computing sets of graded attribute implications with witnessed non redundancy in this paper we extend our previous results on sets of graded attribute implications with witnessed non redundancy we assume finite residuated lattices as structures of truth degrees and use arbitrary idempotent truth stressing linguistic hedges as parameters which influence the semantics of graded attribute implications in this setting we introduce algorithm which transforms any set of graded attribute implications into an equivalent non redundant set of graded attribute implications with saturated consequents whose non redundancy is witnessed by antecedents of the formulas as a consequence we solve the open problem regarding the existence of general systems of pseudo intents which appear in formal concept analysis of object attribute data with graded attributes and linguistic hedges furthermore we show a polynomial time procedure for determining bases given by general systems of pseudo intents from sets of graded attribute implications which are complete in data

minimal representation of directed hypergraphs in this paper the problem of minimal representations for particular classes of directed hypergraphs is analyzed various concepts of minimal representations of directed hypergraphs called minimal equivalent hypergraphs are introduced as extensions to the concepts of transitive reduction and minimum equivalent graph of directed graphs in particular we consider equivalent hypergraphs which are minimal with respect to all parameters which may be adopted to characterize a given hypergraph number of hyperarcs number of adjacency lists required for the representation length of the overall description etc the relationships among the various concepts of minimality are discussed and their computational properties are analyzed in order to derive such results a graph representation of hypergraphs is introduced

latticial structures in data analysis abstract in general lattice theory has helped to simplify unify and generalize many aspects of mathematics and it has suggested many interesting new problems in an era of ever increasing proliferation of scientific results its relative simplicity and unifying influence are certainly healthy and refreshing birkhoff 4 p 1 diagrams prove nothing but bring outstanding features readily to the eye they are therefore no substitute for such critical tests as may be applied to the data but are valuable in suggesting such tests and explaining the conclusions founded upon them fisher 16 p 24 quel que soit le point de depart de l activite scientifique cette activite ne peut pleinement convaincre qu en quittant le domaine de base si elle expeimente il faut raisonner si elle raisonne il faut experimenter bachelard 41 p 3 1 1

comments on the classification of the finite subgroups of su 3 many finite subgroups of su 3 are commonly used in particle physics the classification of the finite subgroups of su 3 began with the work of h f blichfeldt at the beginning of the 20th century in blichfeldt s work the two series c and d of finite subgroups of su 3 are defined while the group series 3n2 and 6n2 which are subseries of c and d respectively have been intensively studied there is not much knowledge about the group series c and d in this work we show that c and d have the structures and respectively furthermore we show that while the c groups can be interpreted as irreducible representations of 3n2 the d groups can in general not be interpreted as irreducible representations of 6n2

optimal compression of propositional horn knowledge bases complexity and approximation abstract horn formulae play a prominent role in artificial intelligence and logic programming in this paper we investigate the problem of optimal compression of propositional horn production rule knowledge bases the standard approach to this problem consisting in the removal of redundant rules from a knowledge base leads to an irredundant but not necessarily optimal knowledge base we prove here that the number of rules in any irredundant horn knowledge base involving n propositional variables is at most n 1 times the minimum possible number of rules therefore the quadratic time transformation of an arbitrary horn production rule base to an equivalent irredundant and prime one presented in 9 provides a reasonable approximation algorithm in order to formalize the optimal compression problem we define a boolean function of a knowledge base as being the function whose set of true points is the set of models of the knowledge base in this way the optimal compression of production rule knowledge bases becomes a problem of boolean function minimization in this paper we prove that the minimization of horn functions i e boolean functions associated to horn knowledge bases is np complete

the joy of implications aka pure horn formulas pure horn clauses have also been called among others functional dependencies strong association rules or simply implications we survey the mathematical theory of implications with an emphasis on the progress made in the last 30 years

self induced stochastic resonance in excitable systems the effect of small amplitude noise on excitable systems with strong time scale separation is analyzed it is found that vanishingly small random perturbations of the fast excitatory variable may result in the onset of a deterministic limit cycle behavior absent without noise the mechanism termed self induced stochastic resonance combines a stochastic resonance type phenomenon with an intrinsic mechanism of reset and no periodic drive of the system is required self induced stochastic resonance is different from other types of noise induced coherent behaviors in that it arises away from bifurcation thresholds in a parameter regime where the zero noise deterministic dynamics does not display a limit cycle nor even its precursor the period of the limit cycle created by the noise has a non trivial dependence on the noise amplitude and the time scale ratio between fast excitatory variables and slow recovery variables it is argued that self induced stochastic resonance may offer one possible scenario of how noise can robustly control the function of biological systems

