a pocket calculator algorithm for the shapiro francia test for non normality an application to medicine an easy to calculate approximation to the shapiro francia w test and its p value is proposed its accuracy is sufficient for practical application in samples of size 5 or larger

the use of cusums and other techniques in modelling continuous covariates in logistic regression the assessment of continuous covariates singly as possible predictors in a multivariable logistic regression model is an important first step in the analysis an approach to plotting which uses a cusum cumulative sum of the binary response variable is described extreme deviation statistics associated with the cusum may be used to detect monotonic and non monotonic trends probability plots of the covariate in the two groups defined by the response variable may help to determine the appropriate scale transformation of the covariate and to anticipate possible problems with the logistic fit the ratio of the variances in the response non response groups is informative about the need for a quadratic term in the logistic model smoothed scatterplots of the response are valuable in displaying the observed and fitted values the techniques are illustrated with two data sets

random convex hulls in a product of balls the convex hull of a set of points sampled independently and uniformly from the cartesian product of balls of various dimensions is investigated bounds on the asymptotic behavior of the expected combinatorial complexity volume and mean width are derived when the distribution is held fixed and the sample size approaches infinity the expected combinational complexity and volume are found to depend up to constant factors only on the greatest dimension of any factor ball and the number of balls of that dimension on the other hand the expected mean width depends only on the number of balls and the dimensions of the product

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

variance asymptotics and scaling limits for gaussian polytopes let k n be the convex hull of i i d random variables distributed according to the standard normal distribution on mathbb r d we establish variance asymptotics as n rightarrow infty for the re scaled intrinsic volumes and k face functionals of k n k in 0 1 ldots d 1 resolving an open problem weil and wieacker handbook of convex geometry vol b pp 1391 1438 north holland elsevier amsterdam 1993 variance asymptotics are given in terms of functionals of germ grain models having parabolic grains with apices at a poisson point process on mathbb r d 1 times mathbb r with intensity e h dh dv the scaling limit of the boundary of k n as n rightarrow infty converges to a festoon of parabolic surfaces coinciding with that featuring in the geometric construction of the zero viscosity solution to burgers equation with random input

state complexity of unambiguous operations on finite automata abstract the paper determines the number of states in finite automata necessary to represent unambiguous variants of the union concatenation and kleene star operations on formal languages for the disjoint union of languages represented by an m state and an n state deterministic finite automata dfa the state complexity is m n 1 for the unambiguous concatenation it is known to be m 2 n 1 2 n 2 daley et al orthogonal concatenation language equations and state complexity j ucs 2010 and this paper shows that this number of states is necessary already over a binary alphabet for the unambiguous star the state complexity function is determined to be 3 8 2 n 1 in the case of a unary alphabet disjoint union requires up to 1 2 m n states in a dfa unambiguous concatenation has state complexity m n 2 and unambiguous star requires n 2 states in the worst case for nondeterministic finite automata as well as for unambiguous finite automata the complexity for disjoint union is m n for unambiguous concatenation square and star the resulting complexities are m n 2n and n 1 respectively and all of them are witnessed by binary languages in the unary nondeterministic or unambiguous case the corresponding complexities are m n for disjoint union m n 1 and 2 n 1 for unambiguous concatenation and square respectively and n 1 for unambiguous star

stratified structure of the universe and burgers equation a probabilistic approach the model of the potential turbulence described by the 3 dimensional burgers equation with random initial data was developped by zeldovich and shandarin in order to explain the existing large scale structure of the universe most of the recent probabilistic investigations of large time asymptotics of the solution deal with the central limit type results the gaussian scenario under suitable moment assumptions on the initial velocity field these results and some open questions are discussed in sect 2 where we concentrate on the gaussian model and the shot noise model in sect 3 we construct a probabilistic model of strong initial fluctuations a zero range shot noise field with high amplitudes which reveals an intermittent large time behaviour with the velocity vec v t x determined by the position of the largest initial fluctuation discounted by the heat kernelg t x in a neighborhood ofx the asymptoties of such local maximum ast can be analyzed with the help of the theory of records sect 4 finally in sect 5 we introduce a global definition of a point process oft local maxima and show the weak convergence of the suitably rescaled process to a non trivial limit ast

limit theorems for fourier transforms of functionals of gaussian sequences limit theorems with a non gaussian in fact nonstable limiting distribution have been obtained under suitable conditions for partial sums of instantaneous nonlinear functions of stationary gaussian sequences with long range dependence analogous limit theorems are here obtained for finite fourier transforms of instantaneous nonlinear functions of stationary gaussian sequences with long range dependence

