asymptotic stability of linear neutral delay differential algebraic equations and runge kutta methods this paper is concerned with asymptotic stability of linear neutral delay differential algebraic equations and runge kutta methods first we give a new equivalent sufficient condition for the neutral delay differential algebraic equations to be delay independent asymptotically stable then we investigate the asymptotic stability of the numerical solutions generated by the runge kutta methods combined with lagrange interpolation some results on the asymptotic stability of runge kutta methods of high order are given finally numerical examples of index 1 and 2 are conducted to confirm our numerical stability result

stability of the rosenbrock methods for the neutral delay differential algebraic equations this paper develops the rosenbrock methods for the neutral delay differential algebraic equations nddaes and proves that the rosenbrock methods equipped with suitable interpolation are gp stable under proper assumption for the linear neutral delay differential algebraic equations with constant coefficients furthermore the gp stability of the runge kutta methods is also considered the discussions are supported by the numerical experiments

on the smoothing property of linear delay partial differential equations abstract we consider linear partial differential equations with an additional delay term which under spatial discretization lead to ordinary differential equations with fixed delay of retarded type this means that the semi discrete solution gains smoothness over time for the concept of classical mild and weak solutions we analyse whether this effect also takes place in the original system we show that some systems behave in a neutral way only as a result the smoothness of the exact solution remains unchanged instead of gaining smoothness over time

topological generalizations of network motifs biological and technological networks contain patterns termed network motifs which occur far more often than in randomized networks network motifs were suggested to be elementary building blocks that carry out key functions in the network it is of interest to understand how network motifs combine to form larger structures to address this we present a systematic approach to define motif generalizations families of motifs of different sizes that share a common architectural theme to define motif generalizations we first define roles in a subgraph according to structural equivalence for example the feedforward loop triad a motif in transcription neuronal and some electronic networks has three roles an input node an output node and an internal node the roles are used to define possible generalizations of the motif the feedforward loop can have three simple generalizations based on replicating each of the three roles and their connections we present algorithms for efficiently detecting motif generalizations we find that the transcription networks of bacteria and yeast display only one of the three generalizations the multi output feedforward generalization in contrast the neuronal network of c elegans mainly displays the multi input generalization forward logic electronic circuits display a multi input multi output hybrid thus networks which share a common motif can have very different generalizations of that motif using mathematical modeling we describe the information processing functions of the different motif generalizations in transcription neuronal and electronic networks

asymptotic stability of linear neutral delay differential algebraic equations and linear multistep methods this paper is concerned with delay independent asymptotic stability of linear neutral delay differential algebraic equations and linear multistep methods we first give some sufficient conditions for the delay independent asymptotic stability of these equations then we study and derive a sufficient and necessary condition for the delay independent asymptotic stability of numerical solutions obtained by linear multistep methods combined with lagrange interpolation finally one numerical example is performed to confirm our theoretical result

projected collocation for higher order higher index differential algebraic equations abstract higher order higher index hessenberg systems of initial and boundary value differential algebraic equations daes are considered these types of systems arise in a variety of applications including multibody systems we extend a class of recently introduced projected implicit runge kutta methods and define a new class of projected piecewise polynomial collocation methods for the solution of these problems stabilizing reformulations are considered as well and a new projected invariant method is proposed the higher order ode part of the dae is collocated directly by a piecewise polynomial a projection modification helps restore all the properties of stability and superconvergence which a corresponding collocation method for an ode possesses higher order collocation at radau points is recommended for initial value problems the projection methods appear to be particularly promising for the solution of dae boundary value problems where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations like collocation at gauss points previously defined symmetric methods have severe limitations when applied to these problems including instability oscillation and loss of accuracy the new methods overcome these difficulties for higher index problems we consider reformulation methods of stabilizing index reduction we propose new methods of projected invariants to handle particularly tough higher index problems the advantages offered by these methods are demonstrated numerically

