compact complete minimal immersions in mathbb r 3 in this paper we find for any arbitrary finite topological type a compact riemann surface m an open domain m c m with the fixed topological type and a conformal complete minimal immersion x m 3 which can be extended to a continuous map x m 3 such that x m is an embedding and the hausdorff dimension of x m is 1 we also prove that complete minimal surfaces are dense in the space of minimal surfaces spanning a finite set of closed curves in 3 endowed with the topology of the hausdorff distance

isoperimetric inequalities for submanifolds jellett minkowski s formula revisited in this paper we provide an extension to the jellett minkowski formula for immersed submanifolds within ambient manifolds which possess a pole and radial curvatures bounded from above or below using this generalized jellett minkowski formula allows us to focus on several isoperimetric problems specifically it becomes possible to concentrate on lower bounds for the isoperimetric quotients of any pre compact domain with a smooth boundary or on the isoperimetric profile of the submanifold and its modified volume in the particular case of a rotationally symmetric model space with strictly decreasing radial curvatures an aleksandrov type theorem is provided

rational recursion operators for integrable differential difference equations in this paper we introduce the concept of prehamiltonian pairs of difference operators demonstrate their connections with nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential difference equations we begin with a rigorous setup of the problem in terms of the skew field of rational pseudo difference operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational pseudo difference operators in particular we give a criteria for a rational operator to be weakly nonlocal a difference operator is called prehamiltonian if its image is a lie subalgebra with respect to the lie bracket on the difference field two prehamiltonian operators form a prehamiltonian pair if any linear combination of them is prehamiltonian then we show that a prehamiltonian pair naturally leads to a nijenhuis operator and a nijenhuis operator can be represented in terms of a prehamiltonian pair this provides a systematic method to check whether a rational operator is nijenhuis as an application we construct a prehamiltonian pair and thus a nijenhuis recursion operator for the differential difference equation recently discovered by adler and postnikov the nijenhuis operator obtained is not weakly nonlocal we prove that it generates an infinite hierarchy of local commuting symmetries we also illustrate our theory on the well known examples including the toda the ablowitz ladik and the kaup newell differential difference equations

a note on well posedness for camassa holm equation in this note we investigate the problem of well posedness for a shallow water equation with data having critical regularity our results are based on the use of besov spaces b2 rs which generalize the sobolev spaces hs with critical index s 3 2

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

small oscillations a new solution based on non noetherian symmetries a new non noetherian symmetry transformation for the small oscillations problem is exhibited a novel way to solve this classical problem based on conservation laws constructed using the new symmetry is presented applications of these results to other fields of physics are outlined

complete proper minimal surfaces in convex bodies of mathbbr 3 consider a convex domain b of r 3 we prove that there exist complete minimal surfaces which are properly immersed in b we also demonstrate that if d and dare convex domains with d bounded and the closure of d contained in dthen any minimal disk whose boundary lies in the boundary of d can be approximated in any compact subdomain of d by a complete minimal disk which is proper in d we apply these results to study the so called type problem for a minimal surface we demonstrate that the interior of any convex region of r 3 is not a universal region for minimal surfaces in the sense explained by meeks and perez in 9

the calabi yau conjectures for embedded surfaces in this talk i will discuss the proof of the calabi yau conjectures for embedded surfaces this is joint work with bill minicozzi cm9 the calabi yau conjectures about surfaces date back to the 1960s much work has been done on them over the past four decades in particular examples of jorge xavier from 1980 and nadirashvili from 1996 showed that the immersed versions were false we will show here that for embedded surfaces i e injective immersions they are in fact true their original form was given in 1965 in ca where e calabi made the following two conjectures about minimal surfaces see also s s chern page 212 of ch conjecture 1 prove that a complete minimal hypersurface in r must be unbounded calabi continued it is known that there are no compact minimal submanifolds of r or of any simply connected complete riemannian manifold with sectional curvature 0 a more ambitious conjecture is conjecture 2 a complete minimal hypersurface in r has an unbounded projection in every n 2 dimensional flat subspace