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

estimating departure from normality departure of a sample from a normal distribution should be assessed by a quantity that is meaningful in terms of the data rather than merely by the p value from a test statistic indices of departure based on the shapiro francia w and the shapiro wilk w statistics are derived and shown to have a natural interpretation in relation to the normal probability plot a new diagnostic plot is proposed an example is given which shows the relationship between one of the new indices and errors in calculated reference ranges due to non normality of the data

l a l b decidability results from complete formal systems abstract the equivalence problem for deterministic pushdown automata is shown to be decidable we exhibit a complete formal system for deducing equivalent pairs of deterministic rational boolean series on the alphabet associated with a dpda m we then extend the result to deterministic pushdown transducers from a free monoid into an abelian group a general algebraic and logical framework inspired by harrison et al theoret comput sci 9 1979 173 205 conrcelle theoret comput sci 6 1978 255 279 and meitus kybernetika 5 1989 14 25 in russian is developed

intrinsic volumes and f vectors of random polytopes on appelle polytope aleatoire k n l enveloppe convexe de n points choisis de facon aleatoire dans un corps convexe k on s interesse a l approximation stochastique de k par k n plus particulierement on obtient des resultats d approximation pour les esperances de v s k v s k n ou v s est le s ieme volume intrinseque du nombre de faces s dimensionnelles de k n et de la distance de hausdorff entre k et k n

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

unrestricted complementation in language equations over a one letter alphabet recently the complete solution of systems of language equations over a one letter alphabet has been derived where the operators are union concatenation and star this paper addresses the question of adding the complementation operator it is known that if no stars are present and if the concatenation operator is restricted such equations need not have solutions but if they do they can be solved explicitly if the constants are regular in which case the solutions are also regular in this paper we study language equations over a one letter alphabet where unrestricted concatenation interacts with complementation union and star we define a procedure for determining a regular solution of the general type of equation assuming that all constants are regular discuss its properties and show examples we then prove that in general equations with complementation and complementation and concatenation in which all constants are regular may have nonregular solutions this shows that language equations can be used to obtain non context free languages from regular languages using only operators under which the regular languages are closed

the stabilized probability plot summary the stabilized probability plot is introduced an attractive feature of the plot that enhances its interpretability is that the variances of the plotted points are approximately equal this prompts the definition of a new and powerful goodness of fit statistic dsp which analogous to the standard kolmogorov smirnov statistic d is defined to be the maximum deviation of the plotted points from their theoretical values using either d or dsp it is shown how to construct acceptance regions for qq pp and the new plots acceptance regions can help remove much of the subjectivity from the interpretation of these probability plots

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

a random approximation of set valued cadlag functions abstract in this paper we study the problem of estimating a cadlag function f whose values are compact convex sets for this purpose a random selection of points in the interval 0 1 is considered and for each selected point x a random sample in f x on the basis of this a sequence of approximants f n m is constructed where n and m are the respective sample sizes under general conditions rates of convergence are obtained for skorokhod s j 1 topology and in case of continuity of the estimated function also for the uniform one

two results concerning asymptotic behavior of solutions of the burgers equation with force we consider the burgers equation with an external force for the case of the force periodic in space and time we prove the existence of a solution periodic in space and time which is the limit of a wide class of solutions ast if the force is the product of a periodic function ofx and white noise in time we prove the existence of an invariant distribution concentrated on the space of space periodic functions which is the limit of a wide class of distributions ast

parsing by matrix multiplication generalized to boolean grammars the well known parsing algorithm for context free grammars due to valiant 1975 25 is analyzed and extended to handle the more general boolean grammars which are context free grammars augmented with conjunction and negation operators in the rules the algorithm reduces construction of a parsing table to computing multiple products of boolean matrices of various sizes its time complexity on an input string of length n is o bmm n logn where bmm n is the number of operations needed to multiply two boolean matrices of size nxn which is o n w with w 2 373 as per the current knowledge a parse tree can be constructed in time mm n log o 1 n where mm n is the complexity of multiplying two integer matrices by applying a known efficient procedure for determining witnesses for boolean matrix multiplication the algorithm has a succinct proof of correctness and is ready to be implemented

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

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

on the expressive power of univariate equations over sets of natural numbers equations of the form email protected x are considered where the unknown x is a set of natural numbers the expression f x may contain the operations of set addition defined as s t m n email protected s email protected t union intersection as well as ultimately periodic constants an equation with a non periodic solution of exponential growth rate is constructed at the same time it is demonstrated that no sets with super exponential growth rate can be represented it is also shown that restricted classes of these equations cannot represent sets with super linearly growing complements nor sets that are additive bases of order 2 the results have direct implications on the power of unary conjunctive grammars with one nonterminal symbol

variance asymptotics and scaling limits for random polytopes abstract let k be a convex set in r d and let k be the convex hull of a homogeneous poisson point process p of intensity on k when k is a simple polytope we establish scaling limits as for the boundary of k in a vicinity of a vertex of k and we give variance asymptotics for the volume and k face functional of k k 0 1 d 1 resolving an open question posed in 17 the scaling limit of the boundary of k and the variance asymptotics are described in terms of a germ grain model consisting of cone like grains pinned to the extreme points of a poisson point process on r d 1 r having intensity d e d h d h d v