continuous parallel block methods and their applications continuous numerical methods have many applications in the numerical solution of discontinuous ordinary differential equations delay differential equations delay differential algebraic equations neutral delay differential equations integro differential equations etc this paper deals with a continuous extension for the discrete approximate solution of ordinary differential equations generated by a class of parallel block methods existence and uniqueness for the continuous extension are discussed convergence and absolute stability of the continuous parallel block methods for ordinary differential equations are studied as applications we adopt the continuous parallel block methods to solve delay differential equations neutral delay differential equations and delay differential algebraic equations and obtain sufficient and necessary conditions for the continuous parallel block methods to be asymptotically stable several numerical experiments are given to illustrate the performance of the continuous parallel block methods

projected implicit runge kutta methods for differential algebraic equations in this paper a new class of numerical methods projected implicit runge kutta methods is introduced for the solution of index 2 hessenberg systems of initial and boundary value differential algebraic equations daes these types of systems arise in a variety of applications including the modeling of singular optimal control problems and parameter estimation for differential algebraic equations such as multibody systems the new methods appear to be particularly promising for the solution of dae boundary value problems where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations previously defined symmetric methods have severe limitations when applied to these problems including instability oscillation and loss of accuracy the new methods overcome these difficulties for linear problems we define an essential underlying boundary value ode and prove well conditioning of the differential or state space solutio

graph animals subgraph sampling and motif search in large networks we generalize a sampling algorithm for lattice animals connected clusters on a regular lattice to a monte carlo algorithm for graph animals i e connected subgraphs in arbitrary networks as with the algorithm in n kashtan et al bioinformatics 20 1746 2004 it provides a weighted sample but the computation of the weights is much faster linear in the size of subgraphs instead of superexponential this allows subgraphs with up to ten or more nodes to be sampled with very high statistics from arbitrarily large networks using this together with a heuristic algorithm for rapidly classifying isomorphic graphs we present results for two protein interaction networks obtained using the tandem affinity purification tap method one of escherichia coli with 230 nodes and 695 links and one for yeast saccharomyces cerevisiae with roughly ten times more nodes and links we find in both cases that most connected subgraphs are strong motifs z scores 10 or antimotifs z scores 10 when the null model is the ensemble of networks with fixed degree sequence strong differences appear between the two networks with dominant motifs in e coli being nearly bipartite graphs and having many pairs of nodes that connect to the same neighbors while dominant motifs in yeast tend towards completeness or contain large cliques we also explore a number of methods that do not rely on measurements of z scores or comparisons with null models for instance we discuss the influence of specific complexes like the 26s proteasome in yeast where a small number of complexes dominate the k cores with large k and have a decisive effect on the strongest motifs with 6 8 nodes we also present zipf plots of counts versus rank they show broad distributions that are not power laws in contrast to the case when disconnected subgraphs are included

robust stability and stabilization for singular systems with state delay and parameter uncertainty considers the problems of robust stability and stabilization for uncertain continuous singular systems with state delay the parametric uncertainty is assumed to be norm bounded the purpose of the robust stability problem is to give conditions such that the uncertain singular system is regular impulse free and stable for all admissible uncertainties while the purpose of the robust stabilization is to design a state feedback control law such that the resulting closed loop system is robustly stable these problems are solved via the notions of generalized quadratic stability and generalized quadratic stabilization respectively necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are derived a strict linear matrix inequality lmi design approach is developed an explicit expression for the desired robust state feedback control law is also given finally a numerical example is provided to demonstrate the application of the proposed method

approximation theory of fuzzy systems based upon genuine many valued implications siso cases it is proved that the single input and single output siso fuzzy systems based upon genuine many valued implications are universal approximators it is shown theoretically that fuzzy control systems based upon genuine many valued implications are equivalent to those based upon t norm implications the general approach to construct fuzzy systems is given it is also shown that defuzzifier based upon center of areas is not appropriate to the fuzzy systems based upon genuine many valued implications

