performance limitations of non minimum phase systems in the servomechanism problem abstract this paper studies the cheap regulator problem and the cheap servomechanism problem for systems which may be non minimum phase the study extends some well known properties of perfect regulation and the perfect tracking and disturbance rejection of minimum phase systems to non minimum phase systems it is shown that perfect rejection to disturbances applied to the plant input can be achieved no matter whether the system is minimum phase or non minimum phase whereas a fundamental limitation exists in the achievable transient performance of tracking and rejection to disturbances applied to the plant output for a non minimum phase system and that this limitation can be simply and completely characterized by the number and locations of those zeros of the system which lie in the right half of the complex plane furthermore this limitation provides a quantitative measure of the degree of difficulty which is inherent in the control of such non minimum phase systems

a tqft from quantum teichmuller theory by using quantum teichmuller theory we construct a one parameter family of tqfts on the categroid of admissible leveled shaped 3 manifolds

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

ratio coordinates for higher teichmuller spaces we define new coordinates for fock goncharov s higher teichmuller spaces for a surface with holes which are the moduli spaces of representations of the fundamental group into a reductive lie group g some additional data on the boundary leads to two closely related moduli spaces the mathscr x space and the mathscr a space forming a cluster ensemble fock and goncharov gave nice descriptions of the coordinates of these spaces in the cases of g mathrm pgl m and g mathrm sl m together with poisson structures we consider new coordinates for higher teichmuller spaces given as ratios of the coordinates of the mathscr a space for g mathrm sl m which are generalizations of kashaev s ratio coordinates in the case m 2 using kashaev s quantization for m 2 we suggest a quantization of the system of these new ratio coordinates which may lead to a new family of projective representations of mapping class groups these ratio coordinates depend on the choice of an ideal triangulation decorated with a distinguished corner at each triangle and the key point of the quantization is to guarantee certain consistency under a change of such choices we prove this consistency for m 3 and for completeness we also give a full proof of the presentation of kashaev s groupoid of decorated ideal triangulations

robust exact differentiation via sliding mode technique abstract the main problem in differentiator design is to combine differentiation exactness with robustness in respect to possible measurement errors and input noises the proposed differentiator provides for proportionality of the maximal differentiation error to the square root of the maximal deviation of the measured input signal from the base signal such an order of the differentiation error is shown to be the best possible one when the only information known on the base signal is an upper bound for lipschitz s constant of the derivative

brief paper simplex sliding mode methods for the chattering reduction control of multi input nonlinear uncertain systems the simplex sliding mode control method is further developed by considering uncertain control systems non affine in the control law in order to reduce chattering effects a set of integrators is added in the input channels the augmented system is then controlled by a switching logic based on the simplex control method as a result the original control vector turns out to be continuous a second order sliding mode observer is used when the sliding output is not available explicit conditions are identified about systems uncertainties and the simplex geometry in order to guarantee the convergence of the proposed methodology

moduli spaces and macromolecules techniques from moduli spaces are applied to biological macromolecules the first main result provides new a priori constraints on protein geometry discovered empirically and confirmed computationally the second main result identifies up to homotopy the natural moduli space of several interacting rna molecules with the riemann moduli space of a surface with several boundary components in each fixed genus applications to rna folding prediction are discussed the mathematical and biological frameworks are surveyed and presented from first principles

tracking nonlinear non minimum phase systems using sliding control nonlinear systems affine in the input and having a well defined vector relative degree are considered the invertibility of the input output dynamics can be used to achieve tracking of smooth desired trajectories if the associated zero dynamics are stable we consider tracking in nonlinear systems with unacceptable zero dynamics i e non minimum phase systems the desired trajectories are assumed to be generated by some exosystem we use sliding control to achieve tracking independent of disturbances entering in the channels of the input the main idea is i to do an output redefinition such that the zero dynamics with respect to this new output are acceptable ii to define a modified desired trajectory for the new output to track such that in the process the original output tracks the original desired trajectory asymptotically

a discrete uniformization theorem for polyhedral surfaces a notion of discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper this discrete conformality is shown to be computable it is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a unique hyperbolic polyhedral metric with a given discrete curvature satisfying gauss bonnet formula furthermore the hyperbolic polyhedral metric with given curvature can be obtained using a discrete yamabe flow with surgery in particular each hyperbolic polyhedral metric on a closed surface with negative euler characteristic is discrete conformal to a unique hyperbolic metric

