stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

well posedness of a highly nonlinear shallow water equation on the circle abstract we present a comprehensive introduction and overview of a recently derived model equation for waves of large amplitude in the context of shallow water waves and provide a literature review of all the available studies on this equation furthermore we establish a novel result concerning the local well posedness of the corresponding cauchy problem for space periodic solutions with initial data from the sobolev space h s on the circle for s 3 2

new exact traveling wave solutions for the modified form of degasperis procesi equation abstract with the aid of symbolic computation a generalized auxiliary equation method is proposed to construct more general exact solutions to some types of nlpdes in this letter using the auxiliary equation method some new solitary and traveling wave solutions of such nonlinear mdp equations given by u t u xxt 4 u 2 u x 3 u x u xx uu xxx are obtained in a very general form several interesting special cases of these general solutions are also discussed

global solutions and blow up phenomena to a shallow water equation abstract a nonlinear shallow water equation which includes the famous camassa holm ch and degasperis procesi dp equations as special cases is investigated the local well posedness of solutions for the nonlinear equation in the sobolev space h s r with s 3 2 is developed provided that 1 x 2 u 0 does not change sign u 0 h s s 3 2 and u 0 l 1 r the existence and uniqueness of the global solutions to the equation are shown to be true in u t x c 0 h s r c 1 0 h s 1 r conditions that lead to the development of singularities in finite time for the solutions are also acquired

an euler relation for valuations on polytopes abstract a locally finite point set such as the set z n of integral points gives rise to a lattice of polytopes in euclidean space taking vertices from the given point set we develop the combinatorial structure of this polytope lattice and derive euler type relations for valuations on polytopes using the language of mobius inversion in this context a new family of inversion relations is obtained thereby generalizing classical relations of euler dehn sommerville and macdonald

boundary control of viscosity degasperis procesi equation abstract this paper investigates the boundary control problem of the viscosity degasperis procesi equation in 0 1 results of the existence of a solution to the viscosity degasperis procesi equation in a short time interval under the boundary condition are given in the paper it proves the global exponential stability under the given boundary condition in l 2 h 1 h 2 h 3 space

centro affine tensor valuations abstract all measurable sl n covariant symmetric tensor valuations on convex polytopes containing the origin in their interiors are completely classified it is shown that essentially the only examples of such valuations are the moment tensor and a tensor derived from l p surface area measures this generalizes and unifies earlier results for the scalar vector and matrix valued case

sl m c equivariant and translation covariant continuous tensor valuations abstract the space of continuous sl m c equivariant m 2 and translation covariant valuations taking values in the space of real symmetric tensors on c m r 2 m of rank r 0 is completely described the classification involves the moment tensor valuation for r 1 and is analogous to the known classification of the corresponding tensor valuations that are sl 2 m r equivariant although the method of proof cannot be adapted

traveling wave solutions of the camassa holm equation all weak traveling wave solutions of the camassa holm equation are classified we show that in addition to smooth solutions there are a multitude of traveling waves with singularities peakons cuspons stumpons and composite waves

exact solitary and periodic wave solutions of the k 2 2 equation defocusing branch an auxiliary elliptic equation method is presented for constructing exact solitary and periodic travelling wave solutions of the k 2 2 equation defocusing branch some known results in the literature are recovered more efficiently and some new exact travelling wave solutions are obtained also new stationary wave solutions are obtained

integral geometry of unitary area measures abstract the existence of kinematic formulas for area measures with respect to any connected closed subgroup of the orthogonal group acting transitively on the unit sphere is established in particular the kinematic operator for area measures is shown to have the structure of a co product in the case of the unitary group the algebra associated to this co product is described explicitly in terms of generators and relations as a consequence a simple algorithm that yields explicit kinematic formulas for unitary area measures is obtained

global existence for a new periodic integrable equation we establish the local well posedness for a new periodic integrable equation we show that the equation has classical solutions that blowup in finite time as well as classical solutions which exist globally in time