topology of technology graphs small world patterns in electronic circuits recent theoretical studies and extensive data analyses have revealed a common feature displayed by biological social and technological networks the presence of small world patterns here we analyze this problem by using several graphs obtained from one of the most common technological systems electronic circuits it is shown that both analogic and digital circuits exhibit small world behavior we conjecture that the small world pattern arises from the compact design in which many elements share a small close physical neighborhood plus the fact that the system must define a single connected component which requires shortcuts connecting different integrated clusters the degree distributions displayed are consistent with a conjecture concerning the sharp cutoffs associated to the presence of costly connections amaral et al proc natl acad sci usa 97 11 149 2000 thus providing a limit case for the classes of universality of small world patterns from real artificial networks the consequences for circuit design are outlined

solving neutral delay differential equations with state dependent delays in this paper we consider a class of neutral delay differential equations with state dependent delays for such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving rise to subsequent points called breaking points where the solution derivative is still discontinuous as a consequence in a right neighbourhood of each such point we have to face a cauchy problem where the equation has a discontinuous right hand side in this case the existence and the uniqueness of the solution is no longer guaranteed to the right of such points and hence the solution of the neutral equation may either cease to exist or bifurcate after illustrating why uniqueness and existence of the solution is no longer guaranteed for general state dependent problems and showing a possible way to detect these occurrences automatically we explain how to generalize regularize the problem in order to suitably extend the solution beyond the breaking point this is important for example when exploring numerically the presence of possible periodic orbits

the numerical solution of delay differential algebraic equations of retarded and neutral type in this paper we consider the numerical solution of initial value delay differential algebraic equations ddaes of retarded and neutral types with a structure corresponding to that of hessenberg daes we give conditions under which the ddae is well conditioned and show how the ddae is related to an underlying retarded or neutral delay ordinary differential equation dode we present convergence results for linear multistep and runge kutta methods applied to ddaes of index 1 and 2 and show how higher index hessenberg ddaes can be formulated in as stable a way as index 2 hessenberg ddaes we also comment on some practical aspects of the numerical solution of these problems

stability of computational methods for constrained dynamics systems many methods have been proposed for numerically integrating the differential algebraic systems arising from the euler lagrange equations for constrained motion these are based on various problem formulations and discretizations we offer a critical evaluation of these methods from the standpoint of stability nconsidering a linear model we first give conditions under which the differential algebraic problem is well conditioned this involves the concept of an essential underlying ode we review a variety of reformulations which have been proposed in the literature and show that most of them preserve the well conditioning of the original problem then we consider stiff and nonstiff discretizations of such reformulated models in some cases the same implicit discretization may behave in a very different way when applied to different problem formulations acting as a stiff integrator on some formulations and as a nonstiff integrator on others we present the approach of projected invariants as a method for yielding problem reformulations which are desirable in this sense

link and subgraph likelihoods in random undirected networks with fixed and partially fixed degree sequences the simplest null models for networks used to distinguish significant features of a particular network from a priori expected features are random ensembles with the degree sequence fixed by the specific network of interest these fixed degree sequence fds ensembles are however famously resistant to analytic attack in this paper we introduce ensembles with partially fixed degree sequences pfds and compare analytic results obtained for them with monte carlo results for the fds ensemble these results include link likelihoods subgraph likelihoods and degree correlations we find that local structural features in the fds ensemble can be reasonably well estimated by simultaneously fixing only the degrees of a few nodes in addition to the total number of nodes and links as test cases we use two protein interaction networks escherichia coli saccharomyces cerevisiae the internet on the autonomous system as level and the world wide web fixing just the degrees of two nodes gives the mean neighbor degree as a function of node degree k k in agreement with results explicitly obtained from rewiring for power law degree distributions we derive the disassortativity analytically in the pfds ensemble the partition function can be expanded diagrammatically we obtain an explicit expression for the link likelihood to lowest order which reduces in the limit of large sparse undirected networks with l links and with k max l to the simple formula p k k kk 2l kk in a similar limit the probability for three nodes to be linked into a triangle reduces to the factorized expression p delta k 1 k 2 k 3 p k 1 k 2 p k 1 k 3 p k 2 k 3

