gauge theories labelled by three manifolds we propose a dictionary between geometry of triangulated 3 manifolds and physics of three dimensional n 2 gauge theories under this duality standard operations on triangulated 3 manifolds and various invariants thereof classical as well as quantum find a natural interpretation in field theory for example independence of the sl 2 chern simons partition function on the choice of triangulation translates to a statement that s 3 b partition functions of two mirror 3d n 2 gauge theories are equal three dimensional n 2 field theories associated to 3 manifolds can be thought of as theories that describe boundary conditions and duality walls in four dimensional n 2 scfts thus making the whole construction functorial with respect to cobordisms and gluing

large n duality lagrangian cycles and algebraic knots we consider knot invariants in the context of large n transitions of topological strings in particular we consider aspects of lagrangian cycles associated to knots in the conifold geometry we show how these can be explicitly constructed in the case of algebraic knots we use this explicit construction to explain a recent conjecture relating study of stable pairs on algebraic curves with homfly polynomials furthermore for torus knots using the explicit construction of the lagrangian cycle we also give a direct a model computation and recover the homfly polynomial for this case

stability of a complex dynamical network model abstract in this paper firstly we propose a complex dynamical network model which is always uniformly asymptotically stable about its equilibrium then we give theoretical analysis of its uniform asymptotical stability by using lyapunov stability theory finally we show a numerical example to verify our theoretical result

three dimensional quantum gravity chern simons theory and the a polynomial we study three dimensional chern simons theory with complex gauge group sl 2 which has many interesting connections with three dimensional quantum gravity and geometry of hyperbolic 3 manifolds we show that in the presence of a single knotted wilson loop in an infinite dimensional representation of the gauge group the classical and quantum properties of such theory are described by an algebraic curve called the a polynomial of a knot using this approach we find some new and rather surprising relations between the a polynomial the colored jones polynomial and other invariants of hyperbolic 3 manifolds these relations generalize the volume conjecture and the melvin morton rozansky conjecture and suggest an intriguing connection between the sl 2 partition function and the colored jones polynomial

synchronization of complex dynamical networks with hybrid coupling delays on time scales by handling multitude kronecker product terms the problem of synchronization is studied for complex dynamical networks with hybrid coupling delays on time scales the hybrid coupling delays consist of both discrete and distributed coupling delays some novel and useful synchronization criteria of complex dynamical networks are derived based on stability theory of error dynamical system by employing the standard lyapunov krasovskii functional matrix expansion method to handle multitude kronecker product terms and the modified jensen s inequalities on time scale new sufficient conditions guaranteeing the global exponential stability of the origin of complex dynamical networks are established interms of linear matrix inequality lmi numerical examples are included to demonstrate the effectiveness of the proposed method

polynomial invariants for torus knots and topological strings we make a precision test of a recently proposed conjecture relating chern simons gauge theory to topological string theory on the resolution of the conifold first we develop a systematic procedure to extract string amplitudes from vacuum expectation values vevs of wilson loops in chern simons gauge theory and then we evaluate these vevs in arbitrary irreducible representations of su n for torus knots we find complete agreement with the predictions derived from the target space interpretation of the string amplitudes we also show that the structure of the free energy of topological open string theory gives further constraints on the chern simons vevs our work provides strong evidence towards an interpretation of knot polynomial invariants as generating functions associated to enumerative problems

combinatorial knot contact homology and transverse knots abstract we give a combinatorial treatment of transverse homology a new invariant of transverse knots that is an extension of knot contact homology the theory comes in several flavors including one that is an invariant of topological knots and produces a three variable knot polynomial related to the a polynomial we provide a number of computations of transverse homology that demonstrate its effectiveness in distinguishing transverse knots including knots that cannot be distinguished by the heegaard floer transverse invariants or other previous invariants