ellipsoids and matrix valued valuations a classification is obtained of borel measurable gl n covariant symmetric matrix valued valuations on the space of n dimensional convex polytopes the only ones turn out to be the moment matrix corresponding to the classical legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by lutwak yang and zhang a classical concept from mechanics is the legendre ellipsoid or ellipsoid of inertia 2k associated with a convex body k r it can be defined as the unique ellipsoid centered at the center of mass of k such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of k if we fix a scalar product x y for x y r 2k can be defined by the moment matrix m2 k of k this is the n n matrix with coefficients

the cramer rao inequality for star bodies associated with each body k in euclidean n space mathbb r sp n is an ellipsoid gamma sb 2k called the legendre ellipsoid of k it can be defined as the unique ellipsoid centered at the body s center of mass such that the ellipsoid s moment of inertia about any axis passing through the center of mass is the same as that of the body in an earlier paper the authors showed that corresponding to each convex body k subset mathbb r sp n is a new ellipsoid gamma sb 2 k that is in some sense dual to the legendre ellipsoid the legendre ellipsoid is an object of the dual brunn minkowski theory while the new ellipsoid gamma sb 2 k is the corresponding object of the brunn minkowski theory the present paper has two aims the first is to show that the domain of gamma sb 2 can be extended to star shaped sets the second is to prove that the following relationship exists between the two ellipsoids if k is a star shaped set then gamma sb 2 k subset gamma sb 2k with equality if and only if k is an ellipsoid centered at the origin this inclusion is the geometric analogue of one of the basic inequalities of information theory the cramer rao inequality

autonomous ovsyannikov theorem and applications to nonlocal evolution equations and systems abstract this work presents an ovsyannikov type theorem for an autonomous abstract cauchy problem in a scale of decreasing banach spaces which in addition to existence and uniqueness of solution provides an estimate about the analytic lifespan of the solution then using this theorem it studies the cauchy problem for camassa holm type equations and systems with initial data in spaces of analytic functions on both the circle and the line which is the main goal of this paper finally it studies the continuity of the data to solution map in spaces of analytic functions

initial boundary value problems for nonlinear dispersive wave equations in this paper we study initial value boundary problems of two types of nonlinear dispersive wave equations on the half line and on a finite interval subject to homogeneous dirichlet boundary conditions we first prove local well posedness of the rod equation and of the b equation for general initial data furthermore we are able to specify conditions on the initial data which on the one hand guarantee global existence and on the other hand produce solutions with a finite life span in the case of finite time singularities we are able to describe the precise blow up scenario of breaking waves our approach is based on sharp extension results for functions on the half line or on a finite interval and several symmetry preserving properties of the equations under discussion

towered waves and anti waves in the generalized degasperis procesi equation new traveling wave solutions of the generalized degasperis procesi equation are investigated the solutions are characterized by three parameters using an improved qualitative method abundant traveling wave solutions such as smooth waves peaked waves cusped waves compacted waves looped waves and fractal like waves are obtained especially some strange composite wave solutions such as towered waves and their anti waves are first given we also study the limiting behavior of all periodic solutions as the parameters trend to some special values

global well posedness and limit behavior of the solutions to the viscous degasperis procesi equation this paper is concerned with the global well posedness and limit behavior of the solutions to the viscous degasperis procesi equation we prove the global existence and uniqueness of solution to the viscous degasperis procesi equation for u0 l2 r the local well posedness for u0 l2 r has been proven based on the energy estimate the strong and weak limit behaviors of solution to the viscous degasperis procesi equation are obtained

well posedness travelling waves and geometrical aspects of generalizations of the camassa holm equation abstract in this paper we consider a five parameter equation including the camassa holm and the dullin gottwald holm equations among others we prove the existence and uniqueness of solutions of the cauchy problem using kato s approach conservation laws of the equation up to second order are also investigated from these conservation laws we establish some properties for the solutions of the equation and we also find a quadrature for it the quadrature obtained is of capital importance in a classification of bounded travelling wave solutions we also find some explicit solutions given in terms of elliptic integrals finally we classify the members of the equation describing pseudo spherical surfaces