asymptotic behavior of nonlinear delay differential algebraic equations and implicit euler methods this paper is concerned with the theory of asymptotic behavior of a class of nonlinear delay differential algebraic equations and implicit euler methods sufficient conditions for the theory of stability and asymptotic stability of the equations are given an example shows these conditions can be applied conveniently to nonlinear equations it is also showed that the implicit euler methods are stable and asymptotically stable

approximation methods for the consistent initialization of differential algebraic equations the algebraic constraints in a system of differential algebraic equations daes impose a consistency requirement on the initial values that can be difficult to satisfy in this thesis the consistency requirement is characterized by a system of equations an approximation method is introduced for these equations and the numerical solution of the resulting system is analyzed for certain important classes of daes finally some numerical experiments are described

differential algebraic equations with after effect in this paper we are concerned with the solution of delay differential algebraic equations these are differential algebraic equations with after effect or constrained delay differential equations the general semi explicit form of the problem consists of a set of delay differential equations combined with a set of constraints that may involve retarded arguments even simply stated problems of this type can give rise to difficult analytical and numerical problems the more tractable examples can be shown to be equivalent to systems of delay or neutral delay differential equations our purpose is to highlight some of the complexities and obstacles that can arise when solving these problems and to indicate problems that require further research

systematic identification of statistically significant network measures we present a graph embedding space i e a set of measures on graphs for performing statistical analyses of networks key improvements over existing approaches include discovery of motif hubs multiple overlapping significant subgraphs computational efficiency relative to subgraph census and flexibility the method is easily generalizable to weighted and signed graphs the embedding space is based on scalars functionals of the adjacency matrix representing the network scalars are global involving all nodes although they can be related to subgraph enumeration there is not a one to one mapping between scalars and subgraphs improvements in network randomization and significance testing we learn the distribution rather than assuming gaussianity are also presented the resulting algorithm establishes a systematic approach to the identification of the most significant scalars and suggests machine learning techniques for network classification

constructive theory for fuzzy systems abstract in this paper a constructive theory is developed to establish the fact that we can build a fuzzy system to approximate any continuous function on a compact set within a prescribed error bound based on the theory an algorithm is described that can actually construct a near minimum fuzzy system for a given function to a desired accuracy

correlation of automorphism group size and topological properties with program size complexity evaluations of graphs and complex networks the systems biology dtc university of oxford uk dated february 25 2014 we show that numerical approximations of kolmogorov complexity k of graphs and networkscapture some group theoretic and topological properties of empirical networks ranging frommetabolic to social networks and of small synthetic networks that we have produced that k andthe size of the group of automorphisms of a graph are correlated opens up interesting connectionsto problems in computational geometry and thus connects several measures and concepts fromcomplexity science we derive these results via two di erent kolmogorov complexity approximationmethods applied to the adjacency matrices of the graphs and networks the methods used are thetraditional lossless compression approach to kolmogorov complexity and a normalised version of ablock decomposition method bdm based on algorithmic probability theory keywords kolmogorov complexity graph automorphism complex networks graph automor phisms algorithmic probability block decomposition method compressibility biological networks network biology i introduction

subgraph ensembles and motif discovery using an alternative heuristic for graph isomorphism a heuristic based on vertex invariants is developed to rapidly distinguish nonisomorphic graphs to a desired level of accuracy the method is applied to sample subgraphs from an escherichia coli protein interaction network and as a probe for discovery of extended motifs the network s structure is described using statistical properties of its n node subgraphs for n ensuremath leqslant 14 the zipf plots for subgraph occurrences are robust power laws that do not change when rewiring the network while fixing the degree sequence although many specific subgraphs exchange rank the exponent for the zipf law depends on n studying larger subgraphs highlights some striking patterns for various n motifs or connected pieces that are overabundant in the ensemble of subgraphs have more edges for a given number of nodes than antimotifs and generally display a bipartite structure or tend toward a complete graph in contrast antimotifs which are underabundant connected pieces are mostly trees or contain at most a single small loop