exponential synchronization of complex networks of linear systems and nonlinear oscillators a unified analysis a unified approach to the analysis of synchronization for complex dynamical networks i e networks of partial state coupled linear systems and networks of full state coupled nonlinear oscillators is introduced it is shown that the developed analysis can be used to describe the difference between the state of each node and the weighted sum of the states of those nodes playing the role of leaders in the networks thus making it feasible to consider the error dynamics for the whole network system different from the other various methods given in the existing literature the analysis employed in this paper is demonstrated successfully in not only providing the consistent convergence analysis with much simpler form but also explicitly specifying the convergence rate

synchronization of singular complex dynamical networks with time varying delays abstract this paper considers delay dependent synchronizations of singular complex dynamical networks with time varying delays a modified lyapunov krasovskii functional is used to derive a sufficient condition for synchronization in terms of lmis linear matrix inequalities which can be easily solved by various convex optimization algorithms numerical examples show the effectiveness of the proposed method

synchronization of a complex dynamical network with coupling time varying delays via sampled data control abstract in this paper the synchronization problem of a complex dynamical network with coupling time varying delays via delayed sampled data controller is investigated in order to make full use of the sawtooth structure characteristic of the sampling input delay a discontinuous lyapunov functional is proposed based on the extended wirtinger inequality from a convex representation of the sector restricted nonlinearity in system dynamics the stability condition based on lyapunov stability theory is obtained by utilization of linear matrix inequality formulation to find the controller which achieves the synchronization of a complex dynamical network with coupling time varying delay finally two numerical examples are given to illustrate the effectiveness of the proposed methods

plane curves associated to character varieties of 3 manifolds consider a compact 3 manifold m with boundary consisting of a single torus the papers cs1 cs2 cgls discuss the variety of characters of sl2 c representations of zl m and some of the ways in which the topological structure of m is reflected in the algebraic geometry of the character variety we will describe in this paper a certain affine algebraic curve dm which is naturally associated to the character variety of m a basis b 9 1 for the peripheral subgroup of m determines an embedding p8 of dm into c x c with coordinates l and m the closure in c 2 of ps dm is a plane algebraic curve and therefore is defined by a polynomial am n0 m that after certain normalizations is uniquely determined up to multiplication by constants the polynomial am b is effectively computable and is an invariant of the manifold m together with the choice of basis b the results in this paper describe how geometric properties of the character variety and hence topological properties of m are reflected by the polynomial am b in the case that m is the complement of a knot k in a homology 3 sphere we may take the basis b to consist of the longitude and meridian of k with the usual orientation conventions this basis is well defined modulo the involution which inverts both the longitude and meridian it will follow from the construction of dm that if the basis b is obtained from b by inverting both generators then the regular maps pb and pb have the same image thus ar am s am b is an invariant of the knot the polynomial am b displays in a striking way information about the incompressible surfaces in m this involves the newton polygon of am b

algebraic a hypergeometric functions we formulate and prove a combinatorial criterion to decide if an a hypergeometric system of differential equations has a full set of algebraic solutions or not this criterion generalises the so called interlacing criterion in the case of hypergeometric functions of one variable

knots bps states and algebraic curves we analyze relations between bps degeneracies related to labastida marino ooguri vafa lmov invariants and algebraic curves associated to knots we introduce a new class of such curves which we call extremal a polynomials discuss their special properties and determine exact and asymptotic formulas for the corresponding extremal bps degeneracies these formulas lead to nontrivial integrality statements in number theory as well as to an improved integrality conjecture which is stronger than the known m theory integrality predictions furthermore we determine the bps degeneracies encoded in augmentation polynomials and show their consistency with known colored homfly polynomials finally we consider refined bps degeneracies for knots determine them from the knowledge of super a polynomials and verify their integrality we illustrate our results with twist knots torus knots and various other knots with up to 10 crossings

synchronization for complex dynamical networks with time delay and discrete time information the synchronization problem is studied for complex dynamical networks cdns with time delay based on discrete time communication pattern a new lyapunov functional is proposed which is positive definite at communication instants but not necessarily positive definite inside the communication intervals and can make full use of the available information about the discrete time communication pattern based on the lyapunov functional an exponential synchronization criterion is proposed to ensure the synchronization of the considered cdns the effectiveness of the developed results are shown by the numerical simulation