zygmund vector fields hilbert transform and fourier coefficients in shear coordinates we parametrize the space cal z of zygmund vector fields on the unit circle in terms of infinitesimal shear functions on the farey tesselation then we express the hilbert transform and the fourier coefficients of the zygmund vector fields in terms of the above parametrization by infinitesimal shear functions finally we compute the weil petersson metric on the teichm uller space of a punctured surface in terms of shears

the decorated teichmuller space of punctured surfaces a principal 5 bundle over the usual teichmuller space of ans times punctured surface is introduced the bundle is mapping class group equivariant and admits an invariant foliation several coordinatizations of the total space of the bundle are developed there is furthermore a natural cell decomposition of the bundle finally we compute the coordinate action of the mapping class group on the total space the total space is found to have a rich equivariant geometric structure we sketch some connections with arithmetic groups diophantine approximations and certain problems in plane euclidean geometry furthermore these investigations lead to an explicit scheme of integration over the moduli spaces and to the construction of a universal teichmuller space which we hope will provide a formalism for understanding some connections between the teichmuller theory the kp hierarchy and the virasoro algebra these latter applications are pursued elsewhere

super riemann surfaces uniformization and teichmuller theory teichmuller theory for super riemann surfaces is rigorously developed using the supermanifold theory of rogers in the case of trivial topology in the soul directions relevant for superstring applications the following results are proven the super teichmuller space is a complex super orbifold whose body is the ordinary teichmuller space of the associated riemann surfaces with spin structure for genusg 1 it has 3g 3 complex even and 2g 2 complex odd dimensions the super modular group which reduces super teichmuller space to super moduli space is the ordinary modular group there are no new discrete modular transformations in the odd directions the boundary of super teichmuller space contains not only super riemann surfaces with pinched bodies but rogers supermanifolds having nontrivial topology in the odd dimensions as well we also prove the uniformization theorem for super riemann surfaces and discuss their representation by discrete supergroups of fuchsian and schottky type and by beltrami differentials finally we present partial results for the more difficult problem of classifying super riemann surfaces of arbitrary topology

decorated super teichmuller space we introduce coordinates for a principal bundle s tilde t f over the super teichmuller space st f of a surface f with s geq 1 punctures that extend the lambda length coordinates on the decorated bundle tilde t f t f times mathbb r s over the usual teichmuller space t f in effect the action of a fuchsian subgroup of psl 2 mathbb r on minkowski space mathbb r 2 1 is replaced by the action of a super fuchsian subgroup of osp 1 vert 2 on the super minkowski space mathbb r 2 1 vert 2 where osp 1 vert 2 denotes the orthosymplectic lie supergroup and the lambda lengths are extended by fermionic invariants of suitable triples of isotropic vectors in mathbb r 2 1 vert 2 as in the bosonic case there is the analogue of the ptolemy transformation now on both even and odd coordinates as well as an invariant even two form on s tilde t f generalizing the weil petersson kahler form this finally solves a problem posed in yuri ivanovitch manin s moscow seminar some thirty years ago to find the super analogue of decorated teichmuller theory and provides a natural geometric interpretation in mathbb r 2 1 vert 2 for the super moduli of s tilde t f

on the cohomological dimension of the moduli space of riemann surfaces the moduli space of riemann surfaces of genus g 2 is up to a finite etale cover a complex manifold and so it makes sense to speak of its dolbeault cohomological dimension the conjecturally optimal bound is g 2 this expectation is verified in low genus and supported by harer s computation of its de rham cohomological dimension and by vanishing results in the tautological intersection ring in this paper we prove that such dimension is at most 2g 2 we also prove an analogous bound for the moduli space of riemann surfaces with marked points the key step is to show that the dolbeault cohomological dimension of each stratum of translation surfaces is at most g in order to do that we produce an exhaustion function whose complex hessian has controlled index in the construction of such a function some basic geometric properties of flat surfaces come into play

n 2 super teichmuller theory abstract based on earlier work of the latter two named authors on the higher super teichmuller space with n 1 a component of the flat o s p 1 2 connections on a punctured surface here we extend to the case n 2 of flat o s p 2 2 connections indeed we construct here coordinates on the higher super teichmuller space of a surface f with at least one puncture associated to the supergroup o s p 2 2 which in particular specializes to give another treatment for n 1 which is simpler than the earlier work the minkowski space in the current case where the corresponding super fuchsian groups act is replaced by the superspace r 2 2 4 and the familiar lambda lengths are extended by odd invariants of triples of special isotropic vectors in r 2 2 4 as well as extra bosonic parameters which we call ratios defining a flat r connection on f as in the pure bosonic or n 1 cases we derive the analogue of ptolemy transformations for all these new variables

