decoupling of linear multivariable systems by unity output feedback compensation the existence and the synthesis of a stabilizing compensator for the diagonal decoupling of a linear multivariable system is examined a new sufficient condition is established which is generically satisfied by any proper plant p and guarantees the existence of a stabilizing compensator that will give rise to a closed loop transfer function matrix which is non singular and diagonal

regulator problem with internal stability a frequency domain solution this paper considers the general regulator problem with internal stability where the measured outputs are not necessarily the same as the regulated outputs using polynomial matrix techniques necessary and sufficient conditions are obtained in terms of skew primeness of two polynomial matrices one of these polynomial matrices represents the disturbance modes whereas the other is the polynomial system matrix representing the system zeros various special cases considered in the literature are also analyzed in terms of these necessary and sufficient conditions

minimal structure in the block decoupling problem with stability abstract in this paper we consider the block decoupling problem with stability we characterize the minimal mcmillan degree achievable for the block decoupled system for this we introduce a new list of integers which are feedback invariant these integers are related in a simple way to the minimal number of unstable and infinite zeros of the block decoupled system

on the computation of proper feedforward internally stabilizing and pole placing compensators by combining two important results from the polynomial and fractional representation approaches we derive a state space formula for the direct computation of a proper feedforward internally stabilizing and pole placing compensator c s

random polytopes and the efron stein jackknife inequality let k be a smooth convex body the convex hull of independent random points in k is a random polytope estimates for the variance of the volume and the variance of the number of vertices of a random polytope are obtained the essential step is the use of the efron stein jackknife inequality for the variance of symmetric statistics consequences are strong laws of large numbers for the volume and the number of vertices of the random polytope a conjecture of barany concerning random and best approximation of convex bodies is confirmed analogous results for random polytopes with vertices on the boundary of the convex body are given

central limit theorems for gaussian polytopes choose n random independent points in r d according to the standard normal distribution their convex hull k n is the gaussian random polytope we prove that the volume and the number of faces of k n satisfy the central limit theorem settling a well known conjecture in the field

decoupling of multivariable control systems over unique factorization domains necessary and sufficient conditions are established for the existence of a state variable feedback decoupling of an m input m output time invariant linear control system over a unique factorization domain an explicit computation is provided for the feedback and the feedforward gain matrix also necessary and sufficient conditions for the existence of a stability preserving state feedback decoupling are given the results are illustrated by some examples

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

on estimation of monotone and concave frontier functions abstract when analyzing the productivity of firms one may want to compare how the firms transform a set of inputs x typically labor energy or capital into an output y typically a quantity of goods produced the economic efficiency of a firm is then defined in terms of its ability to operate close to or on the production frontier the boundary of the production set the frontier function gives the maximal level of output attainable by a firm for a given combination of its inputs the efficiency of a firm may then be estimated via the distance between the attained production level and the optimal level given by the frontier function from a statistical viewpoint the frontier function may be viewed as the upper boundary of the support of the population of firms density in the input and output space it is often reasonable to assume that the production frontier is a concave monotone function then a famous estimator in the univariate input and output case is the data envelopment analysis dea estimato

on the stable exact model matching problem abstract the stable exact model matching problem semmp is investigated we state and prove a number of equivalent necessary and sufficient conditions for the existence of proper solutions to the exact model matching problem that are also stable i e have no poles inside a symmetric forbidden subset of the finite complex plane c these results can be viewed as the counterpart of the results in 3 and 9 for the case of the ring of proper and stable rational functions

feedback representation of precompensators application to decoupling linear multi input multi output plants in this paper we examine the problem of internally stabilizing and simultaneously diagonally decoupling a linear multivariable system by two parameter compensation based on our earlier results boussaid and guerin 1996 we prove that any given plant p of full row rank can be decoupled with stability by the considered configuration computation of the decoupling compensators follows by easy constructions the decoupling problem with a minimum number of unstable zeros in the decoupled system is also formulated and solved in the generic case

random points and lattice points in convex bodies assume k subset of r d is a convex body and x is a large finite subset of k how many convex polytopes are there whose vertices belong to x is there a typical shape of such polytopes how well does the maximal such polytope which is actually the convex hull of x approximate k we are interested in these questions mainly in two cases the first is when x is a random sample of n uniform independent points from k in this case motivation comes from sylvester s famous four point problem and from the theory of random polytopes the second case is when x k boolean and z d where z d is the lattice of integer points in r d and the questions come from integer programming and geometry of numbers surprisingly or not so surprisingly the answers in the two cases are rather similar