the polytope algebra let f be an ordered field and let p denote the family of all convex polytopes in the d dimensional vector space v over f the universal abelian group corresponding to the translation invariant valuations on p has generators p for p p with 0 satisfying the relations v p q p p q whenever p q p q p and t p t p for p p and t v with multiplication induced by m p q p q is almost a graded commutative algebra over f in that dr 0 r with 0 z r a vector space over f r 1 and r s r s r s 0 r 0 forr d the dilatation d p p for p p and f is such that x rx for x r and 0 negative dilatations arise from the euler map e p p f 1 dimf f the sum extending over all faces f of p since x rx for x r and 0 separating group homomorphisms for are the frame functionals which give the volumes of the faces of polytopes determined by successive support hyperplanes in sequences of directions two isomorphisms on are described one related to cones of outer normal vectors and the other to the polytope groups obtained from by discarding polytopes of dimension less than d various applications of the polytope algebra are given including a theory of mixed polytopes which has implications for mixed valuations

on existence and uniqueness of the global weak solution for a shallow water equation abstract a shallow water equation including the camassa holm and degasperis procesi equations as special cases is investigated provided that initial value u 0 h 1 r u 0 l 1 r and 1 x 2 u 0 does not change sign the existence and uniqueness of global weak solution for the equation are established

on the cauchy problem for the periodic generalized degasperis procesi equation abstract we mainly study the cauchy problem of the periodic generalized degasperis procesi equation first we establish the local well posedness for the equation second we give the precise blow up scenario a conservation law and prove that the equation has smooth solutions which blow up in finite time finally we investigate the blow up rate for the blow up solutions

holder continuity of the solution map for the novikov equation the novikov equation ne has been discovered recently as a new integrable equation with cubic nonlinearities that is similar to the camassa holm and degasperis procesi equations which have quadratic nonlinearities ne is well posed in sobolev spaces hs on both the line and the circle for s 3 2 in the sense of hadamard and its data to solution map is continuous but not uniformly continuous this work studies the continuity properties of ne further for initial data in hs s 3 2 it is shown that the solution map for ne is holder continuous in hr topology for all 0 r s with exponent depending on s and r

a new ellipsoid associated with convex bodies it is shown that corresponding to each convex body there is an ellipsoid that is in a sense dual to the legendre ellipsoid of classical mechanics sharp affine isoperimetric inequalities are obtained between the volume of the convex body and that of its corresponding new ellipsoid these inequalities provide exact bounds for the isotropic constant associated with the new ellipsoid among other things this leads to a new approach to establishing ball s maximal shadows conjecture for symmetric convex bodies corresponding to each origin symmetric convex or more general subset of euclidean n space r there is a unique ellipsoid with the following property the moment of inertia of the ellipsoid and the moment of inertia of the convex set is the same about every 1 dimensional subspace of r this ellipsoid is called the legendre ellipsoid of the convex set the legendre ellipsoid and its polar the binet ellipsoid are well known concepts from classical mechanics see milman and pajor mpa1 mpa2 lindenstrauss and milman lim and leichtweis le for some historical references it has slowly come to be recognized that along side the brunn minkowski theory there is a dual theory the nature of the duality between this dual brunn minkowski theory and the brunn minkowski theory is subtle and not yet understood it is easily seen that the legendre and binet ellipsoid is an object of this dual brunnminkowski theory this observation leads immediately to the natural question regarding the possible existence of a dual analog of the classical legendre ellipsoid in the brunn minkowski theory it is the aim of this paper to demonstrate the existence of precisely this dual object in retrospect one may well wonder why research supported in part by nsf grant dms 9803261 typeset by ams tex 1 2 a new ellipsoid associated with convex bodies the new ellipsoid presented in this note was not discovered long ago the simple answer is that the definition of the new ellipsoid becomes obvious only with the notion of l2 curvature in hand however the brunn minkowski theory was only recently extended to incorporate the new notion of lp curvature see l2 l3 a positive definite n n real symmetric matrix a generates an ellipsoid a in r defined by a x r x ax 1 where x ax denotes the standard inner product of x and ax in r associated with a star shaped about the origin set k r is its legedre ellipsoid 2k which is generated by the matrix mij k where mij k n 2 v k k ei x ej x dx with e1 en denoting the standard basis for r and v k denoting the ndimensional volume of k we will associate a new ellipsoid 2k with each convex body k r one approach to defining 2k without introducing new notation is to first define it for polytopes and then use approximation with respect to the hausdorff metric to extend the definition to all convex bodies suppose p r is a polytope that contains the origin in its interior let u1 un denote the outer unit normals to the faces of p let a1 an denote the areas i e n 1 dimensional volumes of the corresponding faces and let h1 hn denote the distances from the origin to the corresponding faces the ellipsoid 2p is generated by the matrix mij p where mij p 1 v p n l 1 al hl ei ul ej ul an alternate definition of the operator 2 will be given after additional notation is introduced the easily established affine nature of the operator 2 is formally stated in lemma 1 if k r is star shaped about the origin then for each gl n

