river landscapes and optimal channel networks we study tree structures termed optimal channel networks ocns that minimize the total gravitational energy loss in the system an exact property of steady state landscape configurations that prove dynamically accessible and strikingly similar to natural forms here we show that every ocn is a so called natural river tree in the sense that there exists a height function such that the flow directions are always directed along steepest descent we also study the natural river trees in an arbitrary graph in terms of forbidden substructures which we call k path obstacles and ocns on a d dimensional lattice improving earlier results by determining the minimum energy up to a constant factor for every d 2 results extend our capabilities in environmental statistical mechanics

universal coalgebra a theory of systems in the semantics of programming finite data types such as finite lists have traditionally been modelled by initial algebras later final coalgebras were used in order to deal with infinite data types coalgebras which are the dual of algebras turned out to be suited moreover as models for certain types of automata and more generally for transition and dynamical systems an important property of initial algebras is that they satisfy the familiar principle of induction such a principle was missing for coalgebras until the work of aczel 1988 on a theory of non wellfounded sets in which he introduced a proof principle nowadays called coinduction it was formulated in terms of bisimulation a notion originally stemming from the world of concurrent programming languages milner 1980 park 1981 using the notion of coalgebra homomorphism the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras aczel and mendler 1989 thus the three basic notions of universal algebra algebra homomorphism of algebras and congruence turn out to correspond to coalgebra homomorphism of coalgebras and bisimulation respectively in this paper the latter are taken as the basic ingredients of a theory called universal coalgebra some standard results from universal algebra are reformulated using the afore mentioned correspondence and proved for a large class of coalgebras leading to a series of results on e g the lattices of subcoalgebras and bisimulations simple coalgebras and coinduction and a covariety theorem for coalgebras similar to birkhoff s variety theorem

exploring network scaling through variations on optimal channel networks metabolic allometry a common pattern in nature is a close to 3 4 power scaling law between metabolic rate and body mass in organisms across and within species an analogous relationship between metabolic rate and water volume in river networks has also been observed optimal channel networks ocns at local optima accurately model many scaling properties of river systems including metabolic allometry ocns are embedded in 2d space this work extends the model to three dimensions in this paper we compare characteristics of 3d ocns with 2d ocns and with organic metabolic networks studying the scaling behaviors of area length volume and energy we find that the 3d ocn has predictable characteristics analogous to those of the 2d version as well as scaling properties similar to metabolic networks in biological organisms

co induction in relational semantics abstract an application of the mathematical theory of maximum fixed points of monotonic set operators to relational semantics is presented it is shown how an important proof method which we call co induction a variant of park s 1969 principle of fixpoint induction can be used to prove the consistency of the static and the dynamic relational semantics of a small functional programming language with recursive functions

simple statistical models for river networks abstract the geometric scaling properties associated with simple river network models have been studied using computer simulations in these models the river networks are comprised of random trajectories which start at randomly selected positions on a square lattice and continue until they either reach an edge of the lattice or join a previous trajectory to form a branched network for those cases where the trajectories are either self avoiding random walks saws or indefinitely growing self avoiding walks igsaws the river basins are compact but have fractal basin boundaries with a dimensionally near to 5 4 for the igsaw model the longest river is a self similar fractal with a fractal dimensionality close to or equal to that of the igsaw itself if the trajectory is a directed walk then the model is very similar to the scheidegger river network model in this case the basin boundary and the longest channel are self affine brownian processes the results obtained from the igsaw and directed walk models can be described in terms of a simple scaling model for the saw models this scaling picture does not apply and it appears that practical simulations are far from the asymptotic large system size limit

statistical properties of aggregation with injection we study a generalized aggregation process in which charged particles diffuse and coalesce randomly on a lattice for one dimensional and mean field models we show that there exists a statistically invariant steady state when randomly charged particles are continuously injected the steady state charge distribution obeys a power law with the exponent depending both on the type of the injection and on the spatial dimension the response of the system to a perturbation i e relaxation is characterized by either a power law decay t 1 or a compressed exponential decay exp t 1

trees networks and hydrology this paper reviews theoretical and observational material on form and function of natural networks appeared in somewhat disparate contexts from physics to biology whose study is related to hydrologic research moving from the exact result that drainage network configurations minimizing total energy dissipation are stationary solutions of the general equation describing landscape evolution we discuss the properties and the dynamic origin of the scale invariant structure of river patterns and its relation to optimal selection we argue that at least in the fluvial landscape nature works through imperfect searches for dynamically accessible optimal configurations and that purely random or deterministic constructs are clearly unsuitable to properly describe natural network forms we also show that optimal networks are spanning loopless configurations only under precise physical requirements that arise under the constraints imposed by continuity in the case of rivers every spanning tree proves a local minimum of total energy dissipation this is stated in a theorem form applicable to generic networks suggesting that other branching structures occurring in nature e g scale free and looping may possibly arise through optimality to different selective pressures we thus conclude that one recurrent self organized mechanism for the dynamic origin of fractal forms is the robust strive for imperfect optimality that we see embedded in many natural patterns chief and foremost hydrologic ones copyright 2006 by the american geophysical union