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

concept lattices with negative information classical formal concept analysis fca extracts represents and manages knowledge from positive information i e its fundamental data model is a binary relation between a set of objects and attributes indicating the presence of a property in an object however some applications require to treat the absence of some property in an object as a negative information to be explicitly represented and managed too although mixed positive and negative information has been addressed in the past in fca such approaches maintain the standard framework which hides the specific semantics and avoids the further use of direct techniques and methods for mixed information in this work the foundations of fca are extended and in particular mixed concept lattices are studied in depth the main result of this work is a characterization theorem specifying in lattice theoretic terms which lattices are isomorphic to a mixed concept lattice

canards in r3 abstract we give a geometric analysis of canard solutions in three dimensional singularly perturbed systems with a folded two dimensional critical manifold by analysing the reduced flow we obtain singular canard solutions passing through a singularity on the fold curve we classify these singularities called canard points as folded saddles folded nodes and folded saddle nodes we prove the existence of canard solutions in the case of the folded saddle we show the existence of canards in the folded node case provided a generic non resonance condition is satisfied and in a subcase of the folded saddle node the proof is based on the blow up method

functional dependencies in horn theories abstract this paper studies functional dependencies in horn theories both when the theory is represented by its clausal form and when it is defined as the horn envelope of a set of models we provide polynomial algorithms for the recognition of whether a given functional dependency holds in a given horn theory as well as polynomial algorithms for the generation of some representative sets of functional dependencies we show that some problems of inferring functional dependencies e g constructing an irredundant fd cover are computationally difficult we also study the structure of functional dependencies that hold in a horn theory showing that every such functional dependency is in fact a single positive term boolean function and prove that for any horn theory the set of its minimal functional dependencies is quasi acyclic finally we consider the problem of condensing a horn theory prove that any horn theory has a unique condensation and develop an efficient polynomial algorithm for condensing horn theories

lattices closures systems and implication bases a survey of structural aspects and algorithms abstract concept lattices and closed set lattices are graphs with the lattice property they have been increasingly used this last decade in various domains of computer science such as data mining knowledge representation databases or information retrieval a fundamental result of lattice theory establishes that any lattice is the concept lattice of its binary table a consequence is the existence of a bijective link between lattices contexts via the table and a set of implicational rules via the canonical direct basis the possible transformations between these objects give rise to relevant tools for data analysis in this paper we present a survey of lattice theory from the algebraic definition of a lattice to that of a concept lattice through closure systems and implicational rules including the exploration of fundamental bijective links between lattices reduced contexts and bases of implicational rules and concluding with the presentation of the main generation algorithms of these objects

a mathematical framework for critical transitions bifurcations fast slow systems and stochastic dynamics abstract bifurcations can cause dynamical systems with slowly varying parameters to transition to far away attractors the terms critical transition or tipping point have been used to describe this situation critical transitions have been observed in an astonishingly diverse set of applications from ecosystems and climate change to medicine and finance the main goal of this paper is to give an overview which standard mathematical theories can be applied to critical transitions we shall focus on early warning signs that have been suggested to predict critical transitions and point out what mathematical theory can provide in this context starting from classical bifurcation theory and incorporating multiple time scale dynamics one can give a detailed analysis of local bifurcations that induce critical transitions we suggest that the mathematical theory of fast slow systems provides a natural definition of critical transitions since noise often plays a crucial role near critical transitions the next step is to consider stochastic fast slow systems the interplay between sample path techniques partial differential equations and random dynamical systems is highlighted each viewpoint provides potential early warning signs for critical transitions since increasing variance has been suggested as an early warning sign we examine it in the context of normal forms analytically numerically and geometrically we also consider autocorrelation numerically hence we demonstrate the applicability of early warning signs for generic models we end with suggestions for future directions of the theory

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

slowly varying control parameters delayed bifurcations and the stability of spikes in reaction diffusion systems we present three examples of delayed bifurcations for spike solutions of reaction diffusion systems the delay effect results as the system passes slowly from a stable to an unstable reg ime and was previously analyzed in the context of ode s in p mandel and t erneux j stat phys 48 5 6 pp 1059 1070 1987 it was found that the instability would not be fully realized until the system had entered well into the unstab le regime the bifurcation is said to have been delayed relative to the threshold value computed directly from a linear stability analysis in contrast to the study of mandel and erneux we analyze the delay effect in systems of partial differential equations pde s in particular for spike solutions of singularly perturbed generalized giere r meinhardt and gray scott models we analyze three examples of delay resulting from slow passage into regimes of oscillatory and competition instability in the first example for the gierer meinhardt model on the infinite real line we analyze the delay resulting from slowly tuning a control parameter through a hopf bifurcation in the second example we consider a hopf bifurcation of the gierer meinhardt model on a finite one dimensional domain in this s cenario as opposed to the extrinsic tuning of a system parameter through a bifurcation value we analyze the delay of a bifurcation triggered by slow intrinsic dynamics of the pde system in the third example we consider competition instabilities triggered by the extrinsic tuning of a feed rate parameter in all three cases we find that the syste m must pass well into the unstable regime before the onset of instability is fully observed indicating delay we also fin d that delay has an important effect on the eventual dynamics of the system in the unstable regime we give analytic predictions for the magnitude of the delays as obtained through the analysis of certain explicitly solvable nonloca l eigenvalue problems nlep s the theory is confirmed by numerical solutions of the full pde systems