system theoretic and algebraic aspects of the rings of stable and proper stable rational functions abstract a new approach in transfer function methods for solving a variety of control theoretic problems is to work with fractional representations over the ring of stable or proper stable rational functions of various transfer matrices the rings of stable and proper stable rational functions are well known to be euclidean domains these rings differ critically from the polynomial ring by the nonuniqueness of the remainders obtained this major difficulty in extending the idea of polynomial models of fuhrmann to the rings of stable and causal stable rational functions is circumvented by choosing a remainder in a special form a natural realization theory is thus developed for matrix fraction representations of transfer function matrices over these rings as an application of the new theory developed linear matrix equations qx ry t and qx yr t over the rings of stable and causal stable rational functions are reduced to finite sets of linear equations over the base field

central limit theorems for random polygons in an arbitrary convex set we study the probability distribution of the area and the number of vertices of random polygons in a convex set kr 2 the novel aspect of our approach is that it yields uniform estimates for all convex sets kr 2 without imposing any regularity conditions on the boundary k our main result is a central limit theorem for both the area and the number of vertices settling a well known conjecture in the field we also obtain asymptotic results relating the growth of the expectation and variance of these two functionals

the equation xr qy phi a characterization of solutions in this paper we consider the solutions of the equation xr qy phi here q r phi are given p times q m times t and p times t polynomial matrices over a field k x and y are p times m and q times t polynomial matrices which are unknown using certain recent results on the realization of matrix fraction descriptions of transfer matrices we give a characterization parametrization of all possible x y which solve this equation this also provides a system theoretic interpretation for this equation

transfer matrices realization and control of continuous time linear time varying systems via polynomial fractional representations abstract an algebraic theory of transfer matrices fractional representations and control for linear continuous time time varying systems based on the realization theory of input output maps is given it is shown for the first time that the realization of such systems specified by an abstract input output map as a module homomorphism over noncommutative polynomial rings can be established using an abstract kalman input output map defined over a ring of skew polynomials with time varying coefficients it is shown that in fact transfer matrices can be defined as formal power

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

central limit theorems for random polytopes let k be a smooth convex set the convex hull of independent random points in k is a random polytope central limit theorems for the volume and the number of i dimensional faces of random polytopes are proved as the number of random points tends to infinity one essential step is to determine the precise asymptotic order of the occurring variances

the block decoupling problem for systems over a ring the decoupling problem by static state feedback for a system with coefficients in a ring is investigated using a geometric approach the notion of a precontrollability submodule which captures the geometric aspects of the notion of controllability subset of the state space in the present context is introduced and it is used for stating solvability conditions for the decoupling problem for a system with coefficients in a noetherian ring this result extends and improves previous results concerning systems with coefficients in a principal ideal domain and it can in particular be used for dealing with decoupling problem involving delay differential systems

brief simultaneous model matching and disturbance rejection with stability by state feedback this work deals with the model matching problem and the disturbance rejection problem with stability by state feedback each one of these control problems has separately received a lot of contributions but to our best knowledge we propose here for the first time a complete structural solution for square same number of control inputs and outputs strictly proper systems our structural conditions nicely enhance the role played by the zeros both finite and infinite

disturbance rejection a polynomial approach a detailed analysis of the disturbance rejection problem in single input single output linear systems is presented the analysis is based on an external polynomial model and employs the algebra of polynomials all solutions are expressed in a parametric form and solvability conditions are given for a number of constraints including those of internal stability and arbitrary pole placement measurable and unmeasurable disturbances are distinguished and the effect of output feedback is investigated a simple design procedure is then proposed which provides a stable dynamical solution and which can be extended so as to minimize its order the procedure consists of solving two linear polynomial equations and yields the compensator in a compact form suitable for direct realization

the combinatorial structure of random polytopes abstract choose n random points in r d let p n be their convex hull and denote by f i p n the number of i dimensional faces of p n a general method for computing the expectation of f i p n i 0 d 1 is presented this generalizes classical results of efron in the case i 0 and renyi and sulanke in the case i d 1 to arbitrary i for random points chosen in a smooth convex body a limit law for f i p n is proved as n for random points chosen in a polytope the expectation of f i p n is determined as n this implies an extremal property for random points chosen in a simplex