margulis spacetimes via the arc complex we study strip deformations of convex cocompact hyperbolic surfaces defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface we prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation in an essentially unique way the infinitesimal version of this result gives a parameterization by the arc complex of the moduli space of margulis spacetimes with fixed convex cocompact linear holonomy as an application we provide a new proof of the tameness of such margulis spacetimes m by establishing the crooked plane conjecture which states that m admits a fundamental domain bounded by piecewise linear surfaces called crooked planes the noninfinitesimal version gives an analogous theory for noncompact complete anti de sitter 3 manifolds

global stabilization for nonlinear uncertain systems with unmodeled actuator dynamics second order sliding mode control 2 smc algorithms are analyzed to assess their global convergence properties while standard first order sliding mode control 1 smc algorithms derive their effectiveness from the global solution of the well known reaching condition ss spl dot spl les k sup 2 s s 0 being the actual sliding manifold 2 smc is based on more complex differential inequalities for which a global solution could not exist the approach presented introduces a suitable commutation logic based on an online simple predictor that prevents an uncontrollable growth of the uncertainties due to this new commutation logic the global convergence of the state trajectory to the designed sliding manifold is ensured

anosov flows surface groups and curves in projective space note that in 10 w goldman gives a complete description of these connected components in the case of finite covers of psl 2 r in the case of psl 2 r two homeomorphic components called teichmuller spaces play a central role these two components are well known to be homeomorphic to a ball of dimension 6g 6 n hitchin generalises this situation to psl n r indeed one of these components when n is odd and two when n is even has a very simple topology let us define an n fuchsian representation to be a representation which can be written as 0 where 0 is a cocompact representation with values in psl 2 r and is the irreducible representation of psl 2 r in psl n r we denote by reph 1 s psl n r a connected component that contains fuchsian representations and call it a hitchin component in fact when n is odd there is one hitchin component and when n is even two isomorphic ones n hitchin proves in 17

isomonodromic deformations and twisted yangians arising in teichmuller theory abstract in this paper we build a link between the teichmuller theory of hyperbolic riemann surfaces and isomonodromic deformations of linear systems whose monodromy group is the fuchsian group associated to the given hyperbolic riemann surface by the poincare uniformization in the case of a one sheeted hyperboloid with n orbifold points we show that the poisson algebra d n of geodesic length functions is the semiclassical limit of the twisted q yangian y q o n for the orthogonal lie algebra o n defined by molev ragoucy and sorba we give a representation of the braid group action on d n in terms of an adjoint matrix action we characterize two types of finite dimensional poissonian reductions and give an explicit expression for the generating function of their central elements finally we interpret the algebra d n as the poisson algebra of monodromy data of a frobenius manifold in the vicinity of a non semi simple point

technical communique quasi continuous hosm control for systems with unmatched perturbations the control of nonlinear systems subject to unmatched perturbations is studied a new design algorithm is proposed based on the block control and quasi continuous higher order sliding modes techniques the proposed method provides for the finite time exact tracking of a smooth desired signal in spite of unmatched perturbations

euclidean decompositions of noncompact hyperbolic manifolds on introduit une methode pour diviser une variete hyperbolique non compacte de volume fini en morceaux euclidiens canoniques

global weak solutions and attractors of the three dimensional maxwell bloch two level laser systems the three dimensional maxwell bloch system governs the multi longitudinal and transverse mode dynamics of two level wide aperture lasers in an optical ring cavity the system is hyperbolic in the propagation direction and dispersive in the transverse directions due to diffraction effects a rich variety of optical patterns and chaos are present in the dynamics we show the global existence of weak solutions inlp 2 p spaces of the maxwell bloch system under both absorbing and periodic boundary conditions the weak solutions are unique within the class of solutions provided by our regularization procedure and approach a universal attractor which has only partial smoothing instead of thec smoothing property found in early works for the longitudinal one dimensional and transverse two dimensional cases the idea of the proof makes essential use of both the hyperbolicity and dispersivity of the system in the case of periodic boundary condition our result depends on a conjectural strichartz inequality

fock goncharov coordinates for rank two lie groups let g be a simply connected simple complex lie group of rank 2 we give explicit fock goncharov coordinates for configurations of triples and quadruples of affine flags in g we show that the action on triples by orientation preserving permutations corresponds to explicit quiver mutations and that the same holds for the flip changing the diagonal in a quadrilateral this gives explicit coordinates on higher teichmuller space and also coordinates for boundary unipotent representations of 3 manifold groups as an application we compute the generic boundary unipotent representations in mathrm sp 4 mathbb c langle i rangle for the figure eight knot complement

