the computational complexity of knot genus and spanning area we investigate the computational complexity of some problems in three dimensional topology and geometry we show that the problem of determining a bound on the genus of a knot in a 3 manifold is np complete using similar ideas we show that deciding whether a curve in a metrized pl 3 manifold bounds a surface of area less than a given constant c is np hard

evolution by mean curvature flow of lagrangian spherical surfaces in complex euclidean plane abstract we describe the evolution under the mean curvature flow of embedded lagrangian spherical surfaces in the complex euclidean plane c 2 in particular we answer the question 4 7 addressed in 10 by a neves about finding out a condition on a starting lagrangian torus in c 2 such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the clifford torus

a second order godunov type scheme for compressible fluid dynamics abstract a second order accurate scheme for the integration in time of the conservation laws of compressible fluid dynamics is presented two related versions are proposed one lagrangian and the second direct eulerian they both share the common ingredient which is a full analytic solution for the time derivatives of flow quantities at a jump discontinuity assuming initial nonvanishing slopes on both sides while this solution is an extension of the solution to the classical riemann problem the resulting schemes are second order extensions of godunov s methods in both cases they are very simple to implement in computer codes several numerical examples are shown where the only additional mechanism is a simple monotonicity algorithm

level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

an adaptive observer design for n 1 coupled linear hyperbolic pdes based on swapping in this paper we use swapping design filters to bring systems of n 1 partial differential equations of the hyperbolic type to static form standard parameter identification laws can then be applied to estimate unknown parameters in the boundary conditions proof of boundedness of the adaptive laws are offered and the results are demonstrated in simulations

boundary observer design for a class of semi linear hyperbolic pde systems with recycle loop abstractthe present manuscript considers the observer design problem for a class of scalar semi linear hyperbolic partial differential equation pde systems with a recycle loop through the boundar

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

pl equivariant surgery and invariant decompositions of 3 manifolds using normal surface theory h j2 we introduce the notion of least weight normal surfaces the weight of a normal surface is a nonnegative integer invariant of the normal isotopy class of the surface if we focus on a particular class of normal surfaces and choose representatives which minimize the weight over the class then we have least weight normal surfaces it is remarkable how these least weight normal surfaces exhibit many of the same useful properties as least area minimal surfaces they provide a piecewise linear pl environment to obtain the recent topological results coming from the analysis and geometry of least area surfaces in an impressive series of papers meeks and yau m y i m y z m y 3 m y m y meeks simon and yau m s y scott s meeks and scott m s and freedman hass and scott f h s introduced the analysis and geometry of least area surfaces into the study of topological questions about 3 manifolds the consequences have led to the resolution of many outstanding problems in the topology of 3 manifolds since most of the problems being solved with these new techniques are topological in nature it has been felt that there should be a more topological approach to the proofs of these results besides the bulk of the work in the theory of

ancient solutions to mean curvature flow for isoparametric submanifolds mean curvature flow for isoparametric submanifolds in euclidean spaces and spheres was studied in liu and terng duke math j 147 1 157 179 2009 in this paper we will show that all these solutions are ancient solutions and study their limits as time goes to negative infinity we also discuss rigidity of ancient mean curvature flows for hypersurfaces in spheres and its relation to the chern s conjecture on the norm of the second fundamental forms of minimal hypersurfaces in spheres

direct reduction and differential constraints direct reductions of partial differential equations to systems of ordinary differential equations are in one to one correspondence with compatible differential constraints the differential constraint method is applied to prove that a parabolic evolution equation admits infinitely many characteristic second order reductions but admits a non characteristic second order reduction if and only if it is linearizable

control of 2 2 linear hyperbolic systems backstepping based trajectory generation and pi based tracking abstract we consider the problems of trajectory generation and tracking for general 2 2 systems of first order linear hyperbolic pdes with anti collocated boundary input and output we solve the trajectory generation problem via backstepping the reference input which generates the desired output incorporates integral operators acting on advanced and delayed versions of the reference output with kernels which were derived by vazquez krstic and coron for the backstepping stabilization of 2 2 linear hyperbolic systems we apply our approach to a wave pde with indefinite in domain and boundary damping for tracking the desired trajectory we employ a pi control law on the tracking error of the output we prove exponential stability of the closed loop system under the proposed pi control law when the parameters of the plant and the controller satisfy certain conditions by constructing a novel non diagonal lyapunov functional we demonstrate that the proposed pi control law compensates in the output the effect of in domain and boundary disturbances we illustrate our results with numerical examples