gaussian fields and random packing consider sequential packing of unit volume balls in a large cube in any dimension and with poisson input we show after suitable rescaling that the spatial distribution of packed balls tends to that of a gaussian field in the thermodynamic limit the results cover related applied models including ballistic deposition and spatial birth growth models

hyperbolic asymptotics in burgers turbulence and extremal processes large time asymptotics of statistical solutionu t x 1 2 of the burgers equation 1 1 is considered where x l x is a stationary zero mean gaussian process depending on a large parameterl 0 so that xi l x sim sigma l eta x l l to infty where sigma l l 2 2 log l 1 2 and x is a given standardized stationary gaussian process we prove that asl the hyperbolicly scaled random fieldsu l2t l2x converge in distribution to a random field with saw tooth trajectories defined by means of a poisson process on the plane related to high fluctuations of x which corresponds to the zero viscosity solutions at the physical level of rigor such asymptotics was considered before by gurbatov malakhov and saichev 1991

formal languages over gf 2 abstract variants of the union and concatenation operations on formal languages are investigated in which boolean logic in the definitions that is conjunction and disjunction is replaced with the operations in the two element field gf 2 conjunction and exclusive or union is thus replaced with symmetric difference whereas concatenation gives rise to a new gf 2 concatenation operation which is notable for being invertible all operations preserve regularity and for a pair of languages recognized by an m state and an n state dfa their gf 2 concatenation is recognized by a dfa with m 2 n states and this number of states is in the worst case necessary similarly the state complexity of gf 2 inverse is 2 n 1 next a new class of formal grammars based on gf 2 operations is defined and it is shown to have the same computational complexity as ordinary grammars with union and concatenation in particular simple parsing in time o n 3 fast parsing in the time of matrix multiplication and parsing in nc2

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

the convex hull of the integer points in a large ball where fk p denotes the number of k dimensional faces of the polytope p the limit as r of the average of r 2 3f0 pr on an interval r r h is determined by balog and deshoullier bd and turns out to be 3 453 h must be large our main result extends 1 1 to any d 2 and to any fk pr with k 0 d 1 theorem 1 for every d 2 there are constants c1 d and c2 d such that for all k 0 d 1

cones generated by random points on half spheres and convex hulls of poisson point processes let u 1 u 2 ldots be random points sampled uniformly and independently from the d dimensional upper half sphere we show that as n rightarrow infty the f vector of the d 1 dimensional convex cone c n generated by u 1 ldots u n weakly converges to a certain limiting random vector without any normalization we also show convergence of all moments of the f vector of c n and identify the limiting constants for the expectations we prove that the expected grassmann angles of c n can be expressed through the expected f vector this yields convergence of expected grassmann angles and conic intrinsic volumes and answers thereby a question of barany et al random struct algorithms 50 1 3 22 2017 https doi org 10 1002 rsa 20644 our approach is based on the observation that the random cone c n weakly converges after a suitable rescaling to a random cone whose intersection with the tangent hyperplane of the half sphere at its north pole is the convex hull of the poisson point process with power law intensity function proportional to vert x vert d gamma where gamma 1 we compute the expected number of facets the expected intrinsic volumes and the expected t functional of this random convex hull for arbitrary gamma 0

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

equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points a general formula is proved which relates the equiaffine inner parallel curves of a plane convex body and the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points this formula is applied to improve some well known results in geometric probability for example an estimate which was established for a special case by l c g rogers is obtained with the best possible bound and an asymptotic formula due to a renyi and r sulanke is extended to an asymptotic expansion

unambiguous conjunctive grammars over a one symbol alphabet abstract it is demonstrated that unambiguous conjunctive grammars over a unary alphabet a have non trivial expressive power and that their basic properties are undecidable the key result is that for every base of positional notation k 11 and for every one way real time cellular automaton operating over the alphabet of base k digits between k 9 4 and k 1 2 the language of all strings a n with the base k representation of the form 1 w 1 where w is accepted by the automaton is described by an unambiguous conjunctive grammar another encoding is used to simulate a cellular automaton in a unary language containing almost all strings these constructions are used to show that for every fixed unambiguous conjunctive language l 0 testing whether a given unambiguous conjunctive grammar generates l 0 is undecidable

random polytopes in a convex polytope independence of shape and concentration of vertices write d for the set of all convex bodies convex compact sets with nonempty interior in d define o g l d as the set of those k e 5 b d with vol k 1 fix k e g i d and choose points x l x e k randomly independently and according to the uniform distribution on k then k c o n v x l xn is a random polytope in k write e k n for the expectation of the random variable v o l k k n e k n shows how well k approximates k in volume on the average groemer grl proved that among all convex bodies k e o g l d the ellipsoids are approximated worst i e

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