polyhedra inscribed in a quadric we study convex polyhedra in three space that are inscribed in a quadric surface up to projective transformations there are three such surfaces the sphere the hyperboloid and the cylinder our main result is that a planar graph gamma is realized as the 1 skeleton of a polyhedron inscribed in the hyperboloid or cylinder if and only if gamma is realized as the 1 skeleton of a polyhedron inscribed in the sphere and gamma admits a hamiltonian cycle this answers a question asked by steiner in 1832 rivin characterized convex polyhedra inscribed in the sphere by studying the geometry of ideal polyhedra in hyperbolic space we study the case of the hyperboloid and the cylinder by parameterizing the space of convex ideal polyhedra in anti de sitter geometry and in half pipe geometry just as the cylinder can be seen as a degeneration of the sphere and the hyperboloid half pipe geometry is naturally a limit of both hyperbolic and anti de sitter geometry we promote a unified point of view to the study of the three cases throughout

differentials on graph complexes abstract we study the cohomology of complexes of ordinary non decorated graphs introduced by m kontsevich we construct spectral sequences converging to zero whose first page contains the graph cohomology in particular these spectral sequences may be used to show the existence of an infinite series of previously unknown and provably non trivial cohomology classes and put constraints on the structure of the graph cohomology as a whole

triangulated riemann surfaces with boundary and the weil petersson poisson structure given a hyperbolic surface with geodesic boundary s the lengths of a maximal system of disjoint simple geodesic arcs on s that start and end at s perpendicularly are coordinates on the teichmuller space t s we express the weil petersson poisson structure of t s in this system of coordinates and we prove that it limits pointwise to the piecewise linear poisson structure defined by kontsevich on the arc complex of s at the same time we obtain a formula for the first order variation of the distance between two closed geodesics under fenchel nielsen deformation

multiple edges in m kontsevich s graph complexes and computations of the dimensions and euler characteristics we provide a generating function for the graded dimensions of m kontsevich s graph complexes of ordinary graphs this generating function can be used to compute the euler characteristic in each loop order furthermore we show that graphs with multiple edges can be omitted from these graph complexes

finite type invariants and fatgraphs abstract we define an invariant g m of pairs m g where m is a 3 manifold obtained by surgery on some framed link in the cylinder i is a connected surface with at least one boundary component and g is a fatgraph spine of in effect g is the composition with the n maps of le murakami ohtsuki of the link invariant of andersen mattes reshetikhin computed relative to choices determined by the fatgraph g this provides a basic connection between 2d geometry and 3d quantum topology for each fixed g this invariant is shown to be universal for homology cylinders i e g establishes an isomorphism from an appropriate vector space h of homology cylinders to a certain algebra of jacobi diagrams via composition g g 1 for any pair of fatgraph spines g g of we derive a representation of the ptolemy groupoid i e the combinatorial model for the fundamental path groupoid of teichmuller space as a group of automorphisms of this algebra the space h comes equipped with a geometrically natural product induced by stacking cylinders on top of one another and furthermore supports related operations which arise by gluing a homology handlebody to one end of a cylinder or to another homology handlebody we compute how g interacts with all three operations explicitly in terms of natural products on jacobi diagrams and certain diagrammatic constants our main result gives an explicit extension of the lmo invariant of 3 manifolds to the ptolemy groupoid in terms of these operations and this groupoid extension nearly fits the paradigm of a tqft we finally re derive the morita penner cocycle representing the first johnson homomorphism using a variant generalization of g

on hilbert fourier and wavelet transforms several new families of wavelets which are derived from earlier work by the author on teichmuller theory are introduced and studied these wavelets are associated with the modular group rather than the affine group of the real line as for classical wavelets in further contrast to the usual setting there are explicit and essentially algebraic formulas relating wavelet and fourier expansions furthermore rather than relying on equally spaced multiscale sampling methods the wavelet expansions introduced here rely instead on an explicit nonuniformly spaced multiscale sampling method that depends upon the farey sequence of classical number theory these ingredients combine to give novel methods for calculating wavelet fourier and hilbert transforms with potential applications to myriad problems in electrical engineering by employing a new characteristic length associated to 2 2 matrices finiteness and convergence properties of the wavelet expansions are derived for functions with more than 5 2 derivatives in l2 one of our wavelet families forms a frame and the hilbert transform is calculable in hyperbolic coordinates 2002 wiley periodicals inc