estimation of boundary parameters in general heterodirectional linear hyperbolic systems abstract in this paper we extend recent results on state and boundary parameter estimation in coupled systems of linear partial differential equations pdes of the hyperbolic type consisting of n rightward and one leftward convecting equations to the general case which involves an arbitrary number of pdes convecting in both directions two adaptive observers are derived based on swapping design where one introduces a set of filters that can be used to express the system states as linear static combinations of the filter states and the unknown parameters standard parameter identification laws can then be applied to estimate the unknown parameters one observer which requires sensing at both boundaries generates estimates of the boundary parameters and system states while the second observer estimates the parameters from sensing limited to the boundary anti collocated with the uncertain parameters proof of boundedness of the adaptive laws is offered and sufficient conditions ensuring exponential convergence are derived the theory is verified in simulations

mean curvature flows and isotopy of maps between spheres let f be a smooth map between unit spheres of possibly different dimensions we prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions a corollary is that any area decreasing map between unit spheres of possibly different dimensions is isotopic to a constant map c 2004 wiley periodicals inc

exact description of simple wave interactions in multicomponent chromatography nonlinear wave interaction processes for a quasilinear hyperbolic homogeneous system of first order partial differential equations multicomponent chromatography are investigated the wave analysis is worked out by extending to the present multicomponent case the leading ideas of a well established method of approach that was developed for solving initial value wave problems in terms of exact solutions to 2 2 hyperbolic homogeneous systems these interaction processes may model different situations concerning the separation of a mixture into its chemical components several numerical plots are also given in order to illustrate the behavior of the exact wave solutions arising from the analysis that is accomplished

a topological proof of the equivariant dehn lemma an elementary topological proof is given for a completely general version of the equivariant dehn lemma in the spirit of the original proof of the nonequivariant version due to c d papakyriakopolous in 1957

embedded minimal surfaces exotic spheres and manifolds with positive ricci curvature let n be a three dimensional riemannian manifold let e be a closed embedded surface in n then it is a question of basic interest to see whether one can deform in its isotopy class to some canonical embedded surface from the point of view of geometry a natural canonical surface will be the extremal surface of some functional defined on the space of embedded surfaces the simplest functional is the area functional the extremal surface of the area functional is called the minimal surface such minimal surfaces were used extensively by meeks yau my21 in studying group actions on three dimensional manifolds in my2 the theory of minimal surfaces was used to simplify and strengthen the classical dehn s lemma loop theorem and the sphere theorem in the setting there one minimizes area among all immersed surfaces and proves that the extremal object is embedded in this paper we minimize area among all embedded surfaces isotopic to a fixed embedded surface in the category of these surfaces we prove a general existence theorem theorem 1 a particular consequence of this theorem is that for irreducible manifolds an embedded incompressible surface is isotopic to an embedded incompressible surface with minimal area we also prove that there exists an embedded sphere of least area enclosing a fake cell provided the complementary volume is not a standard ball and provided there exists no embedded one sided rp2 by making use of the last result and a cutting and pasting argument we are able to settle a well known problem in the theory of three dimensional manifolds we prove that the covering space of any irreducible orientable three dimensional manifold is irreducible it is possible to exploit our existence theorem to study

the asymptotic behavior of symplectic mean curvature flow with pinched curvatures in cp2 abstract it was proved by han li yang 4 that the mean curvature flow of symplectic surfaces with suitably pinched curvatures in c p 2 has a longtime solution and converges to a holomorphic curve as the time approaches infinity in this note we give a refinement of this theorem we prove that the evolving surface becomes more and more umbilical and the limit surface is in fact c p 1 as a consequence the initial surface is symplectically isotopic to c p 1

exact solutions describing soliton like interactions in a nondispersive medium when two solitons collide they interact locally but emerge unscattered from the interaction in this paper we show that there are certain media whose responses are governed by the nonlinear nondispersive wave equation in which any two pulses travelling in opposite directions interact nonlinearly for a finite time when they collide but then part unaffected by the interaction for these media thedependence of the wave speed on the amplitude of the disturbance although special is not pathological in fact by a suitable choice of the constants in the expression for the sound speed the responses of many real materials can be modeled a concise representation is obtained for the exact solution of the initial value problem in terms of the solution of a first order system of linear ordinary differential equations an explicit solution is found for piecewise linear initial data

