curve shortening and the rendezvous problem for mobile autonomous robots if a smooth closed and embedded curve is deformed along its normal vector field at a rate proportional to its curvature it shrinks to a circular point this curve evolution is called euclidean curve shortening and the result is known as the gage hamilton grayson theorem motivated by the rendezvous problem for mobile autonomous robots we address the problem of creating a polygon shortening flow a linear scheme is proposed that exhibits several analogues to euclidean curve shortening the polygon shrinks to an elliptical point convex polygons remain convex and the perimeter of the polygon is monotonically decreasing

on initial boundary value problems in bounded and unbounded domains for a class of nonlinear hyperbolic equations of the third order abstract for the nonlinear hyperbolic equation u 2 1 f x t u u 1 0 u 2 0 u 0 1 u 1 1 problems on bounded solutions in a half strip and in a strip are studied unimprovable conditions of local and global solvability and unique solvability of these problems are established

a dissipative hyperbolic affine curve flow abstract a new hyperbolic affine geometric flow with dissipative term is proposed the equations satisfied by support functions and the graph of the curve under this dissipative flow give rise to fully nonlinear hyperbolic equations we obtain the existence for local solutions of this flow by reducing the flow to a first order system the equations for both perimeter and area of closed curves under the flow are also obtained based on this we show that for a closed curve the solution of this flow converges to a point in finite time furthermore by a method of lefloch smocyk for studying the hyperbolic mean curvature flow global existence of the solution is established

dynamics of one and two dimensional fronts in a bistable equation with time delayed global feedback propagation failure and control mechanisms we study the evolution of fronts in a bistable equation with time delayed global feedback in the fast reaction and slow diffusion regime this equation generalizes the hodgkin grafstein and allen cahn equations we derive a nonlinear equation governing the motion of fronts which includes a term with delay in the one dimensional case this equation is linear we study the motion of one and two dimensional fronts finding a much richer dynamics than for the previously studied cases without time delayed global feedback we explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a hopf bifurcation as the delay time increases we show that for certain delay times the prevailing phase is different from that corresponding to the system in the absence of global coupling numerical simulations of the partial differential equation are in agreement with the analytical predictions

hyperbolic mean curvature flow with a forcing term evolution of plane curves abstract in this paper we study the evolution of closed strictly convex plane curves moving by the hyperbolic mean curvature flow with a forcing term it is shown that the flow admits a unique short time smooth solution and the convexity of the curves is preserved during the evolution when the forcing term is a negative constant we prove the curves either converge to a point or a c 0 curve for a positive constant forcing term the flow has a unique smooth solution in any finite time and expands to infinity as t tends to infinity if the initial curvature is smaller than m the flow will blow up in a finite time if the initial curvature is larger than m

asymptotic closeness to limiting shapes for expanding embedded plane curves we show that for embedded or convex plane curves expansion the difference u x t r t in support functions between the expanding curves t and some expanding circles ct with radius r t has its asymptotic shape as t moreover the isoperimetric difference l2 4 a is decreasing and it converges to a constant mathfrak s 0 if the expansion speed is asymptotically a constant and the initial curve is not a circle for convex initial curves if the expansion speed is asymptotically infinite then l2 4 a decreases to mathfrak s 0 and there exists an asymptotic center of expansion for t

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

geometry of pdes i integral bordism groups in pdes we improve some our previous theorems on the calculation of integral bordism groups of formally integrable and completely integrable pdes emphasizing the role played by singular solutions and weak solutions some applications to interesting pdes defined on finite dimensional manifolds are also considered

convergence in a quasilinear parabolic equation with time almost periodic boundary conditions abstract consider the problem u t a u x u x x f u x x 1 t 0 u x 1 t g t t 0 where g t is an almost periodic function we prove the following convergence results if a time global solution u is bounded then it converges to an almost periodic solution which is unique up to space shift stable and asymptotically stable if u goes to infinity with positive lower average speed then it converges to an almost periodic traveling wave which is unique up to space shift stable asymptotically stable and has positive average speed