succinctness and tractability of closure operator representations it is widely known that closure operators on finite sets can be represented by sets of implications also known as inclusion dependencies as well as by formal contexts in this article we consider these two representation types as well as generalizations of them extended implication sets and context families we discuss the mutual succinctness of these four representations and the tractability of certain operations used to compare and modify closure operators

fundamental study the multiple facets of the canonical direct unit implicational basis the notion of dependencies between attributes arises in many areas such as relational databases data analysis data mining formal concept analysis knowledge structures formalization of dependencies leads to the notion of so called full implicational systems or full family of functional dependencies which is in one to one correspondence with the other significant notions of closure operator and of closure system an efficient generation of a full implicational system or a closure system can be performed from equivalent implicational systems and in particular from the bases for such systems for example the so called canonical basis this paper shows the equality between five other bases originating from different works and satisfying various properties in particular they are unit implicational systems the three main properties of this unique basis are the directness canonical and minimal properties whence the name canonical direct unit implicational basis given to this unit implicational system the paper also gives a nice characterization of this canonical basis and makes precise its link with the prime implicants of the horn function associated to a closure operator it concludes that it is necessary to compare more closely related works made independently and with a different terminology in order to take advantage of the really new results in these works

predictability of critical transitions critical transitions in multistable systems have been discussed as models for a variety of phenomena ranging from the extinctions of species to socioeconomic changes and climate transitions between ice ages and warm ages from bifurcation theory we can expect certain critical transitions to be preceded by a decreased recovery from external perturbations the consequences of this critical slowing down have been observed as an increase in variance and autocorrelation prior to the transition however especially in the presence of noise it is not clear whether these changes in observation variables are statistically relevant such that they could be used as indicators for critical transitions in this contribution we investigate the predictability of critical transitions in conceptual models we study the quadratic integrate and fire model and the van der pol model under the influence of external noise we focus especially on the statistical analysis of the success of predictions and the overall predictability of the system the performance of different indicator variables turns out to be dependent on the specific model under study and the conditions of accessing it furthermore we study the influence of the magnitude of transitions on the predictive performance

deterministic continuation of stochastic metastable equilibria via lyapunov equations and ellipsoids numerical continuation methods for deterministic dynamical systems have been one of the most successful tools in applied dynamical systems theory continuation techniques have been employed in all branches of the natural sciences as well as in engineering to analyze ordinary partial and delay differential equations here we show that the deterministic continuation algorithm for equilibrium points can be extended to track information about metastable equilibrium points of stochastic differential equations we stress that we do not develop a new technical tool but that we combine results and methods from probability theory dynamical systems numerical analysis optimization and control theory into an algorithm that augments classical equilibrium continuation methods in particular we use ellipsoids defining regions of high concentration of sample paths it is shown that these ellipsoids and the distances between them can be efficiently calculated using iterative methods that take advantage of the numer

algorithms for k meet semidistributive lattices in this paper we consider k meet semidistributive lattices and we are interested in the computation of the set colored poset associated to an implicational base the parameter k is of interest since for any finite lattice l there exists an integer k for which l is k meet semidistributive when k 1 they are known as meet semidistributive lattices we first give a polynomial time algorithm to compute an implicational base of a k meet semidistributive lattice from its associated colored poset in other words for a fixed k finding a minimal implicational base of a k meet semidistributive lattice l from a context fca literature of l can be done not just in output polynomial time which is open in the general case but in polynomial time in the size of the input this result generalizes that in 26 second we derive an algorithm to compute a set colored poset from an implicational base which is based on the enumeration of minimal transversals of a hypergraph and turns out to be in polynomial time for k meet semidistributive lattices 13 20 finally we show that checking whether a given implicational base describes a k meet semidistributive lattice can be done in polynomial time

the slow passage through a hopf bifurcation delay memory effects and resonance this paper explores analytically and numerically in the context of the fitzhugh nagumo model of nerve membrane excitability an interesting phenomenon that has been described as a delay or memory effect it can occur when a parameter passes slowly through a hopf bifurcation point and the system s response changes from a slowly varying steady state to slowly varying oscillations on quantitative observation it is found that the transition is realized when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects the dynamic aspect of the parameter variation this delay and its dependence on the speed of the parameter variation are described the model involves several parameters and particular singular limits are investigated one in particular is the slow passage through a low frequency hopf bifurcation where the system s response changes from a slowly varying steady state to slowly varying relaxation oscillations we find in this case the onset o