subgraphs in random networks understanding the subgraph distribution in random networks is important for modeling complex systems in classic erdos networks which exhibit a poissonian degree distribution the number of appearances of a subgraph g with n nodes and g edges scales with network size as approximately n n g however many natural networks have a non poissonian degree distribution here we present approximate equations for the average number of subgraphs in an ensemble of random sparse directed networks characterized by an arbitrary degree sequence we find scaling rules for the commonly occurring case of directed scale free networks in which the outgoing degree distribution scales as p k approximately k gamma considering the power exponent of the degree distribution gamma as a control parameter we show that random networks exhibit transitions between three regimes in each regime the subgraph number of appearances follows a different scaling law approximately nalpha where alpha n g s 1 for gamma gamma c where s is the maximal outdegree in the subgraph and gamma c s 1 we find that certain subgraphs appear much more frequently than in erdos networks these results are in very good agreement with numerical simulations this has implications for detecting network motifs subgraphs that occur in natural networks significantly more than in their randomized counterparts

stability criteria for differential algebraic equations with multiple delays and their numerical solutions abstract this paper is concerned with the asymptotic stability of differential algebraic equations with multiple delays and their numerical solutions first we give a sufficient condition for delay independent stability after characterizing the coefficient matrices that satisfy this stability condition we propose some practical checkable criteria for asymptotic stability then we investigate the stability of numerical solutions obtained by methods and bdf methods finally solvability and stability of a class of weakly regular delay differential algebraic equations are analyzed

a singularity induced bifurcation theorem for infinite dimensional implicit dynamical systems in this note we study a class of partial differential equations whose formal structure resembles the semi explicit index 1 form from the theory of differential algebraic equations it is known from the theory of such problems that one can encounter simple poles in the eigenvalue loci of parameter dependent linearizations leading to the loss of linear stability without the presence of steady state bifurcations we demonstrate how this carries over to implicitly defined problems on a hilbert space

what is a complex graph abstract many papers published in recent years show that real world graphs g n m n nodes m edges are more or less complex in the sense that different topological features deviate from random graphs here we narrow the definition of graph complexity and argue that a complex graph contains many different subgraphs we present different measures that quantify this complexity for instance c 1 e the relative number of non isomorphic one edge deleted subgraphs i e deck size however because these different subgraph measures are computationally demanding we also study simpler complexity measures focussing on slightly different aspects of graph complexity we consider heuristically defined product measures the products of two quantities which are zero in the extreme cases of a path and clique and entropy measures quantifying the diversity of different topological features the previously defined network graph complexity measures medium articulation and offdiagonal complexity odc belong to these two classes we study odc measures in some detail and compare it with our new measures for all measures the most complex graph g c max has a medium number of edges between the edge numbers of the minimum and the maximum connected graph n 1 m g c max n n 1 2 interestingly for some measures c this number scales exactly with the geometric mean of the extremes m g c max n 2 n 1 n 1 5 all graph complexity measures are characterized with the help of different example graphs for all measures the corresponding time complexity is given finally we discuss the complexity of 33 real world graphs of different biological social and economic systems with the six computationally most simple measures including odc the complexities of the real graphs are compared with average complexities of two different random graph versions complete random graphs just fixed n m and rewired graphs with fixed node degrees

retarded differential equations retarded differential equations rdes are differential equations having retarded arguments they arise in many realistic models of problems in science engineering and medicine where there is a time lag or after effect numerical techniques for such problems may be regarded as extensions of dense output methods for ordinary differential equations odes but scalar rdes are inherently infinite dimensional with a richer structure than their ode counterparts we give background material develop a theoretical foundation for the basic numerics and give some results not previously published