manifolds with positive curvature operators are space forms the ricci flow was introduced by hamilton in 1982 h1 in order to prove that a compact three manifold admitting a riemannian metric of positive ricci curvature is a spherical space form in dimension four hamilton showed that compact four manifolds with positive curvature operators are spherical space forms as well h2 more generally the same conclusion holds for compact four manifolds with 2 positive curvature operators che recall that a curvature operator is called 2 positive if the sum of its two smallest eigenvalues is positive in arbitrary dimensions huisken hu described an explicit open cone in the space of curvature operators such that the normalized ricci flow evolves metrics whose curvature operators are contained in that cone into metrics of constant positive sectional curvature hamilton conjectured that in all dimensions compact riemannian manifolds with positive curvature operators must be space forms in this paper we confirm this conjecture more generally we show the following

disturbance rejection in general heterodirectional 1 d linear hyperbolic systems using collocated sensing and control abstract we solve a disturbance rejection problem for a general class of heterodirectional 1 d linear hyperbolic partial differential equations pdes the disturbance enters at one boundary while sensing and actuation is limited to the opposite boundary we do this by combining a separately designed controller and observer into an output feedback controller and proving stability of the overall closed loop system and exponential regulation of the control objective to zero the theory is demonstrated in a simulation

finite group actions on 3 manifolds if g is a finite group acting smoothly on a closed surface f it is well known that g leaves invariant some riemannian metric of constant curvature on f thus any action of g on the 2 sphere s 2 is conjugate in diff s 2 to an orthogonal action if g acts on the torus sx s 1 there is a g invariant flat metric on s a s 1 and if g acts on a surface f with negative euler number then f admits a g invariant hyperbolic metric recently thurston th 1 th 2 th 3 has described the eight 3 dimensional geometries which provide geometric structures for closed 3 manifolds in the same way that the 2 sphere s 2 the euclidean plane e 2 and the hyperbolic plane h 2 provide geometric structures for surfaces see also the survey article by scott sc4 thurston also conjectured that if m is a closed 3 manifold with a geometric structure modelled on one of these eight geometries say x then any smooth action of a finite group g on m should leave invariant some metric on m inducing the geometry x we will say that g preserves the geometric structure on m in this case it should be noted that the restriction to smooth actions of g on m is essential for bing bi showed that there are involutions of s 3 whose fixed set is a wild 2 sphere however in dimension two it was proved by eilenberg ei that any action of a finite group on a surface is conjugate to a smooth action in this paper our main result asserts that thurston s conjecture holds for five of the eight geometries the result is the following

crosscap numbers and the jones polynomial abstract we give sharp two sided linear bounds of the crosscap number non orientable genus of alternating links in terms of their jones polynomial our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several alternating knots with up to twelve crossings we also discuss generalizations of our results for classes of non alternating links

a reduction procedure for generalized riemann problems with application to nonlinear transmission lines generalized simple wave solutions to quasilinear hyperbolic nonhomogeneous systems of pdes are obtained through the differential constraint method these solutions prove to be flexible enough to solve generalized riemann problems where discontinuous initial data are involved within such a theoretical framework the governing model of nonlinear transmission lines is investigated throughout

the existence of embedded minimal surfaces and the problem of uniqueness it was in the early nineteen thirties that douglas and radd solved the plateau problem for a rectifiable jordan curve in euclidean space the minimal surfaces that they obtained are realized by conformal harmonic maps from the unit disk in 1948 morrey generalized the theorem of douglas and radd to minimal surfaces in a homogeneously regular riemannian manifold in both cases the solutions are defective as they may possess branch points it was not until 1968 that osserman 24 was able to prove that interior branch points do not exist on the douglas radd morrey solution osserman s theorem in this full generality was proved by alt 2 3 and gulliver 8 if the jordan curve is real analytic gulliver and lesley 9 showed that the solution surface is free of branch points even on its boundary hence at least in this case we know that the douglas radd morrey solution is an immersion and represents a classical surface however the douglas radd morrey solution may in general have selfintersections it was then a classical problem to find conditions on the jordan curve to guarantee that the solution is an embedded disk in 1932 radd was the first to produce a general condition of this type he proved that if the curve can be projected onto a convex curve in the plane then it bounds a unique embedded minimal disk which is a graph over this plane it was not until the seventies that there was further progress on the problem of embeddedness osserman had conjectured that if a jordan curve is extreme that is lies on the boundary of its convex hull then the douglas rad6 solution is embedded gulliver and spruck 10 proved this conjecture under the additional assumption that the absolute total curvature of the jordan curve is less than 4 z right after this work tomi and tromba 32 showed that an extreme curve must bound an embedded minimal disk which need not be stable in 1976 almgren and simon 1 established the existence for such a curve of an embedded stable minimal disk which need not be a douglas radd solution finally in the same year the authors 18 proved that any douglas rad6 solution must be embedded