lipschitz metric for the novikov equation we consider the lipschitz continuous dependence of solutions for the novikov equation with respect to the initial data in particular we construct a finsler type optimal transport metric which renders the solution map lipschitz continuous on bounded sets of h 1 mathbb r cap w 1 4 mathbb r although it is not lipschitz continuous under the natural sobolev metric from an energy law due to the finite time gradient blowup by an application of thom s transversality theorem we also prove that when the initial data is in an open dense subset of h 1 mathbb r cap w 1 4 mathbb r the solution is piecewise smooth this generic regularity result helps us extend the lipschitz continuous metric to the general weak solutions our method of constructing the metric can be used to treat other kinds of quasi linear equations provided a good knowledge about the energy concentration

a practical method for the design of sliding mode controllers using linear matrix inequalities this paper considers the problem of selecting a sliding surface for a given system in an optimal way such that the performance of the reduced order system is balanced against the control costs required to maintain sliding this approach is quite different from many of the existing schemes in the literature in which no direct account is taken of the control effort two new numerical schemes are presented both involving convex optimization problems three simple examples are given to demonstrate the theory

equivalent lagrangians multidimensional case we generalize a theorem known for one dimensional nonsingular equivalent lagrangians l and to the multidimensional case in particular we prove that the matrix which relates the left hand sides of the euler lagrange equations obtained from l and is such that the trace of all its integer powers are constants of the motion we construct several multidimensional examples in which the elements of are functions of position velocity and time and prove that in some cases equivalence prevails even if det 0

sliding order and sliding accuracy in sliding mode control the synthesis of a control algorithm that stirs a nonlinear system to a given manifold and keeps it within this constraint is considered usually what is called sliding mode is employed in such synthesis this sliding mode is characterized in practice by a high frequency switching of the control it turns out that the deviation of the system from its prescribed constraints sliding accuracy is proportional to the switching time delay a new class of sliding modes and algorithms is presented and the concept of sliding mode order is introduced these algorithms feature a bounded control continuously depending on time with discontinuities only in the control derivative it is also shown that the sliding accuracy is proportional to the square of the switching time delay

stability proof for a well established super twisting parameter setting abstract several sufficient stability conditions exist for the super twisting algorithm for its tuning in practical applications however a popular parameter set is regularly used which is provided in literature without any stability proof this short note provides a novel simple sufficient stability condition which includes this very popular parameter setting

strict lyapunov functions for the super twisting algorithm a method to construct a family of strict lyapunov functions i e with negative definite derivative for the super twisting algorithm without or with perturbations is provided this second order sliding modes algorithm is widely used to design controllers observers and exact differentiators the proposed lyapunov functions ascertain finite time convergence provide an estimate of the convergence time and ensure the robustness of the finite time or ultimate boundedness for a class of perturbations wider than the classical ones for this algorithm since the lyapunov functions and their derivatives are quadratic forms the operation with them is as simple as for linear time invariant systems

improvements on stability analysis and stabilisation of constant and variable gain second order sliding mode algorithms via exact convex representations a novel lyapunov based methodology for both stability analysis and stabilisation of second order sliding mode algorithms in the presence of exogenous disturbances is presented nowadays second ord

a generalized nonisospectral camassa holm equation and its multipeakon solutions abstract motivated by the paper of beals sattinger and szmigielski 2000 3 we propose an extension of the camassa holm equation which also admits the multipeakon solutions the novel aspect is that our approach is mainly based on classic determinant technique furthermore the proposed equation is shown to possess a nonisospectral lax pair

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

homogeneity approach to high order sliding mode design it is shown that a general uncertain single input single output regulation problem is solvable only by means of discontinuous control laws giving rise to the so called high order sliding modes the homogeneity properties of the corresponding controllers yield a number of practically important features in particular finite time convergence is proved and asymptotic accuracy is calculated in a very general way in the presence of input noises discrete measurements and switching delays a robust homogeneous differentiator is included in the control structure thus yielding robust output feedback controllers with finite time convergence it is demonstrated that homogeneity features significantly simplify the design and investigation of a new family of high order sliding mode controllers finally simulation results are presented

some new integrable systems constructed from the bi hamiltonian systems with pure differential hamiltonian operators abstract when both hamiltonian operators of a bi hamiltonian system are pure differential operators we show that the generalized kupershmidt deformation gkd developed from the kupershmidt deformation in 10 offers an useful way to construct new integrable system starting from the bi hamiltonian system we construct some new integrable systems by means of the generalized kupershmidt deformation in the cases of harry dym hierarchy classical boussinesq hierarchy and coupled kdv hierarchy we show that the gkd of harry dym equation gkd of classical boussinesq equation and gkd of coupled kdv equation are equivalent to the new integrable rosochatius deformations of these soliton equations with self consistent sources we present the lax pair for these new systems therefore the generalized kupershmidt deformation provides a new way to construct new integrable systems from bi hamiltonian systems and also offers a new approach to obtain the rosochatius deformation of soliton equation with self consistent sources

