of clear split second peak the centripetal force aects the rate of densication and the mean coordination number but not packing density and radial distribution function c 1999 elsevier science b v all rights reserved

state distributions and minimum relative entropy noise sequences in uncertain stochastic systems the discrete time case this paper is concerned with dissipativity theory and robust performance analysis and design of discrete time stochastic systems driven by statistically uncertain random noise the uncertainty is quantified by the conditional relative entropy of the actual probability law of the noise with respect to a nominal product measure corresponding to a white noise sequence we discuss a balance equation dissipation inequality and superadditivity property for the corresponding conditional relative entropy supply as a function of time the problem of minimizing the supply required to drive the system between given state distributions over a specified time horizon is considered such variational problems involving entropy and probabilistic boundary conditions are known in the literature as schrodinger bridge problems in application to control systems the minimum required conditional relative entropy supply characterizes the robustness of the system with respect to a statistically uncertain random noise we o

ternary composition algebras i structure theorems definite and neutral signatures previous work on ternary composition algebras of definite signature is generalized to the only other case allowed by the axioms namely that of neutral signature one new feature is the presence in the neutral case of counter automorphisms9 the structure of the exceptional eight dimensional algebras is probed and in various guises a fundamental dichotomy is unearthed canonical forms are obtained and various further properties are derived

a six variable identity for a ternary vector cross product in eight dimensional space let x a b c be a ternary vector cross product for eight dimensional euclidean space e an identity is derived which expresses x a b c x u v w in terms of the spin 7 invariant scalar quadruple product phi a b c d a x b c d the proof of the identity is coordinate free and starts out from an explicit expression for x with e viewed as complex four dimensional hilbert space

statistical mechanics of hard ellipsoids i overlap algorithm and the contact function abstract a contact function for two arbitrary ellipsoids is derived the numerical value of the contact function is less than 1 if they overlap and greater than 1 if they do not this extends previous work by vieillard baron who derived an overlap criterion for spheroids without use of a contact function the equivalence of the two criteria has been checked by extensive numerical test with spheroids it is shown that the use of the contact function greatly facilitates the calculation of the pressure in monte carlo simulations

the equivalence between infinite horizon optimal control of stochastic systems with exponential of integral performance index and stochastic differential games a new method based on the theory of large deviations from the invariant measure is introduced for the analysis of stochastic systems with an infinite horizon exponential of integral performance index it is shown that the infinite horizon optimal exponential of integral stochastic control problem is equivalent to a stationary stochastic differential game for an auxiliary system as an application of the developed technique the infinite horizon risk sensitive lqg problem is analyzed for both the completely observed and partially observed case

neighbor list collision driven molecular dynamics simulation for nonspherical hard particles we apply the algorithm presented in the first part of this series of papers to systems of hard ellipses and ellipsoids the theoretical machinery needed to treat such particles including the overlap potentials is developed in full detail we describe an algorithm for predicting the time of collision for two moving ellipses or ellipsoids we present performance results for our implementation of the algorithm demonstrating that for dense systems of very aspherical ellipsoids the novel techniques of using neighbor lists and bounding sphere complexes offer as much as two orders of magnitude improvement in efficiency over direct adaptations of traditional event driven molecular dynamics algorithms the practical utility of the algorithm is demonstrated by presenting several interesting physical applications including the generation of jammed packings inside spherical containers the study of contact force chains in jammed packings and melting the densest known equilibrium crystals of prolate spheroids

ternary composition algebras ii automorphism groups and subgroups we study an eight dimensional ternary composition algebra e e of signature 8 0 or 4 4 such an algebra is associated with an involutary outer automorphism m of the lie algebra so e of the automorphisms and in the 4 4 case counter automorphisms of e are obtained in terms of the d 7 spin group determined by m using m we derive a principle of duplicity for the ternary multiplication related to the well known principle of triality for the associated octonionic binary multiplication however rather than construct e out of the octonions as done in effect by previous authors we give a four dimensional complex construction of e this non octonionic view of e highlights certain 15 dimensional d 6 spin groups rather than the customary 14 dimensional octonionic automorphism groups in the 4 4 case we can choose to interpret complex in terms of the split complex numbers and are thereby led to consider the subgroup chain so 4 4 spin 3 4 sl 4 spin 3 3 instead of the chain so 4 4 spin 3 4 su 2 2 spin 2 4