reasoning with models we develop a model based approach to reasoning in which the knowledge base is represented as a set of models satisfying assignments rather than a logical formula and the set of queries is restricted we show that for every propositional knowledge base kb there exists a set of characteristic models with the property that a query is true in kb if and only if it is satisfied by the models in this set we characterize a set of functions for which the model based representation is compact and provides efficient reasoning these include cases where the formula based representation does not support efficient reasoning in addition we consider the model based approach to abductive reasoning and show that for any propositional kb reasoning with its model based representation yields an abductive explanation in time that is polynomial in its size some of our technical results make use of the monotone theory a new characterization of boolean functions recently introduced the notion of restricted queries is inherent in our approach this is a wide class of queries for which reasoning is efficient and exact even when the model based representation kb provides only an approximate representation of the domain in question moreover we show that the theory developed here generalizes the model based approach to reasoning with horn expressions and captures even the notion of reasoning with horn approximations

pathwise description of dynamic pitchfork bifurcations with additive noise the slow drift with speed of a parameter through a pitchfork bifurcation point known as the dynamic pitchfork bifurcation is characterized by a significant delay of the transition from the unstable to the stable state we describe the effect of an additive noise of intensity by giving precise estimates on the behaviour of the individual paths we show that until time after the bifurcation the paths are concentrated in a region of size around the bifurcating equilibrium with high probability they leave a neighbourhood of this equilibrium during a time interval after which they are likely to stay close to the corresponding deterministic solution we derive exponentially small upper bounds for the probability of the sets of exceptional paths with explicit values for the exponents

a theory of finite closure spaces based on implications abstract although many notions familiar from topology and matroid theory make sense for arbitrary closure spaces we claim that implicational bases are most worthwhile studying they are applied in the theory of relational data bases and in formal concept analysis further applications in combinatorics and algebra are foreseeable the main theorem describes the structure of minimum respectively optimal implicational bases of any finite closure space it has strong corollaries for specific closure spaces e g having a geometric respectively modular respectively lower distributive lattice of closed sets

hunting french ducks in a noisy environment we consider the effect of gaussian white noise on fast slow dynamical systems with one fast and two slow variables containing a folded node singularity in the absence of noise these systems are known to display mixed mode oscillations consisting of alternating large and small amplitude oscillations we quantify the effect of noise and obtain critical noise intensities above which the small amplitude oscillations become hidden by fluctuations furthermore we prove that the noise can cause sample paths to jump away from so called canard solutions with high probability before deterministic orbits do this early jump mechanism can drastically influence the local and global dynamics of the system by changing the mixed mode patterns

on the parametrization of certain finite groups and their representations we discuss a convenient way of parametrizing certain finite groups they comprise groups with abelian normal subgroups and central extensions thereof they include the groups dn t o and their double groups as well as the su 3 subgroups tn 3n2 6n2 72 216 36 and their z3 extensions the method allows for a rapid calculation of representations and coupling coefficients in particular it solves the multiplicity problem of nonsimply reducible groups we illustrate the method treating the hessian group 216 and its central extension

dynamics and absorption properties of stochastic equations with holder diffusion coefficients abstract in this article we characterize the dynamics and absorption properties of a class of stochastic differential equations around singular points where both the drift and diffusion functions vanish according to the holder coefficient of the diffusion function around the singular point we identify different regimes a regime where the solutions almost surely reach the singular point in finite time and regimes of exponential attraction or repulsion from the singular point stability of the absorbing state large deviations for the absorption time existence of stationary or quasi stationary distributions are discussed in particular we show that quasi stationary distributions only exist for 3 4 and for 3 4 1 no quasi stationary distribution is found and numerical simulations tend to show that the process conditioned on not being absorbed initiates an almost sure exponential convergence towards the absorbing state as is demonstrated to be true for 1 these results have several implications in the understanding of stochastic bifurcations and we completely unfold two generic situations the pitchfork and saddle node bifurcations and discuss the hopf bifurcation in the appendix

early warning signals for bifurcations in random dynamical systems with bounded noise abstract we consider discrete time one dimensional random dynamical systems with bounded noise which generate an associated set valued dynamical system we provide necessary and sufficient conditions for a discontinuous bifurcation of a minimal invariant set of the set valued dynamical system in terms of the derivatives of the so called extremal maps we propose an algorithm for reconstructing the derivatives of the extremal maps from a time series that is generated by iterations of the original random dynamical system we demonstrate that the derivative reconstructed for different parameters can be used as an early warning signal to detect an upcoming bifurcation and apply the algorithm to the bifurcation analysis of the stochastic return map of the koper model which is a three dimensional multiple time scale ordinary differential equation used as prototypical model for the formation of mixed mode oscillation patterns we apply our algorithm to data generated by this map to detect an upcoming transition

