asymptotic formation and orbital stability of phase locked states for the kuramoto model abstract we discuss the asymptotic formation and nonlinear orbital stability of phase locked states arising from the ensemble of non identical kuramoto oscillators we provide an explicit lower bound for a coupling strength on the formation of phase locked states which only depends on the diameters of natural frequencies and initial phase configurations we show that when the phases of non identical oscillators are distributed over the half circle and the coupling strength is sufficiently large the dynamics of kuramoto oscillators exhibits two stages transition and relaxation stages in a transition stage initial configurations shrink to configurations whose diameters are strictly less than 2 in a finite time and then the configurations tend to phase locked states asymptotically this improves previous results on the formation of phase locked states by chopra spong 2009 26 and ha ha kim 2010 27 where their attention were focused only on the latter relaxation stage we also show that the kuramoto model is l 1 contractive in the sense that the l 1 distance along two smooth kuramoto flows is less than or equal to that of initial configurations in particular when two initial configurations have the same averaged phases the l 1 distance between them decays to zero exponentially fast for the configurations with different phase averages we use the method of average adjustment and translation invariant of the kuramoto model to show that one solution converges to the translation of the other solution exponentially fast this establishes the orbital stability of the phase locked states our stability analysis does not employ any standard linearization technique around the given phase locked states but instead we use a robust l 1 metric functional as a lyapunov functional in the formation process of phase locked states we estimate the number of collisions between oscillators and lower upper bounds of the transversal phase differences

probabilistic analysis of two heuristics for the 3 satisfiability problem an algorithm for the 3 satisfiability problem is presented and a probabilistic analysis is performed the analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics the algorithm assigns values to variables appearing in a given instance of 3 satisfiability one at a time using the unit clause heuristic and a maximum occurring literal selection heuristic at each step a variable is chosen randomly from a subset of variables which is usually large the algorithm runs in polynomial time and it is shown that the algorithm finds a solution to a random instance of 3 satisfiability with probability bounded from below by a constant greater than zero for a range of parameter values the heuristics studied here can be used to select variables in a backtrack algorithm for 3 satisfiability experiments have shown that for about the same range of parameters as above the backtrack algorithm using the heuristics finds a solution in polynomial average time

probabilistic analysis of a generalization of the unit clause literal selection heuristics for the k satisfiability problem abstract two algorithms for the k satisfiability problem are presented and a probabilistic analysis is performed the analysis is based on an instance distribution which is parametrized to simulate a variety of sample characteristics the algorithms assign values to literals appearing in a given instance of k satisfiability one at a time until a solution is found or it is discovered that further assignments cannot lead to finding a solution one algorithm chooses the next literal from a unit clause if one exists and randomly from the set of remaining literals otherwise the other algorithm uses a generalization of the unit clause rule as a heuristic for selecting the next literal at each step a literal is chosen randomly from a clause containing the least number of literals the algorithms run in polynomial time and it is shown that they find a solution to a random instance of k satisfiability with probability bounded below by a constant greater than zero for two different ranges of parameter values it is also shown that the second algorithm mentioned finds a solution with probability approaching one for a wide range of parameter values

on compactness in bifuzzy topological spaces abstract compactness in a bifuzzy topological space has been studied earlier by abd el monsef and ramadan fuzzy sets and systems 30 1989 165 174 and safiya et al fuzzy sets and systems 62 1994 89 96 as generalizations of p compactness or s compactness in a bitopological space in section 2 of this note we have obtained some interesting results related to pc compactness introduced by safiya et al fuzzy sets and systems 62 1994 89 96 as a concept parallelling that of p compactness of bitopological spaces further in section 3 we have introduced and studied b compactness b compactness in a bifuzzy topological space which are parallel to the notion of b compactness in a bitopological space we have also defied b almost b almost compactness in a bfts which is weaker than b compactness b compactness several related results have been proved

combinatorial sharpness criterion and phase transition classification for random csps we investigate the nature of the phase transition sharp or coarse for random constraint satisfaction problems we first give a sharp threshold criterion specified for csps which is derived from friedgut bourgain s one thus we get a complete and precise classification of the nature of the threshold for symmetric boolean csps in particular we show that it is governed by two local properties strongly related to the problems 1 sat and 2 xor sat

