a network algorithm for performing fisher s exact test in r c contingency tables abstract an exact test of significance of the hypothesis that the row and column effects are independent in an r c contingency table can be executed in principle by generalizing fisher s exact treatment of the 2 2 contingency table each table in a conditional reference set of r c tables with fixed marginal sums is assigned a generalized hypergeometric probability the significance level is then computed by summing the probabilities of all tables that are no larger on the probability scale than the observed table however the computational effort required to generate all r c contingency tables with fixed marginal sums severely limits the use of fisher s exact test a novel technique that considerably extends the bounds of computational feasibility of the exact test is proposed here the problem is transformed into one of identifying all paths through a directed acyclic network that equal or exceed a fixed length some interesting new optimization theorems are developed in the process the numer

on the main channel length area relationship for channel networks motivated by the empirical relationship between the main channel length and the basin area for river networks a similar equation is derived analytically using the random model postulates and the additional assumption that link lengths have a common exponential probability distribution specifically it is shown that in networks with large magnitude m the main channel length l m and the magnitude are related by l m 2 2 where is mean link length this result is utilized to explain another empirical relationship between the distance to the center of gravity lc m and the main channel length lc m l m 2 the observed deviations of the random model prediction regarding the main channel length from empirical observations is used to discuss some important open problems

exact person fit indexes for the rasch model for arbitrary alternatives a new algorithm for obtaining exact person fit indexes for the rasch model is introduced which realizes most powerful tests for a very general family of alternative hypotheses including tests concerning dif as well as model deviating item correlations the method is also used as a goodness of fit test for whole data sets where the item parameters are assumed to be known for tests with 30 items at most exact values are obtained for longer tests a monte carlo algorithm is proposed simulated examples and an empirical investigation demonstrate test power and applicability to item elimination

random self similar river networks and derivations of generalized horton laws in terms of statistical simple scaling recent data analyses of a dozen or so large networks support the presence of simple scaling in the probability distributions of many topologic geometric and hydraulic geometric variables we have labeled them generalized horton laws in contrast to traditional horton laws which are stated only in terms of means these data sets also show that horton s bifurcation ratios are significantly larger than 4 which can not be predicted by the well known random model motivated by the need to understand these data analyses of large river networks within a common theoretical framework a new class of models with statistically self similar network topology is introduced the network construction process can be viewed as a statistical generalization of the type of a recursive process which generates many fractal geometrical objects unlike the well known random model our random channel networks do not belong to the class of binary galton watson stochastic branching processes the generalized horton laws are predicted in the limit of large horton strahler order as a scale parameter for a subclass of random self similar network models a derivation of horton s law of stream numbers as a consequence of the simple scaling of distributions of basin magnitude gives it a new theoretical significance a further subclass also exhibits tokunga self similarity which is an average property of the side tributary structure in river networks comparisons with empirical observations for three large basins analyzed here show that the generalized horton laws are well predicted by this new class of random self similar network models and analytically predicted relationships between topologic and geometric horton ratios are also seen in data

definition of new equivalent indices of horton strahler ratios for the derivation of the geomorphological instantaneous unit hydrograph 1 the horton strahler concept has been important in river basin geomorphology it describes scaling properties and identifies giuh but has also attracted criticisms because it depends on the threshold area s which is used to extract the channel network from digital elevation models dem on the position of the outlet and on the few number of ratios used to identify the scaling laws to overcome these limitations this paper proposes new indices independently of s and which have similar properties than the horton strahler ratios applications were conducted on seven french basins the methodology consists on analyzing the morphometric properties such as the drained area the number of sources and the length of the channel network as a function of s the results show that horton strahler ratios vary considerably with s particularly while the order length l is a staircase function of s results show that the channel network is articulated around two main nodes independent of s which are descriptors of l then we compared the properties of horton strahler laws to those obtained from self similarity analysis which enables to define new indices independent of s these indices are calculated automatically from dem have similar properties as the horton strahler ratios and are used to calculate the giuh independently of s the four new descriptors rbe rle rae and le which have similar properties as rb rl ra and l respectively are equivalent to those of horton strahler s laws and can be used as indicators of hydrological similarity for catchment comparison and regionalization

bck combinators and linear l terms have types abstract a term l is linear when each abstract x m in it has at most one free x in m and each free variable of l occurs free only once bck combinators correspond exactly to linear terms without free variables the main theorem says that all terms of each kind are stratified i e are typable this result is not deep it merely confirms an intuition that non stratification is caused partly by multiple occurrences of variables for bck combinators it was stated as theorem 1 by bunder and meyer in 1 but the proof in 1 seems to contain a gap it uses induction on a definition of level of a combinator but omits to show that the level concept is defined for all bck combinators the proof below proceeds by translating to linear terms for which the induction becomes simply one on length in fact the proof is essentially just an amalgamation of elements from an unpublished proof of m coppo and m dezani ciancaglini and elements from 1 the type schemes of linear terms form a very primitive linear logic and linear logic is of current interest see girard 3 so we shall set out the basic facts about linear terms and bck combinators in some detail these basic facts are not new some appearin komori 6 for example and some are implicit in curry and feys 2