tour of bordered floer theory heegaard floer theory is a kind of topological quantum field theory tqft assigning graded groups to closed connected oriented 3 manifolds and group homomorphisms to smooth oriented four dimensional cobordisms bordered heegaard floer homology is an extension of heegaard floer homology to 3 manifolds with boundary with extended tqft type gluing properties in this survey we explain the formal structure and construction of bordered floer homology and sketch how it can be used to compute some aspects of heegaard floer theory

on ramond decorations we impose constraints on the odd coordinates of super teichmuller space in the uniformization picture for the monodromies around ramond punctures thus reducing the overall odd dimension to be compatible with that of the moduli spaces of super riemann surfaces namely the monodromy of a puncture must be a true parabolic element of the canonical subgroup sl 2 mathbb r of osp 1 2

analysis of second order sliding mode algorithms in the frequency domain a frequency domain analysis of the second order sliding mode algorithms particularly of the twisting algorithm is carried out in the frequency domain with the use of the describing function method and tsypkin s approach it is shown that in the presence of an actuator the transient process may converge to a periodic motion parameters of this periodic motion are analyzed a comparison of the periodic solutions in the systems with higher order sliding mode controllers and the oscillations that occur in classical sliding mode systems with actuators is done

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

the dilogarithmic central extension of the ptolemy thompson group via the kashaev quantization abstract quantization of universal teichmuller space provides projective representations of the ptolemy thompson group which is isomorphic to the thompson group t this yields certain central extensions of t by z called dilogarithmic central extensions we compute a presentation of the dilogarithmic central extension t kash of t resulting from the kashaev quantization and show that it corresponds to 6 times the euler class in h 2 t z meanwhile the braided ptolemy thompson groups t t of funar kapoudjian are extensions of t by the infinite braid group b and by abelianizing the kernel b one constructs central extensions t ab t ab of t by z which are of topological nature we show t kash t ab our result is analogous to that of funar and sergiescu who computed a presentation of another dilogarithmic central extension t cf of t resulting from the chekhov fock goncharov quantization and thus showed that it corresponds to 12 times the euler class and that t cf t ab in addition we suggest a natural relationship between the two quantizations in the level of projective representations

hochschild homology of structured algebras abstract we give a general method for constructing explicit and natural operations on the hochschild complex of algebras over any prop with a multiplication we think of such algebras as a algebras with extra structure as applications we obtain an integral version of the costello kontsevich soibelman moduli space action on the hochschild complex of open tcfts the tradler zeinalian and kaufmann actions of sullivan diagrams on the hochschild complex of strict frobenius algebras and give applications to string topology in characteristic zero our main tool is a generalization of the hochschild complex

nonlinear internal models for output regulation in this note we show how nonlinear internal models can be effectively used in the design of output regulators for nonlinear systems this result provides a significant enhancement of the nonequilibrium theory for output regulation which we have presented in a recent paper

reduced order observer in the sliding mode control of nonlinear nonaffine systems the note considers the variable structure control of nonlinear known nonaffine systems when the state vector is not completely available and the use of observers is required the strategy of introducing integrators in the input channel is exploited to enlarge the class of tractable control systems a new reduced order observer is proposed and conditions are found under which it is proven the convergence to the unique ideal solution of both system and observer the control problem is solved by forcing a sliding regime for the observer while satisfying an exponential stability criterion for the observation error state equation

universal description of laser dynamics near threshold abstract complex order parameter descriptions of large aspect ratio single longitudinal mode two level lasers with flat end reflectors valid near onset of lasing and for small detunings of the laser from the peak gain are given in terms of a complex swift hohenberg equation for class a and c lasers and by a complex swift hohenberg equation coupled to a mean flow for the case of a class b laser the latter coupled system is a physically consistent generalized rate equation model for wide aperture stiff laser systems these universal order parameter equations provide a connection between spatially homogeneous oscillating states of the complex ginzburg landau equation description of the laser system valid for finite negative detunings and traveling wave states described by coupled newell whitehead segel equations valid for finite positive detunings one of the main conclusions of the present paper is that the usual eckhaus instability boundary associated with a long wavelength phase instability and which delineates the region of the stable traveling wave solutions for class a and c lasers no longer defines the stability boundary for the mathematically stiff class b laser instead a short wavelength phase instability appears causing the stability domain to shrink as a function of increasing stiffness of the system this prediction is consistent with the strong spatiotemporal filamentation instabilities experimentally observed in a borad area semiconductor laser a class b system