blow up phenomena for a family of burgers like equations abstract by introducing a stress multiplier we derive a family of burgers like equations we investigate the blow up phenomena of the equations both on the real line r and on the circle s to get a comparison with the degasperis procesi equation on the line r we first establish the local well posedness and the blow up scenario then we use conservation laws of the equations to get the estimate for the l norm of the strong solutions by which we prove that the solutions to the equations may blow up in the form of wave breaking for certain initial profiles analogous results are provided in the periodic case especially we find differences between the burgers like equations and the degasperis procesi equation see remark 4 1

mixture of spherical distributions for single view relighting we present a method for simultaneously estimating the illumination of a scene and the reflectance property of an object from single view images a single image or a small number of images taken from the same viewpoint we assume that the illumination consists of multiple point light sources and the shape of the object is known first we represent the illumination on the surface of a unit sphere as a finite mixture of von mises fisher distributions based on a novel spherical specular reflection model that well approximates the torrance sparrow reflection model next we estimate the parameters of this mixture model including the number of its component distributions and the standard deviation of them which correspond to the number of light sources and the surface roughness respectively finally using these results as the initial estimates we iteratively refine the estimates based on the original torrance sparrow reflection model the final estimates can be used to relight single view images such as altering the intensities and directions of the individual light sources the proposed method provides a unified framework based on directional statistics for simultaneously estimating the intensities and directions of an unknown number of light sources as well as the specular reflection parameter of the object in the scene

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

a type of bounded traveling wave solutions for the fornberg whitham equation abstract in this paper by using bifurcation method we successfully find the fornberg whitham equation u t u x x t u x u u x x x u u x 3 u x u x x has a type of traveling wave solutions called kink like wave solutions and antikink like wave solutions they are defined on some semifinal bounded domains and possess properties of kink waves and anti kink waves their implicit expressions are obtained for some concrete data the graphs of the implicit functions are displayed and the numerical simulation is made the results show that our theoretical analysis agrees with the numerical simulation

valuations on polytopes containing the origin in their interiors we give a classification of non negative or borel measurable sl d invariant homogeneous valuations on the space of d dimensional convex polytopes containing the origin in their interiors the only examples are volume volume of the polar body and the euler characteristic

on the cauchy problem for a two component degasperis procesi system abstract this paper is concerned with the cauchy problem for a two component degasperis procesi system firstly the local well posedness for this system in the nonhomogeneous besov spaces is established then the precise blow up scenario for strong solutions to the system is derived finally two new blow up criterions and the exact blow up rate of strong solutions to the system are presented

limit behavior of the global solutions to the degasperis procesi type equation abstract in this paper we study the degasperis procesi equation with a physically perturbation term a linear dispersion based on the global existence result we show that the solution of the degasperis procesi equation with linear dispersion tends to the solution of the corresponding degasperis procesi equation as the dispersive parameter goes to zero moreover we prove that smooth solutions of the equation have finite propagation speed they will have compact support if its initial data has this property

contravariant minkowski valuations a complete classification of all continuous gl n contravariant minkowski valuations is established as an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical petty projection inequality

the cauchy problem for a generalized camassa holm equation abstract in this paper we are concerned with the cauchy problem of the generalized camassa holm equation using a galerkin type approximation scheme it is shown that this equation is well posed in sobolev spaces h s s 3 2 for both the periodic and the nonperiodic case in the sense of hadamard that is the data to solution map is continuous furthermore it is proved that this dependence is sharp by showing that the solution map is not uniformly continuous the nonuniform dependence is proved using the method of approximate solutions and well posedness estimates moreover it is shown that the solution map for the generalized camassa holm equation is holder continuous in h r topology finally with analytic initial data we show that its solutions are analytic in both variables globally in space and locally in time