model simplification and optimal control of stochastic singularly perturbed systems under exponentiated quadratic cost we study the optimal control of a general class of stochastic singularly perturbed linear systems with perfect and noisy state measurements under positively and negatively exponentiated quadratic cost the expected cost function to be minimized is actually taken as the long term time average of the logarithm of the expected value of an exponentiated quadratic loss we identify appropriate slow and fast subproblems obtain their optimum solutions compatible with the corresponding measurement structure and subsequently study the performances they achieve on the full order system as the singular perturbation parameter epsilon becomes sufficiently small with the expressions given in all cases being exact to within o sqrt epsilon it is shown that the composite controller obtained by appropriately combining the optimum slow and fast controllers achieves a performance level close to the optimal one whenever the full order problem has a solution the slow controller on the other hand achieves asymptotically as epsilon to 0 only a finite performance level but not necessarily optimal provided that the fast subsystem is open loop stable if the intensity of the noise in the system dynamics decreases to zero however the slow controller also achieves a performance level close to the optimal one the paper also presents a more direct derivation than heretofore available of the solution to the linear exponential quadratic gaussian leqg problem under noisy state measurements which allows for a general quadratic cost with cross terms in the exponent and correlation between system and measurement noises and obtains both necessary and sufficient conditions for existence of an optimal solution such a general leqg problem is encountered in the slow fast decomposition of the full order problem even if the original problem does not feature correlated noises in this general context the paper also establishes a complete equivalence between the leqg problem and the hi optimal control problem with measurement feedback though this equivalence does not extend to the slow and fast subproblems arrived at after time scale separation

robust finite horizon minimax filtering for discrete time stochastic uncertain systems we study a finite horizon robust minimax filtering problem for time varying discrete time stochastic uncertain systems the uncertainty in the system is characterized by a set of probability measures under which the stochastic noises driving the system are defined the optimal minimax filter has been found by applying techniques of risk sensitive lqg control the structure and properties of resulting filter are analyzed and compared to h and kalman filters

brief paper risk sensitivity conditions for stochastic uncertain model validation the paper presents sufficient and necessary conditions that verify the relevance of an assumed linear stochastic system model for problems in which probabilistic characteristics of the plant are not known exactly the approach is to establish the existence of an admissible probability space on which the output of the candidate stochastic system model is consistent in a stochastic sense with the noisy output of the plant

neighbor list collision driven molecular dynamics simulation for nonspherical hard particles i algorithmic details in this first part of a series of two papers we present in considerable detail a collision driven molecular dynamics algorithm for a system of non spherical particles within a parallelepiped simulation domain under both periodic or hard wall boundary conditions the algorithm extends previous event driven molecular dynamics algorithms for spheres and is most efficient when applied to systems of particles with relatively small aspect ratios and with small variations in size we present a novel partial update near neighbor list nnl algorithm that is superior to previous algorithms at high densities without compromising the correctness of the algorithm this efficiency of the algorithm is further increased for systems of very aspherical particles by using bounding sphere complexes bsc these techniques will be useful in any particle based simulation including monte carlo and time driven molecular dynamics additionally we allow for a non vanishing rate of deformation of the boundary which can be used to model macroscopic strain and also alleviate boundary effects for small systems in the second part of this series of papers we specialize the algorithm to systems of ellipses and ellipsoids and present performance results for our implementation demonstrating the practical utility of the algorithm

on locally complex algebras and low dimensional cayley dickson algebras the paper begins with short proofs of classical theorems by frobenius and resp zorn on associative and resp alternative real division algebras these theorems characterize the first three resp four cayley dickson algebras then we introduce and study the class of real unital nonassociative algebras in which the subalgebra generated by any nonscalar element is isomorphic to c we call them locally complex algebras in particular we describe all such algebras that have dimension at most 4 our main motivation however for introducing locally complex algebras is that this concept makes it possible for us to extend frobenius and zorn s theorems in a way that it also involves the fifth cayley dickson algebra the sedenions