simple homogeneous sliding mode controller high order sliding mode hosm control is known to provide for finite time exact output regulation of uncertain systems with known relative degrees yet the corresponding universal hosm controllers are typically constructed by special recursive procedures and have complicated form we propose two new families of homogeneous hosm controllers of a very simple form lyapunov functions are provided for a significant part of the first family controllers the second family consists of quasi continuous controllers which can be done arbitrarily smooth everywhere outside of the hosm manifold a regularization procedure ensures high accuracy output regulation by means of control with required smoothness level output feedback controllers are constructed controllers of the orders 3 5 are demonstrated

super twisting algorithm in presence of time and state dependent perturbations abstractin this paper generalized super twisting algorithm is applied to a class of systems whose perturbation and uncertain control coefficient are time and state dependent a non smooth strict lyapunov function is used to obtain the conditions for the global finite time stability and to estimate the time of convergence

an lmi approach for second order sliding set design using piecewise lyapunov functions abstract this paper pursues recent results on second order sliding set design for uncertain linear systems with matched uncertainties a methodology that allows the designer to use a lesser number of derivatives it is based on the direct lyapunov method and its conditions are expressed as linear matrix inequalities which are efficiently solved via convex optimization techniques in contrast with former reports on this matter the class of lyapunov functions in this work is enlarged to the set of those which are piecewise c 1 which naturally leads to design advantages as well as to a more natural adaptation of the lyapunov function to the discontinuities arising in the sliding set design methodology simulation examples are included to show the effectiveness of the proposed approach

well posedness and analyticity for the cauchy problem for the generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem for the generalized camassa holm equation which was proposed by hakkaev and kirchev s hakkaev k kirchev local well posedness and orbital stability of solitary wave solutions for the generalized camassa holm equation comm partial differential equations 30 2005 761 781 we establish the local well posedness for a range of besov spaces moreover with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time which extends some results of danchin r danchin a few remarks on the camassa holm equation differential integral equations 14 2001 953 988 and himonas and misiolek a himonas g misiolek analyticity of the cauchy problem for an integrable evolution equation math ann 327 2003 575 584 to more general equations

geodesic deviation and first integrals of motion generalized killing tensors gkt and generalized conformal killing tensors gckt are defined as the class of totally symmetric tensor fields that generate solutions of a suitable inhomogeneous equation of geodesic deviation along arbitrary and null geodesics respectively a geometric interpretation of these fields as generators of jacobi fields along arbitrary and null geodesics is also given it is shown that well known fields such as killing tensors conformal killing tensors and geodesic collineations belong to the above classes finally first integrals of geodesic motion concomitant with the existence of gkt s and gckt s are determined

on the solvability of the painleve vi equation a rigorous method was introduced by fokas and zhou 1992 for studying the riemann hilbert problem associated with the painleve ii and iv equations the same methodology has been applied to the painleve i iii and v equations in this paper we will apply the same methodology to the painleve vi equation we will show that the cauchy problem for the painleve vi equation admits in general a global meromorphic solution in t furthermore the special solution which can be written in terms of a hypergeometric function is obtained via solving the special case of the riemann hilbert problem

every bordered riemann surface is a complete conformal minimal surface bounded by jordan curves in this paper we find approximate solutions of certain riemann hilbert boundary value problems for minimal surfaces in r and null holomorphic curves in c for any n 3 with this tool in hand we construct complete conformally immersed minimal surfaces in r which are normalized by any given bordered riemann surface and have jordan boundaries we also furnish complete conformal proper minimal immersions from any given bordered riemann surface to any smoothly bounded strictly convex domain of r which extend continuously up to the boundary for n 5 we find embeddings with these properties

symmetries of lagrangians and of their equations of motion a new kind of lagrangian symmetry is defined in such a way that the resulting set of lagrangian symmetries coincides with the set of symmetries of its equations of motion several constants of motion may be associated to each of the new symmetry transformations one example is presented

