cone volume measure of general centered convex bodies abstract we show that the cone volume measure of a convex body with centroid at the origin satisfies the subspace concentration condition this extends former results obtained in the discrete as well as in the symmetric case and implies among others a conjectured best possible inequality for the u functional of a convex body

gauss curvature flow the fate of the rolling stones we prove firey s 1974 conjecture that convex surfaces moving by their gauss curvature become spherical as they contract to points

a probabilistic approach to the geometry of the lᵨⁿ ball this article investigates by probabilistic methods various geometric questions on b n p the unit ball of ln p we propose realizations in terms of independent random variables of several distributions on b n p including the normalized volume measure these representations allow us to unify and extend the known results of the sub independence of coordinate slabs in b n p as another application we compute moments of linear functionals on b n p which gives sharp constants in khinchine s inequalities on b n p and determines the 2 constant of all directions on b n p we also study the extremal values of several gaussian averages on sections of b n p including mean width and l norm and derive several monotonicity results as p varies applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition

integration of the soliton hierarchy with self consistent sources in contrast with the soliton equations the evolution of the eigenfunctions in the lax representation of soliton equation with self consistent sources sescs possesses singularity we present a general method to treat the singularity to determine the evolution of scattering data the akns hierarchy with self consistent sources the mkdv hierarchy with self consistent sources the nonlinear schrodinger equation hierarchy with self consistent sources the kaup newell hierarchy with self consistent sources and the derivative nonlinear schrodinger equation hierarchy with self consistent sources are integrated directly by using the inverse scattering method the n soliton solutions for some sescs are presented it is shown that the insertion of a source may cause the variation of the velocity of soliton this approach can be applied to all other 1 1 dimensional soliton hierarchies

sasa satsuma complex modified korteweg de vries ii and the complex sine gordon ii equation revisited recursion operators nonlocal symmetries and more we present a new symplectic structure and a hereditary recursion operator for the sasa satsuma equation which is widely used in nonlinear optics using an integrodifferential substitution relating this equation to a third order symmetry flow of the complex sine gordon ii equation enabled us to find a hereditary recursion operator and higher hamiltonian structures for the latter equation we also show that both the sasa satsuma equation and the third order symmetry flow for the complex sine gordon ii equation are bi hamiltonian systems and we construct several hierarchies of local and nonlocal symmetries for these systems

a classification of sl n invariant valuations a classification of upper semicontinuous and sl n invariant valuations on the space of n dimensional convex bodies is established as a consequence complete characterizations of cen t ro affine and l p affine surface areas are obtained the proofs make use of a new sl n shaping process for convex bodies

the dual minkowski problem for symmetric convex bodies abstract the dual minkowski problem for even data asks what are the necessary and sufficient conditions on a prescribed even measure on the unit sphere for it to be the q th dual curvature measure of an origin symmetric convex body in r n a full solution to this is given when 1 q n the necessary and sufficient conditions turn out to be an explicit measure concentration condition to obtain the results a variational approach is used where the functional is the sum of a dual quermassintegral and an entropy integral the proof requires two crucial estimates the first is an estimate of the entropy integral which is obtained by using a spherical partition the second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body

classification of limiting shapes for isotropic curve flows with a 5 0 and initial condition x p 0 xzo p this produces a family of curves yt x si t here n is the curvature and n is the outward pointing unit normal vector these equations are particularly natural in that they are isotropic equivariant under rotations in the plane and homogeneous equivariant under dilation of space if time is also scaled accordingly the main aim of this paper is to provide a complete description of the behaviour of embedded convex curves moving by equations of the form 1 1 in certain cases this description has already been provided if a 1 then equation 1 1 is the curve shortening flow for which the following holds

direct perturbation theory for solitons of the derivative nonlinear schrodinger equation and the modified nonlinear schrodinger equation a direct perturbation theory for solitons of the derivative nonlinear schrodinger dnls equation is developed based on a closure of eigenfunctions of the linearized dnls equation around a one soliton solution the slow evolution of soliton parameters and the perturbation induced radiation are obtained under the known simple gaugelike transformation these results are transformed into those for the perturbed modified nonlinear schrodinger mnls equation describing propagation of femtosecond pulses in optical fibers a calculation of the perturbation induced radiation fields for the perturbed dnls and mnls equations is also made our results for the perturbed mnls equation can be reduced perfectly to those for the perturbed nonlinear schrodinger equation in the small nonlinear dispersion limit