two point higher order bvps with singularities in phase variables the existence of solutions for singular higher order differential equations with the lidstone or the n p boundary conditions is proved the right hand sides of differential equations can have singularities in the zero value of their phase variables and so higher derivatives of solutions changing their signs can pass through these singularities proofs are based on the method of a priori estimates the degree theory arguments and on vitali s convergence theorem

convergence in a quasilinear parabolic equation with neumann boundary conditions abstract consider the problem u t a u x u x x f u x x 1 t 0 u x 1 t k t u 1 t t 0 where k are smooth functions which are periodic in both t and u brunovsk et al proved in their paper brunovsk et al 1992 8 that if a time global solution u is bounded then it converges to a periodic solution we prove that if u is unbounded from above then it converges to a periodic traveling wave v x t c t in case k k t or k k u where v is a time periodic function and c 0 in addition the periodic traveling wave is unique up to space shifts or time shifts it is stable and asymptotically stable the average traveling speed c and the instantaneous speed v t c are also studied

blow up rates for the general curve shortening flow abstract the blow up rates of derivatives of the curvature function will be presented when the closed curves contract to a point in finite time under the general curve shortening flow in particular this generalizes a theorem of m e gage and r s hamilton about mean curvature flow in r 2

bv solutions for the cauchy problem of a quasilinear hyperbolic equation with σ finite borel measure and nonlinear source the aim of this paper is to discuss the cauchy problem of the quasilinear hyperbolic equation of the form ut um x up with nonnegative finite borel measures as initial conditions where m 1 0 p 1 are some given real numbers in particular the existence of bv solutions for the above problem is obtained

uniqueness of solutions of degenerate diffusion equations with measures as initial conditions on considere le probleme de cauchy u t u m u p sur r n 0 t u x 0 x sur r n ou m 1 p 1 n 1 0 1 p m 2 n et une mesure finie

piecewise approximation of contours through scale space selection of dominant points this paper describes a method of approximating a shape contour with a polygon the polygon vertices are extracted from the curvature extremes through a scale space description of the contour via linear diffusion these vertices are located on the contour points where the sharper changes of the contour directions occur using a proper strategy a set of extremes that result in a given approximation level is chosen by adding new vertices the approximation level can be improved and a scalable representation of the contour is identified this method results in an approximation that discriminates local from global geometric features and provides a good visual representation of the original contour this polygonal approximation method is used for scalable encoding of the shape contours in this regard an encoding technique suitable for scalable polygonal approximation has been developed we show that encoding the approximated polygons result in a good relation between the distortion and the bitrate finally we show that in addition to coding this method can be efficiently used for shape comparison and shape retrieval

on solvability of boundary value problems for higher order nonlinear hyperbolic equations abstract in the rectangle 0 a 0 b for the nonlinear hyperbolic equation u m n i 0 m 1 h 1 i x u i n k 0 n 1 h 2 k y u m k f x y u u m 1 n 1 the boundary value problems of the type l 1 i u y 0 i 1 m l 2 k u x 0 k 1 n are considered where l 1 i c m 1 0 a r i 1 m and l 2 k c n 1 0 b r k 1 n are linear bounded functionals sufficient conditions of solvability and unique solvability of the general problem and its particular cases nicoletti type dirichlet lidstone and periodic problems are established

minimal surfaces based object segmentation a geometric approach for 3d object segmentation and representation is presented the segmentation is obtained by deformable surfaces moving towards the objects to be detected in the 3d image the model is based on curvature motion and the computation of surfaces with minimal areas better known as minimal surfaces the space where the surfaces are computed is induced from the 3d image volumetric data in which the objects are to be detected the model links between classical deformable surfaces obtained via energy minimization and intrinsic ones derived from curvature based flows the new approach is stable robust and automatically handles changes in the surface topology during the deformation