constructing the exact significance level for a person fit statistic this paper uses an extension of the network algorithm originally introduced by mehta and patel to construct exact tail probabilities for testing the general hypothesis that item responses are distributed according to the rasch model by assuming that item difficulties are known the algorithm is applicable to the statistical tests either given the maximum likelihood ability estimate or conditioned on the total score a simulation study indicates that the network algorithm is an efficient tool for computing the significance level of a person fit statistic based on test lengths of 30 items or less

type theories normal forms and d lambda models abstract a non standard inverse limit model d is constructed which has a non hilbert post complete theory moreover in d a simple semantic characterization of normalizable terms is given these results are proved using the properties of a generalized type assignment system which yields a filter model barendregt coppo and dezani ciancaglini 1983 j symbolic logic 48 931 940 coppo dezani ciancaglini honsell and longo 1983 pp 241 262 logic colloquium 82 north holland amsterdam isomorphic to d the type assignment system is also proved complete with respect to an interpretation of types in the term model of equality based only on normalization properties as an application a class of maximal monoids of normalizable terms is characterized

a reformulation of horton s laws for large river networks in terms of statistical self similarity the well known horton s laws are empirical observations on how the means of measurements for river networks and basins vary with horton strahler order it is now known that these laws are a consequence of an average sense self similarity in the bifurcation structure of river networks in this paper we present a reformulation of horton s laws which generalizes the familiar scaling of first moments or means to scaling of entire distributions we also present extensive data analysis which supports this reformulation and show that this feature is also exhibited by shreve s well known random topology model

directed recursion models for fractal growth fractals constructed by recursive processes are introduced to model growth phenomena these fractals are simultaneously directed and self similar in analogy with patterns growing under diffusion limited conditions the multifractal nature of the harmonic measure associated with laplacian interfaces is qualitatively interpreted using the models calculation of the largest singularity exponent allows one to make conclusions about the behaviour of diffusion limited aggregates

statistical self similarity in river networks parameterized by elevation a stochastic theory is developed to describe spatial variability in link heights in topologically random river networks both the systematic and random spatial variability in link heights are reflected in a scaling invariance property of their probability distributions with drainage area serving as a scale parameter tests of theoretical predictions against empirical observations show that link height distributions exhibit scaling invariance under magnification or reduction of the scale parameter this invariance property is referred to as statistical self similarity it provides a fundamental theoretical basis for some existing empirical relationships on gradients and other river geometries in channel networks and points to important research directions in river basin hydrology

nonparametric goodness of fit tests for the rasch model a monte carlo algorithm realizing a family of nonparametric tests for the rasch model is introduced which are conditional on the item and subject marginals the algorithm is based on random changes of elements of data matrices without changing the marginals most powerful tests against all alternative hypotheses are given for which a monotone characteristic may be computed from the data matrix alternatives may also be composed computation times are long but exactp values are approximated with the quality of approximation only depending on calculation time but not on the number of persons the power and the flexibility of the procedure is demonstrated by means of an empirical example where among others indicators for increased item similarities the existence of subscales violations of sufficiency of the raw score as well as learning processes were found many of the features described are implemented in the program t rasch 1 0 by ponocny and ponocny seliger 1999

the system f of variable types fifteen years later the semantic study of system f stumbles on the problem of variable types for which there was no convincing interpretation we develop here a semantics based on the category theoretic idea of direct limit so that the behaviour of a variable type on any domain is determined by its behaviour on finite ones thus getting rid of the circularity of variable types to do so one has first to simplify somehow the extant semantic ideas replacing scott domains by the simpler and more finitary qualitative domains the interpretation obtained is extremely compact as shown on simple examples the paper also contains the definitions of a very small universal model of lambda calculus and investigates the concept totality

some neglected problems in irt the paper addresses three neglected questions from irt in section 1 the properties of the measurement of ability or trait parameters and item difficulty parameters in the rasch model are discussed it is shown that the solution to this problem is rather complex and depends both on general assumptions about properties of the item response functions and on assumptions about the available item universe section 2 deals with the measurement of individual change or modifiability based on a rasch test a conditional likelihood approach is presented that yields a an ml estimator of modifiability for given item parameters b allows one to test hypotheses about change by means of a clopper pearson confidence interval for the modifiability parameter or c to estimate modifiability jointly with the item parameters uniqueness results for all three methods are also presented in section 3 the mantel haenszel method for detecting dif is discussed under a novel perspective what is the most general framework within which the mantel haenszel method correctly detects dif of a studied item the answer is that this is a 2pl model where however all discrimination parameters are known and the studied item has the same discrimination in both populations since these requirements would hardly be satisfied in practical applications the case of constant discrimination parameters that is the rasch model is the only realistic framework a simple pearsonx2 test for dif of one studied item is proposed as an alternative to the mantel haenszel test moreover this test is generalized to the case of two items simultaneously studied for dif