laurent positivity of quantized canonical bases for quantum cluster varieties from surfaces in 2006 fock and goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed pgl2 local systems on a punctured surface s the moduli space is birational to a cluster mathcal x variety whose positive real points recover the enhanced teichmuller space of s their basis is enumerated by integral laminations on s which are collections of closed curves in s with integer weights around ten years later a quantized version of this basis still enumerated by integral laminations was constructed by allegretti and kim for each choice of an ideal triangulation of s each quantum basis element is a laurent polynomial in the exponential of quantum shear coordinates for edges of the triangulation with coefficients being laurent polynomials in q with integer coefficients we show that these coefficients are laurent polynomials in q with positive integer coefficients our result was expected in a positivity conjecture for framed protected spin characters in physics and provides a rigorous proof of it and may also lead to other positivity results as well as categorification a key step in our proof is to solve a purely topological and combinatorial ordering problem about an ideal triangulation and a closed curve on s for this problem we introduce a certain graph on s which is interesting in its own right

moduli of super riemann surfaces the basic properties of super riemann surfaces are presented and their supermoduli spaces are constructed in a manner suitable for the application of algebro geometric techniques to string theory

canonical extensions of the johnson homomorphisms to the torelli groupoid abstract we prove that every trivalent marked bordered fatgraph comes equipped with a canonical generalized magnus expansion in the sense of kawazumi this magnus expansion is used to give canonical extensions of the higher johnson homomorphisms m for m 1 to the torelli groupoid and we provide a recursive combinatorial formula for tensor representatives of these extensions in particular we give an explicit 1 cocycle in the dual fatgraph complex which extends 2 and thus answer affirmatively a question of morita and penner to illustrate our techniques for calculating higher johnson homomorphisms in general we give explicit examples calculating m for m 3

rigidity of polyhedral surfaces ii we study the rigidity of polyhedral surfaces using variational principles the action functionals are derived from the cosine laws the main focus of this paper is on the cosine law for a nontriangular region bounded by three possibly disjoint geodesics several of these cosine laws were first discovered and used by fenchel and nielsen by studying the derivative of the cosine laws we discover a uniform approach to several variational principles on polyhedral surfaces with or without boundary as a consequence the work of penner bobenko and springborn and thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

perturbative series and the moduli space of riemann surfaces on utilise des techniques de la theorie quantique des champs pour calculer des quantites reliees aux groupes de symetrie de paires f g ou f est une surface et g une epine de f

teichmuller spaces as degenerated symplectic leaves in dubrovin ugaglia poisson manifolds abstract in this paper we study the goldman bracket between geodesic length functions both on a riemann surface g s 0 of genus g with s 1 2 holes and on a riemann sphere 0 1 n with one hole and n orbifold points of order two we show that the corresponding teichmuller spaces t g s 0 and t 0 1 n are realised as real slices of degenerated symplectic leaves in the dubrovin ugaglia poisson algebra of upper triangular matrices s with 1 on the diagonal

the virtual cohomological dimension of the mapping class group of an orientable surface let f f r be the mapping class group of a surface f of genus g with s punctures and r boundary components the purpose of this paper is to establish cohomology properties of f parallel to those of the arithmetic groups if g is a linear algebraic group defined over q x is the symmetric space associated to g and a is an arithmetic subgroup of g then a is virtually torsion free and acts properly discontinuously on x the rational homology of a is the same as that of x a furthermore there is a bordification of x bs to a manifold with corners 3 and an extension of the action of a to a properly discontinuous action on jf so that the quotient x a is compact the boundary of jf is homotopy equivalent to a wedge of spheres say of dimension d and the virtual cohomological dimension of a is n d 1 where n is the dimension of x in the case of the mapping class group there is no analog for g ivanov unpublished has proven that f is not arithmetic nevertheless f acts properly discontinuously on teichmiiller space r which is homeomorphic to euclidean space of dimension 6 g 6 2 s 3 will play the role of the symmetric space the quotient of 9by f is the moduli space of curves whose rational homology is then identified with that of e harvey har has constructed a borel serre bordification j of 3 by analytic methods in the case where f has punctures we will build g by a different combinatorial method in addition we will explicitly describe inside y a cell complex y of dimension 4 g 4 s onto which j may be f equivariantly retracted thus establishing an analog of the constructions for sl n by serre soul6 sol and ash a this complex will be of the lowest possible dimension because we will use j to prove our main result