differential and numerically invariant signature curves applied to object recognition we introduce a new paradigm the differential invariant signature curve or manifold for the invariant recognition of visual objects a general theorem of e cartan implies that two curves are related by a group transformation if and only if their signature curves are identical the important examples of the euclidean and equi affine groups are discussed in detail secondly we show how a new approach to the numerical approximation of differential invariants based on suitable combination of joint invariants of the underlying group action allows one to numerically compute differential invariant signatures in a fully group invariant manner applications to a variety of fundamental issues in vision including detection of symmetries visual tracking and reconstruction of occlusions are discussed

non parametric mean curvature evolution with boundary conditions soit un domaine borne de r n n 2 de frontiere lisse et soit a l operateur de surface minimale donne par a d i a i p a i p i 1 p 2 1 2 on considere des surfaces dont la vitesse en direction de leur normale unite est egale a la courbure moyenne on etudie l equation d evolution qui en resulte u 1 du 2 1 2 au 0 dans 0 t

on the strong maximum principle and the large time behavior of generalized mean curvature flow with the neumann boundary condition abstract a generalized mean curvature flow is considered in a cylindrical domain under the right angle boundary condition its large time behavior is studied by analyzing the level set equation of the flow when the initial hypersurface is not necessarily graph like for a convex domain it is shown that the solution of the level set equation converges to a function whose level sets are perpendicular to the lateral boundary of the domain as time tends to infinity when the initial data is constant outside some bounded set for this purpose a version of the strong maximum principle is established for the level set minimal surface equation although the equation is degenerate

continuity of solutions of a singular parabolic equation which is of parabolic divergence form since 3 0 a e however it is the solution u of 0 1 which will be continuous while the solution u of 0 2 need not be in certain physical applications u corresponds to temperature while u is the enthalpy the difficulty in the analysis of this equation written in the form 0 2 stems from the fact that c need not be bounded or bounded away from zero thus the equation may be of degenerate or singular parabolic type as particular cases of the equations covered by our results we mention the three model problems

unified models from gauged supergroups iii the super 2 sphere osp 3 2 osp 2 2 the formalism of coset space dimensional reduction for internal supersymmetry p d jarvis j math phys 31 1783 1990 p d jarvis and d s mcanally j math phys 33 399 1992 is extended in the supersymmetric case and the homogeneous superspace osp 3 2 osp 2 2 is studied in detail with gauge supergroup osp m 2 2

global regularizing flows with topology preservation for active contours and polygons active contour and active polygon models have been used widely for image segmentation in some applications the topology of the object s to be detected from an image is known a priori despite a complex unknown geometry and it is important that the active contour or polygon maintain the desired topology in this work we construct a novel geometric flow that can be added to image based evolutions of active contours and polygons in order to preserve the topology of the initial contour or polygon we emphasize that unlike other methods for topology preservation the proposed geometric flow continually adjusts the geometry of the original evolution in a gradual and graceful manner so as to prevent a topology change long before the curve or polygon becomes close to topology change the flow also serves as a global regularity term for the evolving contour and has smoothness properties similar to curvature flow these properties of gradually adjusting the original flow and global regularization prevent geometrical inaccuracies common with simple discrete topology preservation schemes the proposed topology preserving geometric flow is the gradient flow arising from an energy that is based on electrostatic principles the evolution of a single point on the contour depends on all other points of the contour which is different from traditional curve evolutions in the computer vision literature

generation and propagation of interfaces for reaction diffusion equations abstract this paper is concerned with the asymptotic behavior as e 0 of the solution ue of the reaction diffusion equation in r n r u t 1 2 0 where is the derivative of a bistable potential we show that if the initial data u 0 has values in both domains of attraction of the potential then an interface will develop in a short time o e 2 ln e we also show that if the wells of the potential are of equal depth then this interface will propagate with normal velocity equal to its mean curvature our result is valid as long as the interface remains smooth if the initial interface is compact and n 2 then the interface will disappear in a finite time but not if n 1 in case the depths of the wells are not equal then in order to obtain get reasonable results we must work on the scaled time s t e slower time scale in this scale we show that the interface moves with a constant speed proportional to the difference of the depths of the two wells along the normal towards the domain of the deeper well this result is valid for all s 0 and does not actually depend on the regularity of the interface we also extend the above results to the homogeneous second initial boundary value problem in case the depths of the wells are equal and the initial interface is orthogonal to the boundary we prove that the interface moves with normal velocity equal to its mean curvature provided that there is a family of hypersurfaces which moves according to their mean curvature and intersect the boundary orthogonally