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

0 efficient triangulations of 3 manifolds efficient triangulations of 3 manifolds are defined and studied it is shown that any triangulation of a closed orientable irreducible 3 manifold m can be modified to a 0 efficient triangulation orm can be shown to be one of the manifolds s 3 rp 3 or l 3 1 similarly any triangulation of a com pact orientable irreducible irreducible 3 manifold can be modified to a 0 efficient triangulation the notion of a 0 efficient ideal tr is de fined it is shown if m is a compact orientable irreducible irreducible 3 manifold having no essential annuli and distinct from the 3 cell then m admits an ideal triangulation furthermore it is shown that any ideal trian gulation of such a 3 manifold can be modified to a 0 efficient ideal triangula tion a 0 efficient triangulation of a closed manifold has only one vertex or the manifold is s 3 and the triangulation has precisely two vertices 0 efficient tri angulations of 3 manifolds with boundary and distinct from the 3 cell have all their vertices in the boundary and then just one vertex in each bound ary component as tools we introduce the concepts of barrier surface and shrinking as well as the notion of crushing a triangulation along a normal sur face a number of applications are given including an algorithm to construct an irreducible decomposition of a closed orientable 3 manifold an algorithm to construct a maximal collection of pairwise disjoint normal 2 spheres in a closed 3 manifold an alternate algorithm for the 3 sphere recognition prob lem results on edges of low valence in minimal triangulations of 3 manifolds and a construction of irreducible knots in closed 3 manifolds

the number of reidemeister moves needed for unknotting there is a positive constant c1 such that for any diagram d representing the unknot there is a sequence of at most 2 c1n reidemeister moves that will convert it to a trivial knot diagram where n is the number of crossings in d a similar result holds for elementary moves on a polygonal knot k embedded in the 1 skeleton of the interior of a compact orientable triangulated p l 3 manifold m there is a positive constant c2 such that for each t 1 if m consists of t tetrahedra and k is unknotted then there is a sequence of at most 2 c 2t elementary moves in m which transforms k to a triangle contained inside one tetrahedron of m we obtain explicit values for c1 and c2

hodograph transformation and differential constraints for wave solutions to 2 2 quasilinear hyperbolic nonhomogeneous systems the differential constraint method is used to work out a reduction approach to determine solutions in a closed form to the highly nonlinear hodograph system arising from 2 2 hyperbolic nonhomogeneous models these solutions inherit all of the features of the standard wave solutions obtainable via the classical hodograph transformation and in the meantime incorporate the dissipative effects induced on wave processes by the source like term involved in the governing equations within such a theoretical framework the problem of integrating the standard linear hodograph system associated with 2 2 homogeneous models is also revisited and a number of results obtained elsewhere of relevant interest in wave problems are recovered as a particular case along the lines of the proposed reduction approach different examples of 2 2 governing models are analysed thoroughly in order to highlight the flexibility of the provided solutions to describe hyperbolic dissipative wave processes

the extension and convergence of mean curvature flow in higher codimension in this paper we first investigate the integral curvature condition to extend the mean curvature flow of submanifolds in a riemannian manifold with codimension d 1 which generalizes the extension theorem for the mean curvature flow of hypersurfaces due to le ysum 12 and the authors 25 26 using the extension theorem we prove two convergence theorems for the mean curvature flow of closed submanifolds in r n d under suitable integral curvature conditions

mean curvature flow of pinched submanifolds to spheres the evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of gerhard huisken s seminal paper hu1 more recently mean curvature flow of higher codimension submanifolds has also received attention in this paper we prove a result analogous to that of hu1 for submanifolds of any codimension let f0 n rn k be a smooth immersion of a compact manifold the mean curvature flow with initial condition f0 is a smooth family of immersions f 0 t rn k satisfying