an overlapping detection algorithm for random sequential packing of elliptical particles abstract random sequential packing of particles is a subject of intense research in many branches of physics and engineering the preponderance of preview work has focused on spherical particles and little is known about non overlapping elliptical particles in the present work a new numerical algorithm is developed to detect overlapping of ellipses by a series of sequential coordinate transformations and a novel golden section search algorithm the accuracy and efficiency of the numerical algorithm are tested and compared with algorithms developed in previous literature its stability is verified by visualizations of random sequential packing of monodispersed and polydispersed ellipses with different boundary conditions then the algorithm is applied to investigate the influence of the shape of ellipses on the random packing fraction as well as to study the influence of the shape and size of ellipses on the wall effect in the particle packing structure and some numerical results are demonstrated finally the reliability of the numerical algorithm extended to the three dimensional space is also evaluated by checking overlapping of ellipsoidal particles

random sequential adsorption of lines and ellipses infinitely many lines of zero thickness may be adsorbed onto a plane without overlap and it is shown that the number of lines adsorbed should scale as 13 where is the number of adsorption trials per unit area of surface this is confirmed by numerical simulation the adsorption of ellipses with major and minor axes 2a 2b respectively is also studied the area coverage theta approaches its maximum value theta max as alpha as to infinity where alpha is typically 1 3 as b a decreases from unity theta max first increases to a maximum value 0 58 or 0 01 when b approximately 0 5 and then decreases a feature not noted in previous work

monte carlo study of the sphere packing problem we employ the monte carlo method to study a constrained optimization problem that is packing spheres with unequal radii into a 3 d bounded region selection of the best fit solution is based on using the boltzmann factor e e t to determine the transition probability which allows us to search for the global optimal solution we determined the least numbers of packed spheres that will occupy the largest volume the optimal occupied volume found is around 44 of a bounded region volume which is obtained within a relative short computing time this suggests that our result could be able to give a good starting point for the radiosurgery treatment plan

absolute valued algebras with involution and infinite dimensional terekhin s trigonometric algebras we prove that if a is an absolute valued algebra in the sense of k urbanik absolute valued algebras with an involution fund math 49 1961 247 258 then the normed space of a becomes a trigonometric algebra in the meaning of p a terekhin trigonometric algebras j math sci new york 95 1999 2156 2160 under the product defined by x y x y y x 2 moreover we show that essentially all infinite dimensional complete trigonometric algebras derive from absolute valued algebras by the above construction method

weakly compact homomorphisms we study the structure of weakly compact homomorphisms between banach algebras in particular it is shown that between many pairs of algebras the only weakly compact homomorphisms are those of finite rank

robust stability and performance of stochastic uncertain systems on an infinite time interval abstract in this paper we consider a robust stability problem for continuous time stochastic uncertain systems the uncertainty in the system is characterized in terms of an uncertain probability distribution on the noise input this uncertainty is assumed to satisfy a certain relative entropy constraint the solution to a specially parametrized risk sensitive performance analysis problem is used to estimate the level of guaranteed performance for the stochastic uncertain system under consideration this solution is obtained by solving an algebraic riccati equation the corresponding performance bound holds for all admissible uncertainties and is nonconservative

absolute valued algebraic algebras are finite dimensional abstract we prove that if a is a non associative algebra over the field of real numbers if there is a norm on a satisfying xy x y for all x y in a and if every one generated subalgebra of a is finite dimensional then a is finite dimensional

robust output feedback stabilization via risk sensitive control we consider a problem of robust linear quadratic gaussian lqg control for discrete time stochastic uncertain systems with partial state measurements for a finite horizon case the problem was recently introduced by petersen et al ieee trans automat control 45 2000 398 in this paper an infinite horizon extension of the results of petersen et al ieee trans automat control 45 2000 398 is discussed we show that for a broad class of uncertain systems under consideration a controller constructed in terms of the solution to a specially parameterized risk sensitive stochastic control problem absolutely stabilizes the stochastic uncertain system