inviscid and viscous stationary waves of gas flow through contracting expanding nozzles abstract in this work we study steady states of one dimensional viscous isentropic compressible flows through a contracting expanding nozzle treating the viscosity coefficient as a singular parameter the steady state problem can be viewed as a singularly perturbed system for a contracting expanding nozzle a complete classification of steady states is given and the existence of viscous profiles is established via the geometric singular perturbation theory particularly interesting is the existence of a maximal sub to super transonic wave and its role in the formation of other complicated transonic waves consisting of a sub to super portion

periodic orbits and chaos in fast slow systems with bogdanov takens type fold points abstract the existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast slow type having bogdanov takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow up method in particular the blow up method is effectively used for analyzing the flow near the bogdanov takens type fold point in order to show that a slow manifold near the fold point is extended along the boutroux s tritronquee solution of the first painleve equation in the blow up space

on functional dependencies in q horn theories this paper studies functional dependencies in q horn theories and discusses their use in knowledge condensation we introduce compact model based representations of q horn theories analyze the structure of functional dependencies in q horn theories and show that every minimal functional dependency in a q horn theory can be expressed either by a single term or by a single clause we also prove that the set of all minimal functional dependencies in is quasi acyclic we then develop polynomial time algorithms for recognizing whether a given functional dependency holds in a q horn theory which is represented either by a q horn cnf or as the q horn envelope of a set of models finally we show that every q horn theory has a unique condensation and can be condensed in polynomial time

predictions of critical transitions with non stationary reduced order models abstract here we demonstrate the ability of stochastic reduced order models to predict the statistics of non stationary systems undergoing critical transitions first we show that the reduced order models are able to accurately predict the autocorrelation function and probability density functions pdf of higher dimensional systems with time dependent slow forcing of either the resolved or unresolved modes second we demonstrate that whether the system tips early or repeatedly jumps between the two equilibrium points flickering depends on the strength of the coupling between the resolved and unresolved modes and the time scale separation both kinds of behaviour have been found to precede critical transitions in earlier studies furthermore we demonstrate that the reduced order models are also able to predict the timing of critical transitions the skill of various proposed tipping indicators are discussed

local analysis near a folded saddle node singularity abstract folded saddle nodes occur generically in one parameter families of singularly perturbed systems with two slow variables we show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems canards and delayed hopf bifurcations we combine techniques from geometric singular perturbation theory the blow up technique and from delayed hopf bifurcation theory complex time path analysis to analyze the flow near such folded saddle nodes in particular we show the existence of canards as intersections of stable and unstable slow manifolds to derive these canard results we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths this enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost furthermore we define a way in way out function describing the maximal delay expected for generic solutions passing through a folded saddle node singularity branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical slow manifold make our analysis significantly different from the classical delayed hopf bifurcation analysis where these eigenvalues are complex only

critical slowing down in one dimensional maps and beyond this is a brief review on critical slowing down near the feigenbaum period doubling bifurcation points and its consequences the slowing down of numerical convergence leads to an operational fractal dimension d 2 3 at a finite order bifurcation point there is a cross over to d 0 0 538 when the order goes to infinity i e to the feigenbaum accumulation point the problem of whether there exists a super scaling for the dimension spectrum d q w that does not depend on the primitive word w underlying the period n tupling sequence seems to remain open

an equivalence between relational database dependencies and a fragment of propositional logic it is known that there is an eqmvalence between functional dependencies m a relatmonal database and a certain fragment of proposmonal logic thins eqmvalence is extended to include both functional and multivalued dependencmes thus for each dependency there is a corresponding statement m proposmonal logic it ms then shown that a dependency funcuonal or multivalued is a consequence of a set of dependencies ff and only ff the corresponding proposiuonal statement s a consequence of the corresponding set of proposmonal statements examples are given to show that these techniques are valuable mn provmdmg much shorter proofs of theorems about dependencies than have been obtained by more tradmonal means it is shown that this eqmvalence cannot be extended to include either join dependencies or embedded multmvalued dependencies

on propositional definability in standard propositional logic logical definability is the ability to derive the truth value of some propositional symbols given a propositional formula and the truth values of some propositional symbols although appearing more or less informally in various ai settings a computation oriented investigation of the notion is still lacking and this paper aims at filling the gap after recalling the two definitions of definability which are equivalent in standard propositional logic while based on different intuitions and defining a number of related notions we give several characterization results and many complexity results for definability we also show close connections with hypothesis discriminability and with reasoning about action and change

probability densities for noisy delay bifurcations the delay of a transition in a nonlinear system due to a slowly varying control parameter can be significantly reduced by very small noise a new asymptotic approximation for the time dependent probability density function gives a complete description of the process into the transition region and is easily interpreted in terms of the noisy dynamics it is also used to calculate mean transition times the method is applied to two nonlinear systems with noise a one dimensional canonical model for a steady bifurcation and the noisy fitzhugh nagumo model