an algorithm for random signed 3 sat with intervals interval k sat k isat is a generalization of classical k sat where the variables can take values in 0 1 instead of 0 1 and the literals are of the form x i for intervals i 0 1 it falls within the class of signed satisfiability problems we propose an algorithm for 3 isat and analyze it on uniformly random formulas the algorithm follows the unit clause paradigm enhanced by a very limited backtracking option using wormald s ode method we prove that if m n 2 3 with high probability our algorithm succeeds in finding an assignment of values to the variables satisfying the formula

a better algorithm for random k sat let boldsymbol phi be a uniformly distributed random k sat formula with n variables and m clauses we present a polynomial time algorithm that finds a satisfying assignment of boldsymbol phi with high probability for constraint densities m n 1 varepsilon k 2 k ln k k where varepsilon k rightarrow0 previously no efficient algorithm was known to find satisfying assignments with a nonvanishing probability beyond m n 1 817 cdot2 k k a frieze and s suen j algorithms 20 1996 pp 312 355

synchronization and transient stability in power networks and nonuniform kuramoto oscillators motivated by recent interest for multiagent systems and smart grid architectures we discuss the synchronization problem for the network reduced model of a power system with nontrivial transfer conductances our key insight is to exploit the relationship between the power network model and a first order model of coupled oscillators assuming overdamped generators possibly due to local excitation controllers a singular perturbation analysis shows the equivalence between the classic swing equations and a nonuniform kuramoto model here nonuniform kuramoto oscillators are characterized by multiple time constants nonhomogeneous coupling and nonuniform phase shifts extending methods from transient stability synchronization theory and consensus protocols we establish sufficient conditions for synchronization of nonuniform kuramoto oscillators these conditions reduce to necessary and sufficient tests for the standard kuramoto model combining our singular perturbation and kuramoto analyses we derive

complete synchronization of kuramoto oscillators with finite inertia abstract we present an approach based on gronwall s inequalities for the asymptotic complete phase frequency synchronization of kuramoto oscillators with finite inertia for given finite inertia and coupling strength we present admissible classes of initial configurations and natural frequency distributions which lead to the complete phase frequency synchronization asymptotically for this we explicitly identify invariant regions for the kuramoto flow and derive second order gronwall s inequalities for the evolution of phase and frequency diameters our detailed time decay estimates for phase and frequency diameters are independent of the number of oscillators we also compare our analytical results with numerical simulations

generalized satisfiability problems minimal elements and phase transitions we develop a probabilistic model on the generalized satisfiability problems defined by schaefer in proceedings of the 10th stoc san diego ca usa association for computing machinery new york 1978 pp 216 226 for which the arity of the constraints is fixed in order to study the associated phase transition we establish new results on minimal elements associated with such generalized satisfiability problems these results are the keys of the exploration we conduct on the location and on the nature of the phase transition for generalized satisfiability we first prove that the phase transition occurs at the same scale for every reasonable problem and we provide lower and upper bounds for the associated critical ratio our framework allows one to get these bounds in a uniform way in particular we obtain a lower bound proportional to the number of variables for k sat without analyzing any algorithm finally we reveal the seed of coarseness for the phase transition of generalized satisfiability 2 xor sat

a compactness in bifuzzy topological spaces abstract compact fuzzy topological spaces were first introduced by chang 3 compact fuzzy topological spaces were introduced by gantner et al 4 abd el monsef et al 1 introduced the notion of almost compact fuzzy topological spaces mashbour et al 8 introduced the notion of bifuzzy topological spaces and defined the notion of compactness in those spaces in this paper we obtain some new results about compact bifuzzy topological spaces we introduce and study the notion of pairwise almost compact bifuzzy topological spaces

on the synchronization theory of kuramoto oscillators under the effect of inertia abstract we investigate the synchronized collective behavior of the kuramoto oscillators with inertia effect both the frequency synchronization for nonidentical case and the phase synchronization for identical case are in view as an application of our general theory we show the global frequency synchronization for the three nonidentical oscillator case

models for random constraint satisfaction problems we introduce a class of models for random constraint satisfaction problems this class includes and generalizes many previously studied models we characterize those models from our class which are asymptotically interesting in the sense that the limiting probability of satisfiability changes significantly as the number of constraints increases we also discuss models which exhibit a sharp threshold for satisfiability in the sense that the limiting probability jumps from 0 to 1 suddenly as the number of constraints increases

compactness and weakly induced bifuzzy topological spaces abstract induced and weakly induced fuzzy topological spaces were introduced by martin weakly induced fuzzy topological spaces j math anal appl 78 1980 634 639 compactness in bifuzzy topological spaces were introduced by mashhour et al bifuzzy topological spaces submitted in this paper we introduce other types of compactness induced and weakly induced bifuzzy topological spaces and use an alteration of lowen s good extension criterion to test those definitions