unified models from gauged supergroups ii exceptional series d 2 1 α g 3 f 4 previously elaborated principles j math phys 31 1783 1990 for dimensional reduction of models of gauged supergroups over supercoset spaces are applied to the exceptional superalgebras the superspace is parametrized by a pair of grassmann coordinates

higgs fields from symmetric connections the bundle picture the derivation forgacs and manton of a classical gauge plus higgs field theory from a purely symmetric gauge theory by dimensional reduction is translated into the pictorial language of symmetric connections on principal bundles the bundle description provides a better understanding of symmetric gauge theories its global nature excludes some gauge group symmetry group combinations and shows that the reduced gauge group is larger than the centralizer ch as previously thought in general it is a certain factor group nk k involving both the gauge and symmetry groups and contains ch as a normal subgroup the commonly used f2 lagrangian of symmetric gauge theories will not prescribe the dynamics of fields associated with this larger group unless these fields are suitably constrained it is shown that a constrained kaluza klein metric is required to construct the f2 lagrangian thus pointing the way to include dynamics for the nk k group

dynamic active contours for visual tracking visual tracking using active contours is usually set in a static framework the active contour tracks the object of interest in a given frame of an image sequence a subsequent prediction step ensures good initial placement for the next frame this approach is unnatural the curve evolution gets decoupled from the actual dynamics of the objects to be tracked true dynamical approaches exist all being marker particle based and thus prone to the shortcomings of such particle based implementations in particular topological changes are not handled naturally in this framework the now classical level set approach is tailored for evolutions of manifolds of codimension one however dynamic curve evolution is at least a codimension two problem we propose an efficient level set based approach for dynamic curve evolution which addresses the artificial separation of segmentation and prediction while retaining all the desirable properties of the level set formulation it is based on a new energy minimization functional which for the first time puts dynamics into the geodesic active contour framework

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

existence and uniqueness for a quasilinear hyperbolic equation with σ finite borel measures as initial conditions abstract the aim of this paper is to discuss the cauchy problem for a quasilinear hyperbolic equation of the form u t u m x u p 0 m 1 p 0 with finite borel measures as initial conditions in particular the existence and uniqueness of global bv solutions with any nonnegative finite borel measures as initial conditions are obtained in the case m p m 1

on a nonclassical problem for some quasilinear hyperbolic equation u 0 t bu l t 0itit 1 3 here t is a positive continuous function on 0 t 0 b 0 p 1 0 and l are real constants equations of type 1 1 arise e g in gas dynamics 10 13 while 1 2 1 3 is a generalization of periodic boundary conditions our aim is to 1nd all global solutions to problem 1 1 1 3 note that the boundary conditions are consistent and may admit several solutions

source type solutions of the porous media equation with absorption the fast diffusion case in this paper we consider the cauchy problem u t u m u p on s t r n 0 t u x 0 0 x r n 0 where n 1 p 1 and 0 m 1

dynamics of an unbounded interface between ordered phases we investigate the evolution of a single unbounded interface between ordered phases in two dimensional ising ferromagnets that are endowed with single spin flip zero temperature glauber dynamics we examine specifically the cases where the interface initially has either one or two corners in both examples the interface evolves to a limiting self similar form we apply the continuum time dependent ginzburg landau equation and a microscopic approach to calculate the interface shape for the single corner system we also discuss a correspondence between the interface and the young diagram that represents the partition of the integers

on a simple maximum principle technique applied to equations on the circle abstract for a class of general quasilinear equations on s 1 we show that by a very simple maximum principle technique as long as the solution stays finite all of its derivatives also remain finite some specific examples are given under suitable assumptions we also derive exponential decay of the derivatives of the solution