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

relaxation oscillations in r3 abstract the existence of periodic relaxation oscillations in singularly perturbed systems with two slow and one fast variable is analyzed geometrically it is shown that near a singular periodic orbit a return map can be defined which has a one dimensional slow manifold with a stable invariant foliation under a natural hyperbolicity assumption on the singular periodic orbit this allows to prove the existence of a periodic relaxation orbit for small values of the perturbation parameter additionally the existence of an invariant torus is proved for the periodically forced van der pol oscillator the analysis is based on methods from geometric singular perturbation theory the blow up method is used to analyze the dynamics near the fold curves

topological equivalence of a plane vector field with its principal part defined through newton polyhedra abstract we introduce a blowing up of singularities of vector fields associated with newton polyhedra in the space of the exponents by means of which we prove a generalization of the hartman grobman theorem namely a plane vector field possessing characteristic orbits is locally topologically equivalent with its principal part under suitable non degeneracy hypotheses under stronger hypotheses a similar equivalence result is proven between a plane vector field and a single quasihomogeneous component of its principal part the case of second order equations is also studied

canonical dichotomous direct bases closure systems are usually characterized in terms of implications the directness property of implicational systems is a key issue in their computational usability in this work we focus on this property studying its connection with the structure of implicational systems and the design of methods for transforming any implicational system into an equivalent direct implicational system we introduce a new paradigm based on the bipartition of the implicational sets into two components according to their behavior wrt the closure in addition we present the notions of two new direct bases named dd basis and canonical dd basis also providing two methods to compute each of them the advantages of the dichotomous approach will be shown both from the theoretical and empirical points of view

a stochastic hopf bifurcation let x t t 0 be the solution of a stochastic differential equation sde in d which fixes 0 and let denote the lyapunov exponent for the linear sde obtained by linearizing the original sde at 0 it is known that under appropriate conditions the sign of controls the stability instability of 0 and the transience recurrence of x t t 0 on d 0 in particular if the coefficients in the sde depend on some parameterz which is varied in such a way that the corresponding lyapunov exponent z changes sign from negative to positive the almost surely stable fixed point at 0 is replaced by an almost surely unstable fixed point at 0 together with an attracting invariant probability measure z on d 0 in this paper we investigate the limiting behavior of z as z converges to 0 from above the main result is that the rescaled measures 1 z z converge in an appropriate weak sense to a non trivial finite measure on d 0

statistical early warning indicators based on autoregressive moving average models we address the problem of defining early warning indicators of critical transitions to this purpose we fit the relevant time series through a class of linear models known as autoregressive moving average arma p q models we define two indicators representing the total order and the total persistence of the process linked respectively to the shape and to the characteristic decay time of the autocorrelation function of the process we successfully test the method to detect transitions in a langevin model and a 2d ising model with nearest neighbor interaction we then apply the method to complex systems namely for dynamo thresholds and financial crisis detection

finite subgroups of su 3 we present a new class of finite subgroups of su 3 of the form zm s zn semidirect product we also apply the methods used to investigate semidirect products to the known su 3 subgroups 3n2 and 6n2 and give analytic formulae for representations characters and clebsch gordan coefficients

support set selection for abductive and default reasoning abstract of all the possible ways of computing abductive explanations the atms procedure is one of the most popular while this procedure is known to run in exponential time in the worst case the proof actually depends on the existence of queries with an exponential number of answers but how much of the difficulty stems from having to return these large sets of explanations here we explore abduction tasks similar to that of the atms but which return relatively small answers the main result is that although it is possible to generate some nontrivial explanations quickly deciding if there is an explanation containing a given hypothesis is np complete as is the task of generating even one explanation expressed in terms of a given set of assumption letters thus the method of simply listing all explanations as employed by the atms probably cannot be improved upon an interesting result of our analysis is the discovery of a subtask we call the support set selection task that is not only at the core of generating explanations but is also at the core of generating extensions in reiter s default logic moreover it is this subtask that accounts for the computational difficulty of both forms of reasoning this establishes for the first time a strong connection between computing abductive explanations and computing extensions in default logic