complete eigenfunctions of linearized integrable equations expanded around a soliton solution complete eigenfunctions for an integrable equation linearized around a soliton solution are the key to the development of a direct soliton perturbation theory in this article we explicitly construct such eigenfunctions for a large class of integrable equations including the kdv nls and mkdv hierarchies we establish the striking result that the linearization operators of all equations in the same integrable hierarchy share the same complete set of eigenfunctions furthermore these eigenfunctions are precisely the squared eigenfunctions of the associated eigenvalue problem the key step in our derivation is to show that the linearization operator of an integrable equation can be factored into a function of the integro differential operator which generates the integrable equation and the linearization operator of the lowest order integrable equation in the same hierarchy we also obtain similar results for the adjoint linearization operator of an integrable equation even though our analysis is conducted only for the kdv nls and mkdv hierarchies similar results are expected for other integrable hierarchies as well we further explicitly present the complete eigenfunctions for the kdv nls and mkdv hierarchy equations and give their inner products thus they can be readily used to develop a direct soliton perturbation theory for any of those hierarchy equations

existence of convex hypersurfaces with prescribed gauss kronecker curvature let f x be a given positive function in rn 1 in this paper we consider the existence of convex closed hypersurfaces x so that its gausskronecker curvature at x x is equal to f x this problem has variational structure and the existence of stable solutions has been discussed by tso j diff geom 34 1991 389 410 using the mountain pass lemma and the gauss curvature flow we prove the existence of unstable solutions to the problem

the squared eigenstates of the sine gordon eigenvalue problem an analysis of the squared sine gordon eigenvalue problem in laboratory coordinates is presented it is shown that unlike the unsquared laboratory coordinate eigenvalue problem the squared laboratory coordinate eigenvalue problem may be cast into the form of a standard eigenvalue problem wherein an eigenvalue independent operator operating on an eigenfunction generates the eigenvalue with this form it becomes rather elementary to obtain the squared eigenfunction expansion of the sine gordon potentials as well as to demonstrate closure

2d toda lattice equation with self consistent sources casoratian type solutions bilinear backlund transformation and lax pair the two dimensional toda lattice equation with self consistent sources is proposed based on its bilinear forms casoratian type solutions and backlund transformation bt for the bilinear forms are presented starting from the bt a lax pair is derived for the 2d toda lattice with self consistent sources

on the uniqueness of lp minkowski problems the constant p curvature case in r3 abstract we study the c 4 smooth convex bodies k r n 1 satisfying k x u x 1 p where x s n k is the gauss curvature of k u is the support function of k and p is a constant in the case of n 2 either when p 1 0 or when p 0 1 in addition to a pinching condition we show that k must be the unit ball this partially answers a conjecture of lutwak yang and zhang about the uniqueness of the l p minkowski problem in r 3 moreover we give an explicit pinching constant depending only on p when p 0 1

the logarithmic minkowski problem for polytopes abstract the logarithmic minkowski problem asks for necessary and sufficient conditions for a finite borel measure on the unit sphere so that it is the cone volume measure of a convex body this problem was solved recently by boroczky lutwak yang and zhang for even measures boroczky et al 2013 8 this paper solves the case of discrete measures whose supports are in general position

the squared eigenfunctions of the massive thirring model in laboratory coordinates we demonstrate a simple procedure for finding the squared eigenfunctions of an integrable system and illustrate the method by deriving the squared eigenfunctions of the massive thirring model we give the formula for the inner products between the direct squared eigenfunction problem and its adjoint we also demonstrate closure and express the variations of the potentials in terms of variations of the scattering data

affine images of isotropic measures necessary and sufficient conditions are given in order for a borel measure on the euclidean sphere to have an affine image that is isotropic a sharp reverse affine isoperimetric inequality for borel measures on the sphere is presented this leads to sharp reverse affine isoperimetric inequalities for convex bodies 2015 international press of boston inc all rights reserved

a positive answer to the busemann petty problem in three dimensions we prove that in e3 the busemann petty problem concerning cen tral sections of centrally symmetric convex bodies has a positive answer together with other results this settles the problem in each dimension

squared eigenfunctions and the perturbation theory for the nondegenerate n n operator a general outline we provide an overview of the soliton perturbation theory for the n n eigenvalue operator key to perturbation studies of integrable systems are the squared eigenfunctions and their adjoints which serve as mappings between variations in the potentials and variations in the scattering data we also address the problem of the normalization of the jost functions how this affects the structure and solvability of the inverse scattering equations and the definition of the scattering data we also discuss the inner products and closure relations for these squared eigenfunctions and their adjoints