adjusted versions of profile likelihood and directed likelihood and extended likelihood several adjusted and invariant versions of profile likelihood and of directed likelihood short for signed log likelihood ratio are discussed these adjusted versions which fall into two groups are closely related to modified profile likelihood and modified directed likelihood the first group has the advantage of not requiring an explicit specification of an ancillary but partly at a cost of a loss of asymptotic accuracy in comparison with the modified versions the second group consists of some asymptotically sharpened versions of modified profile likelihood furthermore a concept of extended likelihood is proposed which associates a likelihood like function to certain combinants

structure smith macmillan form and coprime mfds of a rational matrix inside a region p ω abstract the structure of the smith macmillan form of a rational matrix t s inside a region p where is asymmetric with respect to the real axis subset of the finite complex plane c is determined algorithmic procedures based on elementary row and column operations over the euclidean ring r p s consisting of all rational functions with no poles in p are given coprimeness in p of a pair of rational matrices is studied in detail these results lead to constructive procedures for determining the coprime in p matrix fraction descriptions of t a

the minimal delay decoupling problem feedback implementation with stability in this paper we solve the minimal delay decoupling problem of linear multivariable systems we look for feedback implementable solutions which moreover guarantee closed loop stability we prove that dynamic state feedback decoupling with stability is achievable if the number of independent inputs is large enough to compensate the intrinsic nondecouplability of the system we introduce new feedback invariants characterizing the minimal number of infinite and unstable zeros of the decoupled system

feedback system design the tracking and disturbance rejection problems the problem of designing a compensator for a specified plant which simultaneously stabilizes the resultant feedback system and causes it to track a prescribed family of inputs and or reject prescribed disturbances is considered a set of linear design equations in the space of stable systems is formulated in a general linear systems setting and an explicit parameterization of the resultant solution space is obtained for a class of generalized multivariate systems the theory is illustrated with several single and multivariate examples

stochastische approximation konvexer polygone let k be a convex polygon with perimeter l and area f denote the perimeter of the convex hull of n uniformly and independently distributed points in k by ln and by fn its area as n tends to infinity l e l n c 1 k sqrt n o 1 n 1 varepsilon for any fixed 0 and f e f n c 2 k log n n o 1 n where the constants c1 k and c2 k are given explicitly this generalizes a result of renyi and sulanke

variance asymptotics for random polytopes in smooth convex bodies let k subset mathbb r d be a smooth convex set and let mathcal p lambda be a poisson point process on mathbb r d of intensity lambda the convex hull of mathcal p lambda cap k is a random convex polytope k lambda as lambda rightarrow infty we show that the variance of the number of k dimensional faces of k lambda when properly scaled converges to a scalar multiple of the affine surface area of k similar asymptotics hold for the variance of the number of k dimensional faces for the convex hull of a binomial process in k

necessary and sufficient condition for stability robustness under known additive perturbation a necessary and sufficient condition is derived for the stability robustness of a unity feedback closed loop system involving a strictly proper plant p0 and a stabilizing compensator c0 under the assumption that the plant p0 is perturbed to p0 p0 where p0 is a known strictly proper rational matrix

proper and stable minimal macmillan degrees bases of rational vector spaces the structure of proper and stable bases of rational vector spaces is investigated we prove that if t s is a rational vector space then among the proper bases of 3 s there is a subfamily of proper bases which are 1 stable 2 have no zeros in c bigcup infty and therefore are column row reduced at infinity and 3 their macmillan degree is minimum among the macmillan degrees of all other proper bases of 3 s and it is given by the sum of the macmillan degrees of their columns taken separately it is shown that this notion is the counterpart of forney s concept of a minimal polynomial basis for the family of proper and stable bases of 3 s

predictive likelihood inference with applications summary conditional and marginal predictive likelihood are defined and used in predicting un observed or missing data a distinction is made between predictive and estimative goals of a data analysis such methodology and distinctions are illustrated in the context of outlier theory and mixed model ano va the prediction of unobserved or missing data is considered from a likelihood perspective a general predictive likelihood is proposed for such purposes and contrasted with parametric likelihood the resulting maximum likelihood predictor is also compared with the limiting imputation from the em algorithm the new methodology is illustrated and contrasted with estimative likelihood analysis in the context of outlier theory and mixed model ano va predictions are proposed that are based on conditional and marginal predictive likelihoods for the unobserved data which are predictive adaptations of the methods developed by kalbfleisch and sprott 1970 1973 these authors were concerned with the elimination of nuisance parameters in order to make inference about another set of structural parameters prediction in the frame work of parametric distributions involves the elimination of all parameters in such a way that inference about the unobserved data is possible therefore in the predictive setting all para meters are nuisance parameters and the unobserved data now play a role similar to that of the structural parameters various predictive approaches to statistical inference have been advocated principally by geisser 1971 1975 stone 1974 aitchison and dunsmore 1975 and guttman 1970 a survey is provided by geisser 1980 and a brief overview in a parametric setting is presented in the next section following this a conditional likelihood approach is developed and shown to be closely related to the lauritzen 1974 hinkley 1979 likelihood method but specified in a manner allowing for a wider range of applicability a marginal predictive likelihood procedure is also specified which when applicable essentially agrees with the conditional likelihood procedure for the examples considered for a general account of likelihood based inference see fraser