crofton measures and minkowski valuations a description of continuous rigid motion compatible minkowski valuations is established as an application we present a brunn minkowski type inequality for intrinsic volumes of these valuations

continuous rotation invariant valuations on convex sets the notion of valuation on convex sets can be considered as a generalization of the notion of measure which is deflned only on the class of convex compact sets it is well known that there are important and interesting examples of valuations on convex sets which are not measures in the usual sense as for example the mixed volumes basic deflnitions and some classical examples are discussed in section 2 of this paper for more detailed information we refer to the surveys mc sch and mc3 throughout this paper all the valuations are assumed to be continuous with respect to the hausdorfi metric note that the theory of valuations which are invariant or covariant with respect to translations belongs to the classical part of convex geometry there exists an explicit description of translation invariant continuous valuations on 1 and 2 due to hadwiger h1 the case of 2 is nontrivial continuous rigid motion invariant valuations on d are completely classifled by the remarkable hadwiger theorem as linear combinations of the quermassintegrals cf h2 or for a simpler proof k

on the cauchy problem for a generalized degasperis procesi equation we mainly study the cauchy problem of the generalized degasperis procesi equation first we establish the local well posedness for the equation second we give the precise blow up scenario a conservation law and prove the equation has smooth solutions which blow up in finite time finally we investigate the blow up rate for the blow up solutions

continuity properties of the data to solution map for the generalized camassa holm equation abstract this work studies a generalized camassa holm equation with higher order nonlinearities g kbch the camassa holm the degasperis procesi and the novikov equations are integrable members of this family of equations g kbch is well posed in sobolev spaces h s s 3 2 on both the line and the circle and its solution map is continuous but not uniformly continuous in this work it is shown that the solution map is holder continuous in h s equipped with the h r topology for 0 r s and the holder exponent is expressed in terms of s and r

global existence and blow up phenomena for the degasperis procesi equation this paper is concerned with several aspects of the existence of global solutions and the formation of singularities for the degasperis procesi equation on the line global strong solutions to the equation are determined for a class of initial profiles on the other hand it is shown that the first blow up can occur only in the form of wave breaking a new wave breaking mechanism for solutions is described in detail and two results of blow up solutions with certain initial profiles are established

valuations on lattice polytopes let be a lattice that is a module of finite rank and let denote the family of convex polytopes with vertices in here convexity refers to the underlying rational vector space in this paper it is shown that any valuation on satisfies the inclusion exclusion principle in the strong sense that appropriate extension properties of the valuation hold indeed the core result is that the class of a lattice polytope in the abstract group for valuations on can be identified with its characteristic function in in fact the same arguments are shown to apply to when is a module of finite rank over an ordered ring and more generally to appropriate families of not necessarily bounded polyhedra

well posedness and analytic solutions of a two component water wave equation abstract we first establish the local well posedness for the cauchy problem of a two component water waves system in a critical nonhomogeneous besov space then we derive a continuation criterion for strong solutions to the system finally we prove the existence of analytic solutions to the system

optimal control of the viscous degasperis procesi equation this paper studies the problem for optimal control of the viscous degasperis procesi equation the existence and uniqueness of weak solution to the equation are proved in a short interval the optimal control of the viscous degasperis procesi equation under boundary condition is given and the existence of optimal solution to the equation is proved

global existence of weak solutions for a shallow water equation a nonlinear shallow water equation which includes the famous camassa holm ch and degasperis procesi dp equations as special cases is investigated provided that initial value u email protected h s email protected email protected 32 u email protected l 1 r and email protected x 2 u 0 does not change sign it is shown that there exists a unique global weak solution to the equation