on the number of solutions to the discrete two dimensional l0 minkowski problem abstract our main result shows the uniqueness of planar convex polygons of given outer orientations to the sides and prescribed areas of the triangles formed by the origin with any two consecutive vertices if they exist such polygons are solutions to the discrete l 0 minkowski problem in the plane proposed in a larger generality by lutwak in his extension of the brunn minkowski theory the method exploits the equivalence between the l 0 polygons and the homothetic solutions to appropriate anisotropic crystalline flows hence as a by product we obtain a positive answer to jean taylor s conjecture on the uniqueness of polygonal shapes which are deformed homothetically under the flow by crystalline curvature without any symmetry assumptions

orlicz john ellipsoids abstract the orlicz john ellipsoids which are in the framework of the booming orlicz brunn minkowski theory are introduced for the first time it turns out that they are generalizations of the classical john ellipsoid and the evolved l p john ellipsoids the analog of ball s volume ratio inequality is established for the new orlicz john ellipsoids the connection between the isotropy of measures and the characterization of orlicz john ellipsoids is demonstrated

a direct approach to studying soliton perturbations starting with an integrable nonlinear evolution equation the author investigates perturbations about a one soliton solution through the inversion of a linear equation for the first order correction to the soliton solution this inversion differs from past methods as the proposed method takes place in coordinate space not spectral space while it employs some of the tools of inverse scattering theory the method is applied to the korteweg de vries nonlinear schrodinger and sine gordon equations the first order corrections are then obtained

squared eigenfunctions for the sasa satsuma equation squared eigenfunctions are quadratic combinations of jost functions and adjoint jost functions which satisfy the linearized equation of an integrable equation they are needed for various studies related to integrable equations such as the development of its soliton perturbation theory in this article squared eigenfunctions are derived for the sasa satsuma equation whose spectral operator is a 3 3 system while its linearized operator is a 2 2 system it is shown that these squared eigenfunctions are sums of two terms where each term is a product of a jost function and an adjoint jost function the procedure of this derivation consists of two steps first is to calculate the variations of the potentials via variations of the scattering data by the riemann hilbert method the second one is to calculate the variations of the scattering data via the variations of the potentials through elementary calculations while this procedure has been used before on other integrable equations it is shown here for

an analytic solution to the busemann petty problem on sections of convex bodies we derive a formula connecting the derivatives of parallel section functions of an origin symmetric star body in r n with the fourier transform of powers of the radial function of the body a parallel section function or ni 1 dimensional x ray gives the ni 1 dimensional volumes of all hyperplane sections of the body orthogonal to a given direction this formula provides a new characterization of intersection bodies in r n and leads to a unifled analytic solution to the busemann petty problem suppose that k and l are two origin symmetric convex bodies inr n such that the ni 1 dimensional volume of each central hyperplane section of k is smaller than the volume of the corresponding section of l is the n dimensional volume of k smaller than the volume of l in conjunction with earlier established connections between the busemann petty problem intersection bodies and positive deflnite distributions our formula shows that the answer to the problem depends on the behavior of the ni 2 nd derivative of the parallel section functions the a rmative answer to the busemann petty problem for n 4 and the negative answer for n 5 now follow from the fact that convexity controls the second derivatives but does not control the derivatives of higher orders

intersection bodies and dual mixed volumes the setting for this article is euclidean n space ky n 2 let x denote the set of convex bodies compact convex subsets with nonempty interiors in r and let x denote the subset of x that contains the centered centrally symmetric with respect to the origin bodies for ue s let e denote the hyperplane through the origin that is orthogonal to u and let ki e denote the image of the orthogonal projection of the body ke x onto e we shall use v to denote i dimensional lebesgue measure for v and v we shall usually write v and u shephard 38 poses the question if k l e xe and

nonuniqueness of solutions to the l p minkowski problem abstract in this paper we show by example that there is no uniform estimate for the l p minkowski problem as a result we obtain the nonuniqueness of solutions to the problem

the deift zhou steepest descent method to long time asymptotics for the sasa satsuma equation abstract the initial value problem for the sasa satsuma equation is transformed to a 3 3 matrix riemann hilbert problem with the help of the corresponding lax pair two distinct factorizations of the jump matrix and a decomposition of the vector valued function k are given from which the long time asymptotics for the sasa satsuma equation with decaying initial data is obtained by using the nonlinear steepest descent method

least squares spectral method for the two dimensional stokes interface problems abstract this paper deals with the pseudo spectral least squares method to approximate the solution of the stokes interface problem that is the stokes equations with discontinuous viscosity and singular source term the stokes interface equations are adjusted in each sub domain along with the interface conditions then the equations are reformulated as a first order system by introducing stress tensor as a new independent variable the continuous and discrete norm least squares functionals using legendre and chebyshev weights for the first order system are defined and shown to be equivalent to appropriate product norms the spectral convergence of the proposed method is presented numerical examples are also given to demonstrate the method and analysis