noncommutative marked surfaces abstract the aim of the paper is to attach a noncommutative cluster like structure to each marked surface this is a noncommutative algebra a generated by noncommutative geodesics between marked points subject to certain triangle relations and noncommutative analogues of ptolemy plucker relations it turns out that the algebra a exhibits a noncommutative laurent phenomenon with respect to any triangulation of which confirms its cluster nature as a surprising byproduct we obtain a new topological invariant of which is a free or a 1 relator group easily computable in terms of any triangulation of another application is the proof of laurentness and positivity of certain discrete noncommutative integrable systems

internal dynamics of flat nonlinear siso systems with respect to a non flat output in this article the internal dynamics of flat nonlinear siso systems with respect to a real non flat output is investigated in the coordinates of the flat output and its derivatives the main result constitutes that the parameterization of the real output by the flat output and its derivatives up to a certain order forms a differential equation which is diffeomorphic to the aforementioned internal dynamics

quantisation of super teichmuller theory we construct a quantisation of the teichmuller spaces of super riemann surfaces using coordinates associated to ideal triangulations of super riemann surfaces a new feature is the non trivial dependence on the choice of a spin structure which can be encoded combinatorially in a certain refinement of the ideal triangulation by constructing a projective unitary representation of the groupoid of changes of refined ideal triangulations we demonstrate that the dependence of the resulting quantum theory on the choice of a triangulation is inessential

bending fuchsian representations of fundamental groups of cusped surfaces in pu 2 1 we describe a family of representations of 1 in pu 2 1 where is a hyperbolic riemann surface with at least one deleted point which is obtained by a bending process associated to an ideal triangulation of we give an explicit description of this family by describing a coordinate system on it which is in the spirit of fock and goncharov coordi

the unstable integral homology of the mapping class groups of a surface with boundary we construct a graph complex calculating the integral homology of the bordered mapping class groups we compute the homology of the bordered mapping class groups of the surfaces s 1 1 s 1 2 and s 2 1 using the circle action on this graph complex we build a double complex and a spectral sequence converging to the homology of the unbordered mapping class groups we compute the homology of the punctured mapping class groups associated to the surfaces s 1 1 and s 2 1 finally we use miller s operad to get the first kudo araki and browder operations on our graph complex we also consider an unstable version of the higher kudo araki dyer lashoff operations

2d conformal field theories and holography it is known that the chiral part of any 2d conformal field theory defines a 3d topological quantum field theory quantum states of this tqft are the cft conformal blocks the main aim of this paper is to show that a similar cft tqft relation exists also for the full cft the 3d topological theory that arises is a certain square of the chiral tqft such topological theories were studied by turaev and viro they are related to 3d gravity we establish an operator state correspondence in which operators in the chiral tqft correspond to states in the turaev viro theory we use this correspondence to interpret cft correlation functions as particular quantum states of the turaev viro theory we compute the components of these states in the basis in the turaev viro hilbert space given by colored 3 valent graphs the formula we obtain is a generalization of the verlinde formula the later is obtained from our expression for a zero colored graph our results give an interesting holographic perspective on conformal field theories in two dimensions

the weil petersson symplectic structure at thurston s boundary the weil petersson kahler structure on the teichmuller space f of a punctured surface is shown to extend in an appropriate sense to thurston s symplectic structure on the space m f 0 of measured foliations of compact support on the surface we introduce a space m f 0 of decorated measured foliations whose relationship to m f 0 is analogous to the relationship between the decorated teichmuller space f and f m f 0 is parametrized by a vector space and there is a naturl piecewise linear embedding of m f 0 in m f 0 which pulls back a global differential form to thurston s symplectic form we exhibit a homeomorphism between f and m f 0 which preseres the natural two forms on these spaces following thurston we finally consider the space y of all suitable classes of metrics of constant gaussian curvature on the surface form a natural completion y of y and identify y y with m f 0

cyclic symmetries of an quiver representations abstract this note contains a combinatorial construction of symmetries arising in symplectic geometry partially wrapped or infinitesimal fukaya categories algebraic geometry derived categories of singularities and k theory waldhausen s s construction our specific motivation in the spirit of expectations of kontsevich is a combinatorial construction of categorical quantizations of lagrangian skeleta of symplectic manifolds the main result of this paper gives an immediate solution in the one dimensional case of ribbon graphs

observation and output adaptive tracking for a class of nonlinear non minimum phase systems abstractin this paper the output tracking problem for a class of systems with unstable zero dynamics is addressed the state is assumed not measurable the output of the dynamical system to be controlled has to track a signal which is the sum of a known number of sinusoids with unknown frequencies amplitudes and phases the non minimum phase nature of the considered systems prevents the direct tracking by standard sliding mode methods which are known to generate unstable behaviours of the internal dynamics the proposed method relies on the availability of a flat output and its time derivatives which are functions of the unavailable state therefore a nonlinear observer is needed due to the uncertainty in the frequencies and in the parameters defining the relationship between the output of the system and the flat states adaptive indirect methods are applied