on the structure at infinity of linear square decoupled systems the purpose of this note is to demonstrate that a square invertible linear system c a b is decouplable by static state feedback if and only if its infinite zero orders are respectively equal to the infinite zero orders of the subsystems c i a b where c i denotes the i th row of c until now only the necessity of this condition has been established 1

linear feedback decoupling transfer function analysis the problem of linear system decoupling is examined based on recent results on linear feedback new insight is obtained through which resolution of the decoupliug problem is accomplished by calculations performed directly on the given transfer matrix computation of the decoupling compensators follows by easy constructions the problem of feedback block decoupling with internal stability is also formulated and resolved

equivalence transformations of rational matrices and applications the known theories of transformations between polynomial matrices are extended to the case of rational matrices specifically equivalence between rational matrices having possibly different dimensions is defined and this has the property of preserving the zero structure of rational matrices in the region c some implications of this new equivalence transformation for linear system theory are also provided

cascade control of systems over integral domains necessary and sufficient conditions for the existence of cascade compensators and dynamic equivalence of linear systems over integral domains arc derived the considerations lead to constructive procedures for dynamic compensation

on coprime rational function matrices abstract the concept of coprimeness of matrices with elements in a field of fractions is introduced we focus on the field of rational functions and define when two rational matrices are coprime with respect to different rings the definition of coprimeness at infinity of polynomial matrices is obtained as a particular case moreover the fact that coprimeness is a local property is proved and it is shown how coprimeness changes by performing mobius transformations

necessary and sufficient conditions for zero assignment by constant squaring down the problem of zero assignment by constant squaring down czap is studied for minimal systems described by a transfer function g s rm l s m l using tools from exterior algebra and algebraic geometry conditions for its solvability are given in terms of the numbers m l the forney dynamical order and the rank of the plucker matrix p for the cases l 1 or l m 1 it is shown that g s is completely zero assignable cza if and only if 1 for the cases l 1 m 1 it is proved that g s is cza or generically zero assignable gza under a complex squaring down if and only if l m l 1 and 1 the latter conditions also provide necessary conditions for the existence of a real solution for a generic g s sufficient conditions for solvability of czap by a real squaring down are that l m l 1 and that the number g a0 al 1 1 a0 al 1 i j ai aj is odd for some set a0 al 1 satisfying 1 l 1i 0ai l l 1 2 and 0 a0 a1 al 1 m 1 apart from the existence results the present approach also allows the computation of the solutions whenever they exist finally it is proved that the only fixed zeros of czap are the zeros of g s and a procedure for computing the solutions of czap whenever such solutions exist based on an optimization problem is given

differential representations of multivariable linear systems with disturbances and polynomial matrix equations px ry w and px yn v a unified state space approach to polynomial matrix equations px r y w px yn v based on a generalization of the wolovich guidorzi formula has been presented in this paper a connection is shown between the solutions of the above equations and differential representations for multivariable linear systems with disturbances applications are presented for stable synthesis of certain types of compensators

variance asymptotics and central limit theorems for generalized growth processes with applications to convex hulls and maximal points we show that the random point measures induced by vertices in the convex hull of a poisson sample on the unit ball when properly scaled and centered converge to those of a mean zero gaussian field we establish limiting variance and covariance asymptotics in terms of the density of the poisson sample similar results hold for the point measures induced by the maximal points in a poisson sample the approach involves introducing a generalized spatial birth growth process allowing for cell overlap

comments on parameterization of stabilizing compensators by using reduced order observers a counterexample of the above mentioned paper fujimori ieee trans automat contr vol 38 p 1435 39 1993 is given it is shown that the parameterization of all proper compensators internally stabilizing the plant is incomplete modified parameterization of all proper stabilizing compensators is also presented