underconstrained jammed packings of nonspherical hard particles ellipses and ellipsoids continuing on recent computational and experimental work on jammed packings of hard ellipsoids donev et al science 303 990 2004 we consider jamming in packings of smooth strictly convex nonspherical hard particles we explain why an isocounting conjecture which states that for large disordered jammed packings the average contact number per particle is twice the number of degrees of freedom per particle overline z 2 d f does not apply to nonspherical particles we develop first and second order conditions for jamming and demonstrate that packings of nonspherical particles can be jammed even though they are underconstrained hypoconstrained overline z l2 d f we apply an algorithm using these conditions to computer generated hypoconstrained ellipsoid and ellipse packings and demonstrate that our algorithm does produce jammed packings even close to the sphere point we also consider packings that are nearly jammed and draw connections to packings of deformable but stiff particles finally we consider the jamming conditions for nearly spherical particles and explain quantitatively the behavior we observe in the vicinity of the sphere point

minimax lqg control of stochastic partially observed uncertain systems we consider an infinite horizon linear quadratic minimax optimal control problem for stochastic uncertain systems with output measurement a new description of stochastic uncertainty is introduced using a relative entropy constraint for the stochastic uncertain system under consideration a connection between the worst case control design problem and a specially parametrized risk sensitive stochastic control problem is established using this connection a minimax optimal lqg controller is constructed which is based on a pair of algebraic matrix riccati equations arising in risk sensitive control it is shown that this minimax optimal controller absolutely stabilizes the stochastic uncertain system

observability of linear stochastic uncertain systems a concept of observability of uncertain systems has received some attention in the recent literature in this note we extend this concept to stochastic uncertain systems we show that the observability of a stochastic uncertain system can be characterized using a relative entropy functional which plays the same role as the observability gramian plays in linear system analysis also we establish a connection between the observability of a linear stochastic uncertain system under consideration and risk sensitive performance of an associated reference system

dichotomy and absolute stability of nonlinear systems with periodically nonstationary linear part abstract systems with periodically nonstationary linear part are considered inputs and outputs of nonlinearities satisfy a certain integral quadratic constraint the conditions necessary and sufficient of dichotomy and absolute stability are obtained

robust guaranteed cost state estimation for nonlinear stochastic uncertain systems via an iqc approach abstract this paper presents an approach to robust nonlinear state estimation based on the use of integral quadratic constraints and minimax lqg control the approach involves a class of state estimators which includes copies of the system nonlinearities in the state estimator the nonlinearities being considered are those which satisfy a certain global lipschitz condition the linear part of the state estimator is synthesized using minimax lqg control theory which is closely related to h control theory and this leads to a nonlinear state estimator which gives an upper bound on the estimation error cost

two graded absolute valued algebras abstract we study two graded absolute valued algebras these are two graded algebras satisfying the absolute value multiplicative property only on homogeneous elements thus hexagonions also called sedenions and other sixteen dimensional algebras arise as examples of these algebras the even parts of two graded absolute valued algebras are the absolute valued algebras while the odd parts are exactly the absolute valued triple systems so in a way these two graded algebras give a unifying viewpoint of both structures we also study the simplicity and give several ways to construct two graded absolute valued algebras we also provide a description of isomorphism classes for two graded absolute valued algebras of dimensions 1 2 and 4

guaranteed cost lqg control for uncertain systems with a normalized coprime factor uncertainty structure the paper extends the robust minimax lqg control design methodology to stochastic uncertain systems with a general uncertainty structure which includes normalized coprime factor uncertainty passive uncertainty and sector norm bounded uncertainty as special cases the derivation of the result uses a special parameter dependent girsanov measure transformation the efficacy of the proposed methodology is illustrated using the problem of frequency locking of an optical cavity which occurs in the area of experimental quantum optics

sur les algebres absolument valuees qui verifient l identite x x x 0 let a denote a prehilbert absolute valued real algebra such that x x x 0 for all x a for this algebra we obtain the same results we have previously obtained for the flexible absolute valued algebra our main theorem is a has a finite dimension 1 2 4 or 8 and is isotopic to r c h or c one of the results concerning the isomorphism between a and r c c h or c shows that if for every two idempotents e1 and e2 in a e1 e2 14 then a is isomorphic to r c c h or c the example of infinite dimensional hilbert absolute valued algebra given by urbanik and wright indicates that the assumption x x x 0 for all x a is essential