asymptotic stabilization of minimum phase nonlinear systems how a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown more precisely conditions under which for every compact set of initial states it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described the theory includes the development of globally defined transformations of the system equations to their global normal form

enumeration of chord diagrams on many intervals and their non orientable analogs two types of connected chord diagrams with chord endpoints lying in a collection of ordered and oriented real segments are considered here the real segments may contain additional bivalent vertices in one model but not in the other in the former case we record in a generating function the number of fatgraph boundary cycles containing a fixed number of bivalent vertices while in the latter we instead record the number of boundary cycles of each fixed length second order non linear algebraic partial differential equations are derived which are satisfied by these generating functions in each case giving efficient enumerative schemes moreover these generating functions provide multi parameter families of solutions to the kp hierarchy for each model there is furthermore a non orientable analog and each such model likewise has its own associated differential equation the enumerative problems we solve are interpreted in terms of certain polygon gluings as specific applications we discuss models of several interacting rna molecules we also study a matrix integral which computes numbers of chord diagrams in both orientable and non orientable cases in the model with bivalent vertices and the large n limit is computed using techniques of free probability

exact tracking using backstepping control design and high order sliding modes in this technical note a controller ensuring exponential exact tracking in the presence of matched and unmatched disturbances for the nonlinear systems in the block controllable form is proposed the controller is designed using the backstepping procedure and combines the feedback linearization technique with the high order sliding modes the matched and unmatched disturbances are compensated by the injection of a continuous term generated by the robust exact high order sliding modes differentiator the obtained control law is differentiable and can be applied directly to the system simulations verify the performance of the proposed controller

the euler characteristic of the moduli space of curves let fg 1 g 1 be the mapping class group consisting of all isotopy classes of base point and orientation preserving homeomorphisms of a closed oriented surface f of genus g let 1 be its euler characteristic in the sense of wall that is z f i fgl f l z e f where f is any torsion free subgroup of finite index in f 1 and e is a contractible space on which f acts freely and properly discontinuously an example of such a space is the teichmiiller space 1 and g f can be interpreted as the orbifold euler characteristic of f i join the moduli space of curves of genus g with base point the purpose of this paper is to prove the following formula for f i

brief tracking in a class of nonminimum phase systems with nonlinear internal dynamics via sliding mode control using method of system center a method of asymptotic output tracking in a class of causal nonminimum phase uncertain nonlinear systems is considered local asymptotic stability of output tracking error dynamics are provided for the specified class of systems with the nonlinear hyperbolic at the origin internal dynamics forced by a reference output profile and external disturbances defined by a known linear exosystem the nonlinear vector field of the internal dynamics is expanded in a power series obtained at a selected operational point in the internal dynamics state space the presented technique employs some linear algebraic methods and sliding mode control approach the solution is a complete constructive algorithm

dynamic output feedback for observed variable stucture control systems abstract a global exponential observer is built explicitly for some classes of nonlinear variable structure observed control systems by using piecewise continuous output feedback control laws or the continuous equivalent control for the observer the unavailable state vector is asymptotically kept on the sliding manifold with exponential decaying error some explicit examples are presented

graded manifolds supermanifolds and infinite dimensional grassmann algebras a new approach to superdifferentiable functions of grassmann variables is developed which avoids ambiguities in odd derivatives this is used to give an improved definition of supermanifold over a finite dimensional grassmann algebra a natural embedding of super manifolds over grassmann algebras with increasing number l of generators is developed and thus a limit asl tends to infinity is possible a correspondence between graded manifolds and supermanifolds is constructed extending results of 5 and 8

a lie backlund approach to equivalence and flatness of nonlinear systems a new system equivalence relation using the framework of differential geometry of jets and prolongations of infinite order is studied in this setting two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives this is a lie backlund isomorphism the authors prove that although the state dimension is not preserved the number of input channels is kept fixed they also prove that a lie backlund isomorphism can be realized by an endogenous feedback the differentially flat nonlinear systems introduced by the authors 1992 via differential algebraic techniques are generalized and the new notion of orbitally flat systems is defined they correspond to systems which are equivalent to a trivial one with time preservation or not the endogenous linearizing feedback is explicitly computed in the case of the vtol aircraft to track given reference trajectories with stability