on traveling wave solutions of the θ equation of dispersive type abstract traveling wave solutions to a class of dispersive models u t u t x x u u x u u x x x 1 u x u x x are investigated in terms of the parameter including two integrable equations the camassa holm equation 1 3 and the degasperis procesi equation 1 4 as special models it was proved in h liu and z yin 2011 39 that when 1 2 1 smooth solutions persist for all time and when 0 1 2 strong solutions of the equation may blow up in finite time yielding rich traveling wave patterns this work therefore restricts to only the range 0 1 2 it is shown that when 0 only periodic travel wave is permissible and when 1 2 traveling waves may be solitary periodic or kink like waves for 0 1 2 traveling waves such as periodic solitary peakon peaked periodic cusped periodic or cusped soliton are all permissible

global weak solutions to the weakly dissipative degasperis procesi equation abstract the global weak solutions for the degasperis procesi equation with weakly dissipative term are investigated provided that u 0 h 1 r y 0 1 x 2 u 0 m r supp y 0 x 0 and supp y 0 x 0 the existence and uniqueness of global weak solutions for the equation are shown to be true in the space w l o c 1 r r l l o c r h 1 r

global weak solutions and blow up structure for the degasperis procesi equation in this paper we study several qualitative properties of the degasperis procesi equation we first established the precise blow up rate and then determine the blow up set of blow up strong solutions to this equation for a large class of initial data we finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions

well posedness and blow up phenomena for the generalized degasperis procesi equation abstract in this paper we mainly study the cauchy problem of the generalized degasperis procesi equation we establish the local well posedness and give the precise blow up scenario for the equation then we show that the equation has smooth solutions which blow up in finite time

separating reflection components of textured surfaces using a single image in inhomogeneous objects highlights are linear combinations of diffuse and specular reflection components to our knowledge all methods that use a single input image require explicit color segmentation to deal with multicolored surfaces unfortunately for complex textured images current color segmentation algorithms are still problematic to segment correctly consequently a method without explicit color segmentation becomes indispensable and this work presents such a method the method is based solely on colors particularly chromaticity without requiring any geometrical information one of the basic ideas is to iteratively compare the intensity logarithmic differentiation of an input image and its specular free image a specular free image is an image that has exactly the same geometrical profile as the diffuse component of the input image and that can be generated by shifting each pixel s intensity and maximum chromaticity nonlinearly unlike existing methods using a single image all processes in the proposed method are done locally involving a maximum of only two neighboring pixels this local operation is useful for handling textured objects with complex multicolored scenes evaluations by comparison with the results of polarizing filters demonstrate the effectiveness of the proposed method

tensor valuations on lattice polytopes abstract the ehrhart polynomial and the reciprocity theorems by ehrhart macdonald are extended to tensor valuations on lattice polytopes a complete classification is established of tensor valuations of rank up to eight that are equivariant with respect to the special linear group over the integers and translation covariant every such valuation is a linear combination of the ehrhart tensors which is shown to no longer hold true for rank nine

efficient and robust specular highlight removal a robust and effective specular highlight removal method is proposed in this paper it is based on a key observation the maximum fraction of the diffuse colour component in diffuse local patches in colour images changes smoothly the specular pixels can thus be treated as noise in this case this property allows the specular highlights to be removed in an image denoising fashion an edge preserving low pass filter e g the bilateral filter can be used to smooth the maximum fraction of the colour components of the original image to remove the noise contributed by the specular pixels recent developments in fast bilateral filtering techniques enable the proposed method to run over 200 times faster than state of the art techniques on a standard cpu and differentiates it from previous work

blow up phenomena and local well posedness for a generalized camassa holm equation in the critical besov space abstract in this paper we mainly study the cauchy problem for a generalized camassa holm equation in a critical besov space first by using the littlewood paley decomposition transport equations theory logarithmic interpolation inequalities and osgood s lemma we establish the local well posedness for the cauchy problem of the equation in the critical besov space b 2 1 1 2 then we give a new blow up criterion for the cauchy problem of the equation finally we present a new blow up result and the exact blow up rate of strong solutions to the equation by making use of the conservation law and the obtained blow up criterion

traveling wave solutions of the degasperis procesi equation we classify all weak traveling wave solutions of the degasperis procesi equation in addition to smooth and peaked solutions the equation is shown to admit more exotic traveling waves such as cuspons stumpons and composite waves