geometric properties of random disk packings random packings ofn 2000 rigid disks in the plane subject to periodic boundary conditions on a square primitive cell have been generated by a concurrent construction which treats all disks on an equal footing as opposed to previously investigated sequential constructions the particles start with random positions and velocities and as they move about they grow uniformly in size from points to jammed disks the collection of packings displays several striking geometric features these include for largen typically polycrystalline textures with irregular grain boundaries and linear shear fractures the packings occasionally contain monovacancies and trapped but unjammed rattler disks the latter appear to be confined to the grain boundaries the linear shear fractures preserve bond orientational order but disrupt translational order within the crystalline grains a new efficient event driven simulation algorithm is employed to generate the histories of colliding and jamming disks on a computer which can process one million floating point instructions per second the algorithm processes more than one million pairwise collisions per hour

absolute stabilization and minimax optimal control of uncertain systems with stochastic uncertainty this paper is concerned with existence and optimality properties of so called guaranteed cost controllers for an uncertain system subject to structured uncertainty the uncertainty in the system is assumed to have a stochastic character and to satisfy certain stochastic integral constraints it is shown that a minimax optimal guaranteed cost state feedback controller for a stochastic system can be synthesized as a state feedback controller absolutely stabilizing this system for each initial state of the system this controller can be found by parametric optimization of solutions of a parameter dependent generalized matrix riccati equation arising in stochastic h infty theory

robust controllability of linear stochastic uncertain systems the paper extends the concept of robust controllability via linear state feedback to stochastic uncertain systems we show that the controllability of a stochastic uncertain system can be characterized using solutions to a game type differential riccati equation

nonlinear versus linear control in the absolute stabilizability of uncertain systems with structured uncertainty this note considers a stabilization problem for a class of uncertain linear systems containing structured uncertainty described by a certain integral quadratic constraint the notion of stabilizability considered is that of absolute stabilizability the main result gives a necessary and sufficient condition for the absolute stabilizability of this class of uncertain systems in terms of the existence of a solution to a corresponding diagonally scaled h sup spl infin control problem it follows from this result that absolute stabilizability via nonlinear control implies absolute stabilizability via linear control

robust filtering of stochastic uncertain systems on an infinite time horizon in this paper we consider a robust filtering problem for continuous time stochastic uncertain systems the uncertainty in the system is characterized in terms of an uncertain probability distribution on the noise input this uncertainty is assumed to satisfy a certain relative entropy constraint the solution to a specially parameterized risk sensitive stochastic filtering problem is used to construct a filter for the original uncertain system which guarantees an optimal worst case filtering error the corresponding minimax optimal filter is obtained by solving a pair of algebraic riccati equations

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

minimax optimal control of stochastic uncertain systems with relative entropy constraints this paper considers a new class of discrete time stochastic uncertain systems in which the uncertainty is described by a constraint on the relative entropy between a nominal noise distribution and the perturbed noise distribution this uncertainty description is a natural extension to the case of stochastic uncertain systems of the sum quadratic constraint uncertainty description this paper solves problems of worst case robust performance analysis and output feedback minimax optimal controller synthesis in a general nonlinear setting specializing these results to the linear case leads to a minimax linear quadratic gaussian lqg optimal controller this controller is defined by riccati difference equations and a kalman filter like state equation the paper also shows that the minimax lqg problem will have a solution if and only if a corresponding h sup spl infin control problem has a solution a linear example is presented to illustrate the minimax lqg methodology

computer simulation of random packing of unequal particles a monte carlo simulation model for the random packing of unequal spherical particles is presented in this paper with this model the particle radii obeying a given distribution are generated and randomly placed within a cubic packing domain with high packing density and many overlaps then a relaxation iteration is applied to reduce or eliminate the overlaps while the packing space is gradually expanded the simulation is completed once the mean overlap value falls below a preset value to simulate the random close packing a vibration process is applied after the relaxation iteration for log normal distributed particles the effect of particle size standard deviation and for bidisperse particles the effects of particle size ratio and the volume fraction of large particles on packing density and on coordination number are investigated simulation results show good agreement with that obtained by experiments and by other simulations the randomness homogeneity and isotropy which have not been evaluated before for packing of distributed particles are also examined using statistical measures