the many null distributions of person fit indices this paper deals with the situation of an investigator who has collected the scores ofn persons to a set ofk dichotomous items and wants to investigate whether the answers of all respondents are compatible with the one parameter logistic test model of rasch contrary to the standard analysis of the rasch model where all persons are kept in the analysis and badly fittingitems may be removed this paper studies the alternative model in which a small minority ofpersons has an answer strategy not described by the rasch model such persons are called anomalous or aberrant from the response vectors consisting ofk symbols each equal to 0 or 1 it is desired to classify each respondent as either anomalous or as conforming to the model as this model is probabilistic such a classification will possibly involve false positives and false negatives both for the rasch model and for other item response models the literature contains several proposals for a person fit index which expresses for each individual the plausibility that his her behavior follows the model the present paper argues that such indices can only provide a satisfactory solution to the classification problem if their statistical distribution is known under the null hypothesis that all persons answer according to the model this distribution however turns out to be rather different for different values of the person s latent trait value this value will be called ability parameter although our results are equally valid for rasch scales measuring other attributes

universal scaling laws in surface water bodies and their zones of influence 1 topologically water bodies are the first level topographic regions that get flooded and as the flood level gets higher adjacent water bodies merge the looplike network that forms along all these merging points represents zones of influence of each water body these two topologically interdependent phenomena follow the universal scaling laws similar to certain other environmental and biological phenomena despite morphological variations water bodies and their influence zones of varied sizes and shapes have different sets of scaling exponents thereby determining that they belong to different universality classes

new results for self similar trees with applications to river networks in a little known series of papers beginning in 1966 tokunaga introduced an infinite class of tree graphs based on the strahler ordering scheme as recognized by tokunaga 1984 these trees are characterized by a self similarity property so we will refer to them as self similar trees or ssts ssts are defined in terms of a generator matrix which acts as a blueprint for constructing different trees many familiar tree constructions are absorbed as special cases however in tokunaga s work an additional assumption is imposed which restricts from ssts to a much smaller class we will refer to this subclass as tokunaga s trees this paper presents several new and unifying results for ssts in particular the conditions under which ssts have well defined horton strahler stream ratios are given as well as a general method for computing these ratios it is also shown that the diameters of ssts grow like m where m is the number of leaves in contrast to many other tree constructions here need not equal 1 2 thus ssts offer an explanation for hack s law finally it is demonstrated that large discrepancies exist between the predictions of shreve s well known model and detailed measurements for large river networks while other ssts fit the data quite well other potential applications of the sst framework include diffusion limited aggregation dla lightning bronchial passages neural networks and botanical trees

approximating the conditional distribution of person fit indexes for checking the rasch model we present a simple approximation to the conditional distribution of goodness of fit statistics for the rasch model assuming that the item difficulties are known the approximation is easily programmed and gives relatively accurate assessments of conditionalp values for tests of length 10 or more a few generalizations are discussed

power law distribution of discharge in ideal networks data for several river networks show algebraically decaying distribution functions for the mean annual discharge and the dissipation rate rodriguez iturbe et al 1992 we derive a relation between the exponent a in the integrated mean annual discharge distribution p q q q and the topological dimension dt of the network 1 1 dt using the experimentally determined value dt 1 8 tarboton et al 1988 we find 0 45 in good agreement with the data of rodriguez iturbe et al 1992 for the random model shreve 1967 1969 smart and werner 1974 we find an exponent of 1 2

filter models with polymorphic types abstract using ideas and results from barendrecht 1983 and coppo 1984 on intersection types a comparable theory is developed for second order polymorphic types the set of filters constructed with polymorphic type forms with inclusion a continuous lattice which yields a model of what we call expansion i e the value of a term increases under reduction but not of conversion combining intersection with polymorphic types does give filter models but the two standard ways of interpreting terms do not coincide

self similarity and structure of dla and viscous fingering clusters diffusion limited aggregation clusters and the structures observed in viscous fingering experiments at high capillary numbers are tree like fractals the different branches may be assigned a branch order in a way that exhibits scaling and permits a self similar characterisation in terms of the bifurcation ratio r and the length ratio r of branches of different orders the fractal dimension is given by d ln rn ln l rl good agreement between experiments and simulations is found a crossover function characterises the branch orders

an exact and optimal standardized person test for assessing consistency with the rasch model the rasch model predicts that an individual s ability level is invariant over subtests of the total test and thus all subtests measure the same latent trait a person test of this invariance hypothesis is discussed that is uniformly most powerful and standardized in the sense that the conditional distribution of the test statistic given a particular level of ability does not depend on the absolute value of the examinee s ability parameter the test can be routinely performed by applying a computer program designed by and obtainable from the author finally a suboptimal test is derived that is extremely easy to use and an overall group test of the invariance hypothesis discussed all tests considered do not rely on asymptotic approximations hence they may be applied when the test is of only moderate length and the group of examinees is small