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

lower bounds for random 3 sat via differential equations it is widely believed that the probability of satisfiability for random k sat formulae exhibits a sharp threshold as a function of their clauses to variables ratio for the most studied case k 3 there have been a number of results during the last decade providing upper and lower bounds for the threshold s potential location all lower bounds in this vein have been algorithmic i e in each case a particular algorithm was shown to satisfy random instances of 3 sat with probability 1 o 1 if the clauses to variables ratio is below a certain value we show how differential equations can serve as a generic tool for analyzing such algorithms by rederiving most of the known lower bounds for random 3 sat in a simple uniform manner

on the complete synchronization of the kuramoto phase model abstract we discuss the asymptotic complete phase frequency synchronization for the kuramoto phase model with a finite size n we present sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states our new sufficient conditions and decay rate depend only on the coupling strength and the diameter of initial phase and natural frequency configurations but they are independent of the system size n hence they can be used for the mean field limit for the complete synchronization estimates we estimate the time evolution of the phase and frequency diameters for configurations the initial phase configurations for identical oscillators located on the half circle will converge to the complete synchronized states exponentially fast in contrast for the non identical oscillators the complete frequency synchronization will occur exponentially fast for some restricted class of initial phase configurations our estimates are based on the monotonicity arguments of extremal phase and frequencies which do not employ any linearization procedure of nonlinear coupling terms and detailed information on the eigenvalue of the linearized system around the complete synchronized states we compare our analytical results with numerical simulations

partial entrainment in the finite kuramoto sakaguchi model abstract although modifications of the kuramoto model have been the subject of extensive research the model itself is not yet fully understood we offer several results and observations some analytic others through simulations we derive a sufficient condition for the existence of a solution exhibiting partial entrainment with respect to a given subset of oscillators the result also implies persistence of the entrainment behavior under perturbations the critical values of the coupling strength defining the transitions between different forms of partial entrainment are predicted by an analytical approximation based on the fact that oscillators with large differences in their natural frequencies have little influence on each other s entrainment behavior the predictions agree with the actual values obtained by simulations we indicate by simulations that entrainment can disappear with increasing coupling strength and that in arrays of josephson junctions a similar phenomenon can be observed where it is also possible that a junction leaving one entrained subset joins another entrained subset

complete synchronization of kuramoto oscillators kuramoto oscillator networks represent a standard model of synchronization analysis extensive studies have shown that static interactions can only bring about a partial synchronization phase locking this paper investigates the complete synchronization where the oscillator outputs follow exactly a common trajectory it is shown that oscillators with individual natural frequencies can be synchronized by dynamic couplings with integrator dynamics a necessary and sufficient synchronization condition is derived which shows that oscillators with positive parameters are completely synchronized for arbitrary coupling strength

on exponential synchronization of kuramoto oscillators in this technical note we study the problem of exponential synchronization for one of the most popular models of coupled phase oscillators the kuramoto model we consider the special case of finite oscillators with distinct bounded natural frequencies our first result derives a lower bound on the coupling gain which is necessary for the onset of synchronization this bound improves the one derived by jadbabaie we then calculate a lower bound on the coupling gain that is sufficient to guarantee oscillator synchronization and derive further sufficient conditions to ensure exponential synchronization of the angular frequencies of all oscillators to the mean natural frequency of the group we also characterize the coupling gain that is sufficient for the oscillator phase differences to approach any desired compact set in finite time

inertia effects on periodic synchronization in a system of coupled oscillators we study analytically the synchronization phenomena in a set of globally coupled oscillators under external periodic driving with emphasis on the effects of small inertia we examine in detail both the integer and fractional mode locking present in the system with inertia and derive the self consistency equation for the order parameter which reveals variation of the magnitude of the order parameter according to the external periodic driving in particular it is found that the inertia induces discontinuous transitions between the coherent and incoherent states