two sided equations and skew primeness for n d polynomial matrices abstract the two sided equations in n d polynomial matrices are studied which are often encountered in control problems the solvability conditions are derived the general and minimum degree solutions are investigated and two algorithms of solution are proposed the related concept of skew primeness is generalized for n d polynomials

limit theorems for functionals of convex hulls in 4 a central limit theorem for the number of vertices of the convex hull of a uniform sample from the interior of a convex polygon is derived this is done by approximating the process of vertices of the convex hull by the process of extreme points of a poisson point process and by considering the latter process of extreme points as a markov process for a particular parametrization we show that this method can also be applied to derive limit theorems for the boundary length and for the area of the convex hull this extents results of renyi and sulanke 1963 and buchta 1984 and shows that the boundary length and the area have a strikingly different probabilistic behavior

structure and smith macmillan form of a rational matrix at infinity the use of special row and column operations in the reduction of a rational matrix to its smith macmillan form at infinity is investigated the connections between this reduction procedure and the valuation approach are established a graphical method for finding the smith macmillan form of a rational matrix at infinity from its bode magnitude array and some new results on realization theory for polynomial matrices are presented

decentralized diagonal dynamic stabilization of linear multivariable systems abstract the diagonal stabilization problem dsp is defined over the ring of proper rational functions which have no poles inside a prescribed region of the finite complex plane solvability is intimately related to systems which exhibit the property of cyclicity necessary and sufficient conditions are established for the existence of solutions to the dsp a complete parametrization of stabilizing controllers for the 2 2 case is given conditions of nonsolvability and hence nonstabilizability yield an explicit expression for the fixed modes of the system

realization theoretic solution of two analytic matrix equations with application to stabilization of infinite dimensional systems a realization theoretic approach is proposed for the solutions of two matrix analytic equations arising in the control of infinite dimensional systems the bezout type equationqx ry which arises in the stabilization of infinite dimensional systems and the skew matrix equation qy xr which arises in the tracking stabilization in the presence of disturbances where q r and are given analytic matrices the realization proposed here is based on the operator theoretic machinery described by fuhrmann 1981 with this form of realization we extend to infinite dimensional systems the solutions given by emre 1980 and emre and silverman 1981 for finite dimensional systems a systematic characterization of all h2 matrix solutions of these equations is given when q r satisfies the bezout corona condition explicit formulae for the h2 solutions of the bezout type equation are given in terms of a realization and the hankel map of the given system h2 solutions are shown to be an intermediate

feedback representation of precompensators the general case using the transfer approach we give under internal stability conditions a necessary and sufficient condition of equivalence between open loop precom pensation and a closed loop configuration that consists of dynamic output feedback and inside loop precompensation our result is the generalization of hammer s result hammer 1983 to the general case this result emphases the use of skew prime relationships the parametrization of this problem s solutions will throw new light on multipurpose controllers

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

on the area and perimeter of a random convex hull in a bounded convex set suppose k is a compact convex set in 2 and x i 1 i n is a random sample of points in the interior of k under general assumptions on k and the distribution of the x i we study the asymptotic properties of certain statistics of the convex hull of the sample

limit theorems for the typical poisson voronoi cell and the crofton cell with a large inradius in this paper we are interested in the behavior of the typical poisson voronoi cell in the plane when the radius of the largest disk centered at the nucleus and contained in the cell goes to infinity we prove a law of large numbers for its number of vertices and the area of the cell outside the disk moreover for the latter we establish a central limit theorem as well as moderate deviation type results the proofs deeply rely on precise connections between poisson voronoi tessellations convex hulls of poisson samples and germ grain models in the unit ball besides we derive analogous facts for the crofton cell of a stationary poisson line process in the plane

supporting points processes and some of their applications we introduce a stochastic point process of s supporting points and prove that upon rescaling it converges to a gaussian field the notion of s supporting points specializes for adequately chosen s to pareto or more generally cone extremal points or to vertices of convex hulls or to centers of generalized voronoi tessellations in the models of large scale structure of the universe based on burgers equation the central limit theorems proven here imply i a the asymptotic normality for the number of convex hull vertices in large poisson sample from a simple polyhedra or for the number of pareto vector extremal points in poisson samples with independent coordinates

factorization of linear systems a generalized framework abstract a generalized module theoretic framework for the study of linear time invariant systems is developed crucial in the present discussion are two new notions the generalized order and the adapted bases these notions form generalizations of the classical concepts of order or degree and of proper bases employed in the theory of linear systems the resulting framework is then applied to obtain explicit conditions for system factorization and to study output feedback systems in which the feedback compensator is stable