type inference for polymorphic references abstract the hindley milner discipline for polymorphic type inference in functional programming languages is not sound if used on functions that can create and update references pointers we have found that the reason is a simple technical point concerning the capture of free type variables in store typings we present a modified type inference system and prove its soundness using operational semantics it is decidable whether given an expression e any type can be inferred for e if some type can be inferred for e then a principal type can be inferred principal types are found using unification the ideas extend to polymorphic exceptions and have been adopted in the definition of the programming language standard ml

on hack s law hack s law is reviewed emphasizing its implications for the elongation of river basins as well as its connections with their fractal characteristics the relation between hack s law and the internal structure of river basins is investigated experimentally through digital elevation models it is found that hack s exponent elongation and some relevant fractal characters are closely related the self affine character of basin boundaries is shown to be connected to the power law decay of the probability of total contributing areas at any link and to hack s law an explanation for hack s law is derived from scaling arguments from the results we suggest that a statistical framework referring to the scaling invariance of the entire basin structure should be used in the interpretation of hack s law

compressibility a property of topological spaces related to abstract convexity abstract an abstract operation of hull in a set y associates with every finite set f y a set hull f y a topological space x is said to be compressible into a topological space y equipped with an operation of hull if for every sequence a1 an of open sets covering x and every y1 yn yn there exists a continuous function x y such that for every x x x hull y i x a i the concept of compressibility is extremely useful when dealing with problems of the existence of fixed points continuous selections or with kkm property the obtained results generalize several known theorems they also provide a broad range of new original results

analytical and numerical study of optimal channel networks we analyze the optimal channel network model for river networks using both analytical and numerical approaches this is a lattice model in which a functional describing the dissipated energy is introduced and minimized in order to find the optimal configurations the fractal character of river networks is reflected in the power law behavior of various quantities characterizing the morphology of the basin in the context of a finite size scaling ansatz the exponents describing the power law behavior are calculated exactly and show mean field behavior except for two limiting values of a parameter characterizing the dissipated energy for which the system belongs to different universality classes two modified versions of the model incorporating quenched disorder are considered the first simulates heterogeneities in the local properties of the soil and the second considers the effects of a nonuniform rainfall in the region of mean field behavior the model is shown to be robust for both kinds of perturbations in the two limiting cases the random rainfall is still irrelevant whereas the heterogeneity in the soil properties leads to different universality classes results of a numerical analysis of the model are reported that confirm and complement the theoretical analysis of the global minimum the statistics of the local minima are found to resemble more strongly observational data on real rivers

minimum energy dissipation river networks with fractal boundaries the effects of the entire drainage area boundary on the drainage basins and their size distributions have been studied using the minimum energy dissipation model for river networks a simple scaling relationship ensuremath tau 1 d 2 between the values of the exponents ensuremath tau in the power law basin size distibutions and the fractal dimension d of the boundary of the entire drainage area has been established the scaling relationship has been tested using simulations with external perimeters d 4 3 or hulls d 7 4 of invasion percolation clusters as the boundaries for the entire drainage areas

scaling in weighted networks and complex systems abstract network systems evolving by preferential attachment of new vertices and by link loading are considered the probability of node attaching is assumed to depend linearly on degrees of existing nodes to which new ones are connected the rule of the network loading allows both deterministic and random processes of distributing weights among links specifically the rule is determined by assigning to each link a weight dependent on randomly modified intervals between times at which linked vertices are connected to the system this enables the investigation of the influence of local and global loading processes on statistical properties of weighted networks it is demonstrated that for different values of parameters controlling these processes the distribution of weights concentrated at vertices of such networks displays a large variety of scale free and truncated scale free behaviors it is also shown that statistical properties of the studied weighted networks and statistical properties of some complex evolving spatial systems exhibit a close relationship

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

probabilistic structure of the distance between tributaries of given size in river networks we analyze the distribution of the distances between tributaries of a given size or of sizes larger than a given area draining along either an open boundary or the mainstream of a river network by proposing a description of the distance separating prescribed merging contributing areas we also address related variables like mean or bankfull flow rates and channel and riparian area widths which are derived under a set of reasonable hydrologic assumptions the importance of such distributions lies in their ecological hydrologic and geomorphic implications on the spreading of species along the ecological corridor defined by the river network and on the propagation of infections due to water borne diseases particularly in view of exact theoretical predictions explicitly using the alongstream distribution of confluences carrying a given flow use is made here of real river networks suitably extracted from digital elevation models optimal channel networks and exactly solved tree like constructs like the peano and the scheidegger networks the results obtained redefine theoretically in a coherent and general manner and verify observationally the distribution function of the above distances and thus provide the general probabilistic structure of tributaries in river networks specifically we find that the probability of exceedence of the alongstream distance d of tributaries of size larger than a has the explicit form p d exp cd a h 1 h where c is a constant that depends on the choice of boundary conditions and h 1 is the hurst exponent