explicit construction of the exceptional superalgebras f 4 and g 3 a general method for constructing the extension of an ordinary lie algebra a0 to a superalgebra a0 a1 is given once one knows in which representation of a0 the odd generators a1 are explicit matrix representations for the superalgebras f 4 and g 3 and for ordinary algebras e8 f4 and g2 are presented

source solutions and asymptotic behavior in conservation laws abstract we study the uniqueness of the solutions to the scalar conservation law u t u x 0 when the initial datum is a finite measure the case of a dirac mass is particularly emphasized it is shown how it provides a description of the asymptotic behavior of the solutions initiated by an arbitrary integrable function this behavior is proved to depend on one parameter in the case when is odd while it depends on two when is convex

unified models from gauged supergroups i dimensional reduction formalism unitary and orthosymplectic series principles are elaborated for constructing consistent grand unified models gut s plus higgs sectors from dimensional reduction of gauged supergroups over graded coset spaces a search of possible gut groups is made for unitary and orthosymplectic supergroups over a space parametrized by a single pair of grassmann coordinates

the hele shaw problem and area preserving curve shortening motions we prove existence locally in time of a solution of the following hele shaw problem given a simply connected curve contained in a smooth bounded domain find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve in and whose boundary value on the curve equals its curvature we show that this motion is a curve shortening motion which does not change the area of the region enclosed by the curve in case is the whole plane 2 we also show that if the initial curve is close to an equilibrium curve i e to a circle then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

the variational problem for a certain space time functional defined on planar closed curves abstract we consider a variational problem for a certain space time functional defined on planar closed curves the functional is related to the functional appeared in bellettini and mugnai 2008 4 the variational problem is stated as follows let 0 and 1 denote planar closed curves and t be a positive constant minimize the space time functional over family of planar closed curves which change from 0 at time t 0 into 1 at time t t concerning the variational problem we prove the existence of minimizer in a radially symmetric class and determine all the minimizers for each initial final data moreover we show that there exists a unique non radially symmetric critical point in a neighborhood of a certain radially symmetric minimizer

existence and uniqueness of periodic solutions for higher order hyperbolic partial differential equations abstract assuming the smoothness and a generalized lipschitz condition we establish the existence and uniqueness of the periodic solutions of higher order nonlinear hyperbolic partial differential equations

on the planar dual minkowski problem abstract in this paper we resolve the planar dual minkowski problem proposed by huang et al 2016 31 for all positive indices without any symmetry assumption more precisely given any q 0 and function f on s 1 bounded by two positive constants we show that there exists a convex body in the plane containing the origin in its interior whose dual curvature measure c q has density f in particular if f is smooth then is also smooth

homogenization limit of a parabolic equation with nonlinear boundary conditions abstract consider the parabolic equation e u t a u x u x x f u x 1 x 1 t 0 with nonlinear boundary conditions u x 1 t g u 1 t e nbc u x 1 t g u 1 t e where e 0 is a parameter g is a function which takes values near its supremum frequently each almost periodic function is a special example of g we consider a time global solution u e of e nbc and show that its homogenization limit as e 0 is the solution of e with linear boundary conditions lbc x 1 t sup g x 1 t sup g provided moves upward monotonically when g is almost periodic lou preprint 21 obtained the unique almost periodic traveling wave u e of e nbc this paper proves that the homogenization limit of u e is a classical traveling wave of e lbc

deforming convex curves with fixed elastic energy abstract this paper presents a curve flow which preserves the elastic energy of the evolving curve if the initial curve is a planar simple and smooth curve with positive curvature then the local and global existence of the flow is proved under this flow the evolving curve will converge to a finite circle in the c metric as time goes to infinity

representations of the superalgebra f4 and young supertableaux the authors present a method for constructing typical as well as atypical finite dimensional representations of the superalgebra f4 for such a purpose young supertableaux are introduced

the vallee poussin problem for higher order nonlinear hyperbolic equations for higher order nonlinear hyperbolic equations nonclassical boundary value problems of vallee poussin type are studied and unimprovable in a sense sufficient conditions of solvability unique solvability and well posedness are established