one fixed point actions and homology 3 spheres one of the best known three dimensional manifolds is the poincare homology three sphere 1 2 3 5 this closed 3 manifold has the same homology groups as the standard sphere s3 but is not simply connected see pnc p 106 or br2 section 1 8 numerous investigations during the past century have shown that this manifold has many remarkable properties several equivalent descriptions of 1 2 3 5 are summarized in kis from the viewpoint of transformation groups one noteworthy property is that 1 2 3 5 is the only nonsimply connected homology sphere admitting a transitive action of a compact lie group brl the manifold 1 2 3 5 also figures importantly in regularity questions for group actions on spherelike manifolds linear or orthogonal actions of finite groups on spheres are generally viewed as the simplest and most regular examples of such actions and one of the themes of transformation groups is to determine the extent to which arbitrary actions on spherelike manifolds resemble linear actions one distinguishing feature of linear actions is that their fixed point sets if nonempty are also spheres the well known results of p a smith cf br2 show that continuous p group actions on homology spheres all resemble linear actions in this respect i e the fixed point sets are always zp homology cohomology spheres on the other hand outside the family of p groups there are many exotic group actions perhaps the most elementary example of this type involves 1 2 3 5 if a is the subgroup of the rotation group so3 given by the symmetries of a regular icosahedron so that a is isomorphic to the alternating group a5 then the quotient space s03 a is the poincare homology 3 sphere and the induced action of a5 a has exactly one fixed point this example suggests that general actions of finite groups on homology spheres can be quite different from linear actions and much of the work on exotic finite group actions during the past three decades was at least partially motivated by the existence of this group action

the computational complexity of knot and link problems we consider the problem of deciding whether a polygonal knot in 3 dimensional euclidean space is unknotted ie capable of being continuously deformed without self intersection so that it lies in a plane we show that this problem unknotting problem is in np we also consider the problem splitting problem of determining whether two or more such polygons can be split or continuously deformed without self intersection so that they occupy both sides of a plane without intersecting it we show that it also is in np finally we show that the problem of determining the genus of a polygonal knot a generalization of the problem of determining whether it is unknotted is in pspace we also give exponential worst case running time bounds for deterministic algorithms to solve each of these problems these algorithms are based on the use of normal surfaces and decision procedures due to w haken with recent extensions by w jaco and j l tollefson

marcum q functions and explicit kernels for stabilization of 2 2 linear hyperbolic systems with constant coefficients abstract we find the exact analytical solution to a goursat pde system governing the kernels of a backstepping based boundary control law that stabilizes a constant coefficient 2 2 system of first order hyperbolic linear pdes the solution to the goursat system is related to the solution of a simpler explicitly solvable goursat system through a suitable infinite series of powers of partial derivatives which is summed explicitly in terms of special functions including bessel functions and the generalized marcum q functions of the first order the marcum functions are common in certain applications in communications but have not appeared previously in control design problems the dependence of the explicit solutions with respect to system parameters is analyzed through several examples including the stabilization of a constant equilibrium for a quasi linear plant

deforming hypersurfaces of the sphere by their mean curvature is satisfied in 6 we studied hypersurfaces moving along their mean curvature vector in a general riemannian manifold n 1 it was shown that all hypersurfaces mo satisfying a suitable convexity condition will contract to a single point in finite time during this evolution here we want to show that in a spherical spaceform some convergence results can be obtained without assuming convexity for the initial hypersurface mo in particular we will see that some hypersurfaces do not contract during this flow but straighten out and become totally geodesic i e in case n 1 s 1 they converge to a big s to be precise let g gij and a hi j be the induced metric and the second fundamental form on m and denote by h giih j a 12 h jh j the mean curvature and the squared norm of the second fundamental form respectively

least area incompressible surfaces in 3 manifolds let m be a riemannian manifold and let f be a closed surface a map f f m is called least area if the area of f is less than the area of any homotopic map from f to m note that least area maps are always minimal surfaces but that in general minimal surfaces are not least area as they represent only local stationary points for the area function the existence of least area immersions in a homotopy class of maps has been established when the homotopy class satisfies certain injectivity conditions on the fundamental group 18 17 in this paper we shall consider the possible singularities of such immersions our results show that the general philosophy is that least area surfaces intersect least meaning that the intersections and self intersections of least area immersions are as small as their homotopy classes allow when measured correctly one should note that evidence supporting this view had been found by meeks yau in their embedding theorems for minimal disks and 2 spheres 13 143 our main result asserts that if a least area immersion is homotopic to an embedding then it has no self intersections which clearly exemplifies the above philosophy the precise result is the following