internal stabilization and decoupling in linear multivariable systems by unity output feedback compensation we consider the problem of internally stabilizing and simultaneously diagonally decoupling a linear multivariable system by unity output feedback compensation a sufficient condition is derived for the existence of a cascade proper compensator c s such that when employed in a unity feedback loop involving the proper transfer function matrix p o of a free of unstable hidden modes system sigma p o will not only internally stabilize the feedback closed loop system sigma p o c but will also give rise to a closed loop transfer function matrix h yr diag which is nonsingular diagonal and has desired poles based on this analysis an algorithmic procedure for the computation of such a compensator is presented

on the variance of random polytopes abstract a random polytope is the convex hull of uniformly distributed random points in a convex body k a general lower bound on the variance of the volume and f vector of random polytopes is proved also an upper bound in the case when k is a polytope is given for polytopes as for smooth convex bodies the upper and lower bounds are of the same order of magnitude the results imply a law of large numbers for the volume and f vector of random polytopes when k is a polytope

on the polynomial matrix equation ax yb c in this note an algorithm for finding a unique solution x y of the bilateral polynominal matrix equation ax yb c is presented the solution is obtained by solving a set of linear equations relations between the bilateral polynomial equation and the unilateral one are also discussed

random convex hulls and extreme value statistics in this paper we study the statistical properties of convex hulls of n random points in a plane chosen according to a given distribution the points may be chosen independently or they may be correlated after a non exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem we present a unifying approach based on the notion of support function of a closed curve and the associated cauchy s formulae that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics as an example of correlated points we study here in detail the case when the points represent the vertices of n independent random walks in the continuum time limit this reduces to n independent planar brownian trajectories for which we compute exactly for all n the mean perimeter and the mean area of their global convex hull our results have relevant applications in ecology in estimating the home range of a herd of animals some of these results were announced recently in a short communication phys rev lett 103 140602 2009

classification of proper bases of rational vector spaces minimal macmillian degree bases abstract the algebraic structure of the get of all proper rational vectors contained in a given rational vector space x1d4af s is shown to be that of a noothorian pr s motiulo m ropf pr a is the ring of proper rational functions the proper submodules mt of m form an ascending chain of submodules partially ordered by an invariant of mt defined as the valuation at s ofmt tho various bases of mt are examined and classified according to their property of column reduceness at s the concept of a prime column reduced at basis of mt is introduced it is shown that tho prime bases of mt can be further classified by their macmillan degrees and the existence of minimal macmillan degree bases for mt is established a prime and minimal macmillan degree basis ofmt extends forney s concept of a minimal polynomial basis of s for the rpr mt module the macmillan degrees of the columns of such bases form a set of invariants mt for which are defined as the x1d4af s generalized invariant dynamicol indices o

decoupling without prediction of linear systems with delays a structural approach we exhibit using a new structural approach specific for linear systems with delays a necessary and sufficient condition for static row by row decoupling without prediction moreover the decoupling problem is analyzed through two different concepts the total one without any time assumption and the partial one in a temporal mean which can be considered if the total one has no solution

the generalized sylvester equation in polynomial matrices jordan chains of polynomial matrices are used to characterize the solvability of the generalized sylvester equation

decoupling linear multiinput multioutput plants by dynamic output feedback an algebraic theory this paper presents an algebraic theory for the design of a decoupling compensator for linear time invariant multiinput multioutput systems the design method uses a two input one output compensator which gives a convenient parametrization of all diagonal input output i o maps and all disturbance to output d o maps achievable by a stabilizing compensator for a given plant it is shown that this method has two degrees of freedom any achievable diagonal i o map and any achievable d o map can be realized simultaneously by a choice of an appropriate compensator the difference between all achievable diagonal and nondiagonal i o maps and the cost of decoupling is discussed for some particular algebraic settings

linear systems of pivotals and associated pivotal likelihoods with applications summary a complete linear conditionally sufficient system of pivotal quantities for a vector parameter and the associated pivotal likelihoods for the parameter and its separable subvectors are defined and exemplified the application of these pivotal likelihoods to setting up likelihood confidence intervals is illustrated

limit theorems for convex hulls it is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally after rescaling to a strongly mixing markov process as the sample size tends to infinity the structure of the limiting markov process is determined explicitly and from this a central limit theorem for the number of vertices of the convex hull is derived similar results are given for uniform samples from the unit disk