single and multi solitary wave solutions to a class of nonlinear evolution equations abstract in this paper an effective discrimination algorithm is presented to deal with equations arising from physical problems the aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations many examples are given to illustrate the algorithm at the same time some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations such as camassa holm equation kolmogorov petrovskii piskunov equation and so on then a direct constructive method called multi auxiliary equations expansion method is described to derive the multi solitary wave solutions of nonlinear evolution equations finally a class of novel multi solitary wave solutions of the 2 1 dimensional asymmetric version of the nizhnik novikov veselov equation are given by three direct methods the algorithm proposed in this paper can be steadily applied to some other nonlinear problems

moments and valuations all measurable and rm sl n covariant vector valued valuations on convex polytopes containing the origin in their interiors are completely classified the moment vector is shown to be essentially the only such valuation

optimal control of a class of nonlocal dispersive equations abstract in this paper the optimal control problem for a class of nonlocal dispersive equations is considered we introduce a kind of the definition of a weak solution to this equation and prove the existence of a unique weak solution in a special hilbert space s 0 t we also discuss the existence of optimal control solution to a class of nonlocal dispersive equations with the cost functional adopting the dubovitskii milyutin functional analytical approach we prove the necessary optimality condition of optimal control local maximum principle for the investigative system in fixed final horizon case

a valuation theoretic approach to translative equidecomposability abstract all simple translation invariant valuations on polytopes are classified as a direct consequence the well known conditions for translative equidecomposability are recovered furthermore a simplified proof of the classification of continuous simple translation invariant valuations is presented

exponential decay and symmetry of solitary waves to degasperis procesi equation abstract we improve the decay argument by bona and li 1997 5 for solitary waves of general dispersive equations and illustrate it in the proof for the exponential decay of solitary waves to steady degasperis procesi equation in the nonlocal formulation in addition we give a method which confirms the symmetry of solitary waves including those of the maximum height finally we discover how the symmetric structure is connected to the steady structure of solutions to the degasperis procesi equation and give a more intuitive proof for symmetric solutions to be traveling waves the improved argument and new method above can be used for the decay rate of solitary waves to many other dispersive equations and will give new perspectives on symmetric solutions for general evolution equations

ehrhart tensor polynomials abstract the notion of ehrhart tensor polynomials a natural generalization of the ehrhart polynomial of a lattice polytope was recently introduced by ludwig and silverstein we initiate a study of their coefficients in the vector and matrix cases we give pick type formulas in terms of triangulations of a lattice polygon as our main tool we introduce h r tensor polynomials extending the notion of the ehrhart h polynomial and for matrices investigate their coefficients for positive semidefiniteness in contrast to the usual h polynomial the coefficients are in general not monotone with respect to inclusion nevertheless we are able to prove positive semidefiniteness in dimension two based on computational results we conjecture positive semidefiniteness of the coefficients in higher dimensions furthermore we generalize hibi s palindromic theorem for reflexive polytopes to h r tensor polynomials and discuss possible future research directions

a non local approach to waves of maximal height for the degasperis procesi equation abstract we consider the non local formulation of the degasperis procesi equation u t u u x l 3 2 u 2 x 0 where l is the non local fourier multiplier operator with symbol m 1 2 1 we show that all l pointwise travelling wave solutions are bounded above by the wave speed and that if the maximal height is achieved they are peaked at those points otherwise they are smooth for sufficiently small periods we find the highest peaked travelling wave solution as the limiting case at the end of the main bifurcation curve of p periodic solutions the results imply that there are no l travelling cuspon solutions to the degasperis procesi equation

blow up phenomena and peakons for the b family of forq mch equations abstract this paper is devoted to studying the modified b family of equations with cubic nonlinearity called the b family of forq mch equations which includes the cubic camassa holm equation also called fokas olver rosenau qiao equation as a special case we first study the local well posedness for the cauchy problem of the equation and then make good use of fine structure of the equation we derive the precise blow up scenario and a new blow up result with respect to initial data finally peakon solutions are derived

new sine ellipsoids and related volume inequalities abstract corresponding to the legendre ellipsoid and the lyz ellipsoid two new sine ellipsoids are introduced in this paper these four ellipsoids are closely related in the pythagorean relation and duality several volume inequalities and the valuation properties are obtained for two new ellipsoids