monodromy for the hypergeometric function nfn 1 la fonction hypergeometrique le groupe hypergeometrique la forme hermitienne invariante le cas imprimitif theorie de galois differentielle fonctions hypergeometriques algebriques

local exponential synchronization in complex dynamical networks with time varying delay and hybrid coupling in this paper the local exponential synchronization of complex dynamical networks with time varying delay and hybrid coupling is investigated we assume that the coupling configuration matrices are asymmetric and the inner coupling matrices are not restricted to be diagonal or positive definite through constructing the appropriate lyapunov krasovskii functional and applying the linear matrix inequality lmi technique several novel delay depend sufficient conditions and the decay rate are deduced for the local exponential synchronization moreover the derivative of the time varying delay is extended to any value finally the numerical simulations are given to show the effectiveness of the theoretical results

synchronization of discrete dynamical networks with non delayed and delayed coupling in this paper synchronization of discrete dynamical networks with both non delayed and delayed couplings are studied firstly a class of complex networks with invariable topology structure are considered secondly a class of complex networks with variable topology structure are discussed based on lyapunov function method and linear matrix inequalities the sufficient conditions for asymptotically local synchronization of the above two classes of networks are presented respectively finally numerical examples are provided to verify our results

knot invariants from topological recursion on augmentation varieties using the duality between wilson loop expectation values of su n chern simons theory on s 3 and topological open string amplitudes on the local mirror of the resolved conifold we study knots on s 3 and their invariants encoded in colored homfly polynomials by means of topological recursion in the context of the local mirror calabi yau threefold of the resolved conifold we generalize the topological recursion of the remodelled b model in order to study branes beyond the class of toric harvey lawson special lagrangians as required for analyzing non trivial knots on s 3 the basic ingredients for the proposed recursion are the spectral curve given by the augmentation variety of the knot and the calibrated annulus kernel encoding the topological annulus amplitudes associated to the knot we present an explicit construction of the calibrated annulus kernel for torus knots and demonstrate the validity of the topological recursion we further argue that if an explicit form of the calibrated annulus kernel is provided for any other knot the proposed topological recursion should still be applicable we study the implications of our proposal for knot theory which exhibit interesting consequences for colored homfly polynomials of mutant knots

traffic of molecular motors through tube like compartments the traffic of molecular motors through open tube like compartments is studied using lattice models these models exhibit boundary induced phase transitions related to those of the asymmetric simple exclusion process asep in one dimension the location of the transition lines depends on the boundary conditions at the two ends of the tubes three types of boundary conditions are studied a periodic boundary conditions which correspond to a closed torus like tube b fixed motor densities at the two tube ends where radial equilibrium holds locally and c diffusive motor injection at one end and diffusive motor extraction at the other end in addition to the phase diagrams we also determine the profiles for the bound and unbound motor densities using mean field approximations and monte carlo simulations our theoretical predictions are accessible to experiments

the refined bps index from stable pair invariants a refinement of the stable pair invariants of pandharipande and thomas for non compact calabi yau spaces is introduced based on a virtual bialynicki birula decomposition with respect to a mathbb c action on the stable pair moduli space or alternatively the equivariant index of nekrasov and okounkov this effectively calculates the refined index for m theory reduced on these calabi yau geometries based on physical expectations we propose a product formula for the refined invariants extending the motivic product formula of morrison mozgovoy nagao and szendroi for local mathbb p 1 we explicitly compute refined invariants in low degree for local mathbb p 2 and local mathbb p 1 times mathbb p 1 and check that they agree with the predictions of the direct integration of the generalized holomorphic anomaly and with the product formula the modularity of the expressions obtained in the direct integration approach allows us to relate the generating function of refined pt invariants on appropriate geometries to nekrasov s partition function and a refinement of chern simons theory on a lens space we also relate our product formula to wall crossing