structure identification in relational data this paper presents several investigations into the prospects for identifying meaningful structures in empirical data namely structures permitting effective organization of the data to meet requirements of future queries we propose a general framework whereby the notion of identifiability is given a precise formal definition similar to that of learnability using this framework we then explore if a tractable procedure exists for deciding whether a given relation is decomposable into a constraint network or a cnf theory with desirable topology and if the answer is positive identifying the desired decomposition finally we address the problem of expressing a given relation as a horn theory and if this is impossible finding the best k horn approximation to the given relation we show that both problems can be solved in time polynomial in the length of the data

representations and clebsch gordan coefficients of z metacyclic groups we investigate finite groups of the form zmszn and give all irreducible representations and clebsch gordan coefficients in analytic form two subclasses are considered which seem to be inportant for applications the m metacyclic groups which are important for spin systems and the k metacyclic groups which are the smallest finite groups that have an irreducible representation of dimension p 1 where p is prime

singular hopf bifurcations and mixed mode oscillations in a two cell inhibitory neural network abstract recent studies of a firing rate model for neural competition as observed in binocular rivalry and central pattern generators r curtu a shpiro n rubin j rinzel mechanisms for frequency control in neuronal competition models siam j appl dyn syst 7 2 2008 609 649 showed that the variation of the stimulus strength parameter can lead to rich and interesting dynamics several types of behavior were identified such as fusion equivalent to a steady state of identical activity levels for both neural units oscillations due to either an escape or a release mechanism and a winner take all state of bistability the model consists of two neural populations interacting through reciprocal inhibition each endowed with a slow negative feedback process in the form of spike frequency adaptation in this paper we report the occurrence of another complex oscillatory pattern the mixed mode oscillations mmos they exist in the model at the transition between the relaxation oscillator dynamical regime and the winner take all regime the system distinguishes itself from other neuronal models where mmos were found by the following interesting feature there is no autocatalysis involved as in the examples of voltage gated persistent inward currents and or intrapopulation recurrent excitation and therefore the two cells in the network are not intrinsic oscillators the oscillations are instead a combined result of the mutual inhibition and the adaptation we prove that the mmos are due to a singular hopf bifurcation point situated in close distance to the transition point to the winner take all case we also show that in the vicinity of the singular hopf other types of bifurcations exist and we construct numerically the corresponding diagrams

the role of slow system dynamics in predicting the degeneracy of slow invariant manifolds the case of vdp relaxation oscillations abstract the development of dissipative time scales which are much faster than the rest force the solution of a system of 1st order odes to land and then evolve on a slow invariant manifold the slow evolution on this manifold can be approximated by a reduced system which is constructed with asymptotic analysis and is free of the fast scales in this work the dynamics of the reduced system are compared to the fast and slow dynamics of the original system the comparison is based on the van der pol system which is typical of those exhibiting relaxation oscillation behavior and is characterized by successive dissipative and explosive phases special attention is given to the dissipative phase during which the solution evolves on a slow invariant manifold and the fast slow time scale gap tends to contract it is shown that one cannot predict from the dynamics of the van der pol system the approaching explosive phase and the associated degeneracy of the manifold such a prediction is possible only through the dynamics of the reduced system which involves an explosive component that is absent in the original system this explosive component accelerates as higher order effects are incorporated in the reduced system resulting in a smaller gap between the fast time scale of the original system and the fast time scale of the reduced one this feature sets a limit on the accuracy singular perturbation techniques can deliver even in the presence of a reasonably large fast slow time scale gap in the dynamics of the original system given a fast slow system predicting forthcoming changes in the dimensions of the manifold is very important when it is desired to identify the underlying major physical mechanisms or to control the process

horn approximations of empirical data abstract formal ai systems traditionally represent knowledge using logical formulas sometimes however a model based representation is more compact and enables faster reasoning than the corresponding formula based representation the central idea behind our work is to represent a large set of models by a subset of characteristic models more specifically we examine model based representations of horn theories and show that there are large horn theories that can be exactly represented by an exponentially smaller set of characteristic models we show that deduction based on a set of characteristic models requires only polynomial time as it does using horn theories more surprisingly abduction can be performed in polynomial time using a set of characteristic models whereas abduction using horn theories is np complete finally we discuss algorithms for generating efficient representations of the horn theory that best approximates a general set of models

a methodology for analysis of concept lattice reduction independent methodology for analysis of concept lattice reduction we use sets of proper implications holding in the original and reduced structure highlight the kinds of changes propitiated by the different classes of techniques four reduction techniques were used to illustrate the proposed methodology formal concept analysis fca is a mathematical theory of data analysis with applications in many areas the problem of obtaining a concept lattice of an appropriate size was identified in several applications as one of the most important problems of fca in order to deal with this problem several techniques with different characteristics were proposed for concept lattice reduction however there are currently no adequate methods to assess what types of knowledge transformations can result from a reduction a methodology for analysis of concept lattice reduction is presented here it is based on the use of sets of proper implications holding in the original and reduced formal contexts or concept lattices working with both sets of implications the methodology is able to show what is preserved eliminated inserted or transformed by a reduction technique three classes of reduction techniques are analyzed from the standpoint of the methodology in order to highlight techniques of each class have in common with respect to the transformations performed such analysis is followed by specific examples in each class