symmetries of space time and geodesic symmetries it is shown how to construct symmetries of the geodesic equation starting from space time symmetries the constants of motion associated with space time symmetries are recovered and a few new ones are found using non noetherian conservation theorems explicit examples are presented

a jordan curve spanned by a complete minimal surface in this paper we construct complete conformal minimal immersions f mathbb d longrightarrow mathbb r 3 which admit continuous extensions to the closed disk f overline mathbb d longrightarrow mathbb r 3 moreover f mathbb s 1 mathbb s 1 rightarrow f mathbb s 1 is a homeomorphism and f mathbb s 1 is a non rectifiable jordan curve with hausdorff dimension 1

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

nonlinear sliding mode control design an lmi approach abstract this paper presents a novel nonlinear sliding mode control methodology for systems with both matched and unmatched perturbations including parametric uncertainties instead of traditional approaches where uncertainties and nonlinearities are coped with via linear nominal models and linear sliding surfaces the proposed approach incorporates exact convex expressions to represent both the nonlinear surface and the system thus allowing a significant chattering reduction moreover thanks to the convex form of the nonlinear nominal model when combined with the direct lyapunov method it leads to linear matrix inequalities which are efficiently solved via convex optimization techniques illustrative examples are provided

an integrable semi discretization of the camassa holm equation and its determinant solution an integrable semi discretization of the camassa holm ch equation is presented the keys of its construction are bilinear forms and determinant structure of solutions of the ch equation determinant formulas of n soliton solutions of the continuous and semi discrete camassa holm equations are presented based on determinant formulas we can generate multi soliton multi cuspon and multi soliton cuspon solutions numerical computations using the integrable semi discrete camassa holm equation are performed it is shown that the integrable semi discrete camassa holm equation gives very accurate numerical results even in the cases of cuspon cuspon and soliton cuspon interactions the numerical computation for an initial value condition which is not an exact solution is also presented

limit sets for complete minimal immersions in this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a domain g subset mathbb r 3 with the boundary partial g subset c 2 we prove that the second fundamental form of the surface g is nonnegatively defined at every point of the limit set of such immersions

variable gain super twisting sliding mode control in this note a novel lyapunov based variable gain super twisting algorithm sta is proposed it ensures for linear time invariant systems the global finite time convergence to the desired sliding surface when the matched perturbations uncertainties are lipschitz continuous functions of time that are bounded together with their derivatives by known functions the proposed algorithm has similar properties to the variable gain first order sliding mode control but it provides alleviation to the chattering phenomenon the results are verified experimentally

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

reaching time estimation for super twisting second order sliding mode controller via lyapunov function designing a method of a lyapunov functions design based on resolving of the first order partial differential equation of a special type is presented a successful resolution of this equation guaranties a finite time convergence for the corresponding dynamics given by an ordinary differential equation with a discontinuous right hand side it is also provides an analytical formula for an upper bound of the corresponding reaching time the suggested method is applied to the problem of a stability analysis and a reaching time estimation of the so called ldquosuper twistingrdquo controller the estimation accuracy is confirmed by numerical example

finite propagation speed for the camassa holm equation we prove that any classical solution of the camassa holm equation will have compact support if its initial data has this property

well posedness of a class of solutions to an integrable two component camassa holm system abstract in this paper we study a class of solutions to the cauchy problem for an integrable two component camassa holm system with the initial data u 0 0 1 h 1 r w 1 r l 2 r l r based on characteristics we study a corresponding ode and obtain a unique local solution by applying the contraction mapping principle we prove local existence and uniqueness of the solution to the camassa holm system by constructing a solution obtained from the ode and studying the regularity of the solution finally we show continuous dependence of the solution on the initial data in some weak sense

geodesic first integrals with explicit path parameter dependence in riemannian space times in a riemannian space vn general formulas are obtained for geodesic first integrals which are mth order polynomials in the tangent vector and which are assumed to depend explicitly on the path parameter s it is found that such first integrals must also be polynomials in s necessary and sufficient conditions are found for the existence of these first integrals the existence of many well known symmetries such as homothetic motions scale change affine collineations conformal motions projective collineations conformal collineations or special curvature collineations are shown to be sufficient for the existence of such first integrals with explicit path parameter dependence to illustrate the theory geodesic first integrals of this type have been calculated for four riemannian space times of general relativity