minimum energy and fractal structures of drainage networks this paper explores the similarities of digital elevation maps dems of natural river basins and optimal channel network ocn configurations obtained minimizing the total rate of energy expenditure in the system as a whole and in its parts striking similarities are observed for natural and optimal networks in their fractal aggregation structure and in certain multifractal structures found to be characteristic of river basins our results suggest upon critical assessment of the reliability of the identification of the attractor of the underlying dynamics implied by our optimality concepts that fractal structures are indeed possibly a product of least energy dissipation power laws emerging in the description of the distribution of aggregated quantities from both dems and ocns suggest a link with the framework of self organized criticality in the dynamics of natural channel network formation also the geomorphological description of ocns reveals surprising analogies with well known empirical or experimental results a comparison of peano s basins with ocns suggests that nature seems to reject the type of strict self similarity exhibited by peano s construct in favor of different shapes implying statistical self similarity not only because of chance acting through random conditions but also because of necessity as reflected by least energy expenditure considerations

exact analysis of the peano basin the peano basin is analysed as a deterministic model for river networks the fundamental distributions characterizing the real basins morphology can be explicitly calculated the recently proposed finite size scaling ansatz is tested apart from oscillatory amplitudes typical of deterministic fractals

energy dissipation theories and optimal channel characteristics of river networks the effects of energy dissipation on channel properties of a river network are explored on the basis of a local and global hypothesis of optimality in energy expenditure we investigate the relationships between channel hydraulic geometry flow velocity channel bed slope and streamflow conditions in optimal river networks expressions for the rate of energy dissipation per unit channel area pa are derived as functions of cumulative drainage area and river network parameters optimal channel characteristics are developed that satisfy the hypothesis of local optimality and provide constant pa throughout the river network we show that these optimal channel characteristics are remarkably similar to those of many natural river systems in their downstream hydraulic geometry exponents channel bed slope scaling spatial distribution of average flow velocity boundary shear resistance to flow etc optimal combinations of channel downstream hydraulic geometry and basin topography were analyzed on data from goodwin creek we found ranges of optimality for the combination of the downstream hydraulic geometry exponent for width of leopold and maddock 1953 0 32 b 0 74 and the channel bed slope scaling exponent 0 65 z 0 29 and argue that river networks develop average channel properties within these ranges in order to attain constant pa throughout the network we propose that the hypothesis of local optimality is a central principle that explains the average behavior and adjustment of channel characteristics in natural river systems

a co induction principle for recursively defined domains abstract this paper establishes a new property of predomains recursively defined using the cartesian product disjoint union partial function space and convex powerdomain constructors we prove that the partial order on such a recursive predomain d is the greatest fixed point of a certain monotone operator associated to d this provides a structurally defined family of proof principles for these recursive predomains to show that one element of d approximates another it suffices to find a binary relation containing the two elements that is a post fixed point for the associated monotone operator the statement of the proof principles is independent of any of the various methods available for explicit construction of recursive predomains following milner and tofte 1991 the method of proof is called co induction it closely resembles the way bisimulations are used in concurrent process calculi milner 1989 two specific instances of the co induction principle already occur in the work of abramsky 1990 1991 in the form of internal full abstraction theorems for denotational semantics of sccs and lazy lambda calculus in the first case post fixed binary relations are precisely abramsky s partial bisimulations whereas in the second case they are his applicative bisimulations the co induction principle also provides an apparently useful tool for reasoning about equality of elements of recursively defined datatypes in strict or lazy higher order functional programming languages

species survival and scaling laws in hostile and disordered environments in this work we study the likelihood of survival of single species in the context of hostile and disordered environments population dynamics in this environment as modeled by the fisher equation is characterized by negative average growth rate except in some random spatially distributed patches that may support life in particular we are interested in the phase diagram of the survival probability and in the critical size problem i e the minimum patch size required for surviving in the long time dynamics we propose a measure for the critical patch size as being proportional to the participation ratio of the eigenvector corresponding to the largest eigenvalue of the linearized fisher dynamics we obtain the extinction survival phase diagram and the probability distribution function pdf of the critical patch sizes for two topologies namely the one dimensional system and the fractal peano basin we show that both topologies share the same qualitative features but the fractal topology requires higher spatial fluctuations to guarantee species survival we perform a finite size scaling and we obtain the associated scaling exponents in addition we show that the pdf of the critical patch sizes has an universal shape for the 1d case in terms of the model parameters diffusion growth rate etc in contrast the diffusion coefficient has a drastic effect on the pdf of the critical patch sizes of the fractal peano basin and it does not obey the same scaling law of the 1d case

universal scaling of optimal current distribution in transportation networks transportation networks are inevitably selected with reference to their global cost which depends on the strengths and the distribution of the embedded currents we prove that optimal current distributions for a uniformly injected d dimensional network exhibit robust scale invariance properties independently of the particular cost function considered as long as it is convex we find that in the limit of large currents the distribution decays as a power law with an exponent equal to 2d 1 d 1 the current distribution can be exactly calculated in d 2 for all values of the current numerical simulations further suggest that the scaling properties remain unchanged for both random injections and by randomizing the convex cost functions