mixed mode oscillations with multiple time scales mixed mode oscillations mmos are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes mmos have been observed and studied for over thirty years in chemical physical and biological systems few attempts have been made thus far to classify different patterns of mmos in contrast to the classification of the related phenomena of bursting oscillations this paper gives a survey of different types of mmos concentrating its analysis on mmos whose small amplitude oscillations are produced by a local multiple time scale mechanism recent work gives substantially improved insight into the mathematical properties of these mechanisms in this survey we unify diverse observations about mmos and establish a systematic framework for studying their properties numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper

discovery of the d basis in binary tables based on hypergraph dualization discovery of strong association rules or implications is an important task in data management and it finds application in artificial intelligence data mining and the semantic web we introduce a novel approach for the discovery of a specific set of implications called the d basis that provides a representation for a reduced binary table based on the structure of its galois lattice at the core of the method are the d relation defined in a lattice theoretic framework and the hypergraph dualization algorithm that allows us to effectively produce the set of transversals for a given sperner hypergraph the latter algorithm first developed by specialists from rutgers center for operations research has already found numerous applications in solving optimization problems in data base theory artificial intelligence and game theory one application of the method is for analysis of gene expression data related to a particular phenotypic variable and some initial testing is done for data provided by the university of hawaii cancer center

on the structure of armstrong relations for functional dependencies an armstrong relation for a set of functional dependencies fds is a relation that satisfies each fd implied by the set but no fd that is not implied by it the structure and size number of tuples of armstrong relatsons are investigated upper and lower bounds on the size of minimal sized armstrong relations are derived and upper and lower bounds on the number of distinct entries that must appear m an armstrong relation are given it is shown that the time complexity of finding an armstrong relation gwen a set of functional dependencies is precisely exponential in the number of attributes also shown s the falsity of a natural conjecture which says that almost all relations obeying a given set of fds are armstrong relations for that set of fds finally armstrong relations are used to generahze a result obtained by demetrovics using quite complicated methods about the possible sets of keys for a relauon

geometric singular perturbation theory for stochastic differential equations abstract we consider slow fast systems of differential equations in which both the slow and fast variables are perturbed by noise when the deterministic system admits a uniformly asymptotically stable slow manifold we show that the sample paths of the stochastic system are concentrated in a neighbourhood of the slow manifold which we construct explicitly depending on the dynamics of the reduced system the results cover time spans which can be exponentially long in the noise intensity squared that is up to kramers time we obtain exponentially small upper and lower bounds on the probability of exceptional paths if the slow manifold contains bifurcation points we show similar concentration properties for the fast variables corresponding to non bifurcating modes we also give conditions under which the system can be approximated by a lower dimensional one in which the fast variables contain only bifurcating modes

directed hypergraphs just as ordinary hypergraphs are a generalization of graphs directed hypergraphs dh are a natural generalization of digraphs a dh consists of a set of vertices v and a set of hyperarcs h where a hyperarc is a pair s v s non empty subset of v and v v dhs have a variety of applications they have been used to represent functional dependency in databases horn formulae in propositional logic and or graphs context free grammars etc in the paper after providing a brief historical introduction on the notion of dh and some relevant applications various problems regarding dhs are surveyed and analyzed in particular we consider the complexity of the reachability problem together with its application in the related satisfiability problem for horn cnf formulae and the computation of transitive closure and transitive reduction of directed hypergraphs together with its application to the computation of minimum coverings for a set of functional dependencies finally a short introduction to the problem of computing shortest hyperpaths in directed hypergraphs is provided

some comments on finite subgroups of su 3 a recently published set of finite subgroups of su 3 is shown to contain some groups which are not subgroups of su 3 the others are subgroups of one of the dihedral like family of su 3 subgroups of order 3n2 some comments are made also on the structure of other finite subgroups previously listed

continuation of probability density functions using a generalized lyapunov approach techniques from numerical bifurcation theory are very useful to study transitions between steady fluid flow patterns and the instabilities involved here we provide computational methodology to use parameter continuation in determining probability density functions of systems of stochastic partial differential equations near fixed points under a small noise approximation key innovation is the efficient solution of a generalized lyapunov equation using an iterative method involving low rank approximations we apply and illustrate the capabilities of the method using a problem in physical oceanography i e the occurrence of multiple steady states of the atlantic ocean circulation