two bubble dynamics for threshold solutions to the wave maps equation we consider the energy critical wave maps equation mathbb r 1 2 rightarrow mathbb s 2 in the equivariant case with equivariance degree k ge 2 it is known that initial data of energy 8 pi k and topological degree zero leads to global solutions that scatter in both time directions we consider the threshold case of energy 8 pi k we prove that the solution is defined for all time and either scatters in both time directions or converges to a superposition of two harmonic maps in one time direction and scatters in the other time direction in the latter case we describe the asymptotic behavior of the scales of the two harmonic maps the proof combines the classical concentration compactness techniques of kenig merle with a modulation analysis of interactions of two harmonic maps in the absence of excess radiation

stability and asymptotic stability in the energy space of the sum of n solitons for subcritical gkdv equations we prove in this paper the stability and asymptotic stability in h1 of a decoupled sum of n solitons for the subcritical generalized kdv equations the proof of the stability result is based on energy arguments and monotonicity of the local l2 norm note that the result is new even for p 2 the kdv equation the asymptotic stability result then follows directly from a rigidity theorem in 16

stability and instability of solitary waves of korteweg de vries type considered herein are the stability and instability properties of solitary wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear dispersive long waves special cases for which our analysis is decisive include equations of the korteweg de vries and benjamin ono type necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable

valuations on sobolev spaces all affinely covariant convex body valued valuations on the sobolev space w 1 1 bbb r n are completely classified it is shown that there is a unique such valuation for blaschke addition this valuation turns out to be the operator which associates with each function f in w 1 1 bbb r n the unit ball of its optimal sobolev norm

sections of convex bodies the generalized busemann petty problem asks if the volume of i dimensional central section of a centrally symmetric convex body in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is smaller than that of another such body is the volume of the body also smaller it is proved that the answer is negative if 2 i n the case of a 2 dimensional section remains open the proof uses techniques in functional analysis and radon transforms on grassmannians it also requires the notion of an i intersection body which generalizes the notion of an intersection body inequalities among the volumes of projection bodies polar projection bodies and their central sections are proved they are related to the maximal slice problem

description of two soliton collision for the quartic gkdv equation in this paper we give the rst description of the collision of two solitons for a nonintegrable equation in a special regime we consider solutions of the quartic gkdv equation tu x 2 xu u 4 0 which behave as t 1 like u t x qc1 x c1t qc2 x c2t t x where qc x ct is a soliton andk t kh1k qc2kh1k qc1kh1 the global behavior of u t is given by the following stability result for all t 2 r u t x qc1 t x y1 t qc2 t x y2 t t x where k t kh1k qc2kh1 and limt 1c1 t c limt 1c2 t c in the case where u t is a pure 2 soliton solution as t 1 i e limt 1 k t kh1 0 we obtain c c1 c 0 therefore in contrast with the integrable kdv equation or mkdv equation no global pure 2 soliton solution exists and the collision is inelastic a dierent notion of global 2 soliton is then proposed

iterations of the projection body operator and a remark on petty s conjectured projection inequality abstract we prove that if a convex body has an absolutely continuous surface area measure whose density is sufficiently close to a constant function then the sequence m k of convex bodies converges to the ball with respect to the banach mazur distance as m here denotes the projection body operator our result allows us to show that the ellipsoid is a local solution to the conjectured inequality of petty and to improve a related inequality of lutwak

threshold phenomenon for the quintic wave equation in three dimensions for the critical focusing wave equation square u u 5 rm on mathbb r 3 1 in the radial case we establish the role of the center stable manifold sigma constructed in krieger and schlag am j math 129 3 843 913 2007 near the ground state w 0 as a threshold between blowup and scattering to zero establishing a conjecture going back to numerical work by bizon et al nonlinearity 17 6 2187 2201 2004 the underlying topology is stronger than the energy norm

the second mixed projection problem and the projection centroid conjectures abstract we provide partial answers to the open problems 4 5 4 6 of 3 and 12 9 of 12 regarding the classification of fixed points of the second mixed projection operator and iterates of the projection and centroid operators

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

construction of type ii blow up solutions for the energy critical wave equation in dimension 5 abstract we consider the semilinear wave equation with focusing energy critical nonlinearity in space dimension n 5 t t u u u 4 3 u with radial data it is known 7 that a solution u t u which blows up at t 0 in a neighborhood in the energy norm of the family of solitons w decomposes in the energy space as u t t u t w t u 0 u 1 o 1 where lim t 0 t t 0 and u 0 u 1 h 1 l 2 we construct a blow up solution of this type such that the asymptotic profile u 0 u 1 is any pair of sufficiently regular functions with u 0 0 0 for these solutions the concentration rate is t t 4 we also provide examples of solutions with concentration rate t t 1 for 8 related to the behavior of the asymptotic profile near the origin

anisotropic fractional sobolev norms abstract bourgain brezis mironescu showed that with suitable scaling the fractional sobolev s seminorm of a function f w 1 p r n converges to the sobolev seminorm of f as s 1 the anisotropic s seminorms of f defined by a norm on r n with unit ball k are shown to converge to the anisotropic sobolev seminorm of f defined by the norm with unit ball z p k the polar l p moment body of k the limiting behavior for s 0 is also determined extending results by maz ya shaposhnikova

continuity and representation of valuations on star bodies abstract it is shown that every continuous valuation defined on n dimensional star bodies has an integral representation in terms of the radial function our argument is based on the non trivial fact that continuous valuations are uniformly continuous on bounded sets we also characterize continuous valuations on the n dimensional star bodies that arise as restriction of a measure on r n

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

dual orlicz brunn minkowski theory dual orlicz lϕ affine and geominimal surface areas abstract this paper aims to develop the basic theory for the dual orlicz l affine and geominimal surface areas of star bodies basic properties of these new affine invariants are established moreover related orlicz affine isoperimetric cyclic santalo type and alexandrov fenchel type inequalities are obtained

valuations and busemann petty type problems projection and intersection bodies define continuous and gl n contravariant valuations they played a critical role in the solution of the shephard problem for projections of convex bodies and its dual version for sections the busemann petty problem we consider the question whether fkfl implies v k v l where f is a homogeneous continuous operator on convex or star bodies which is an so n equivariant valuation important previous results for projection and intersection bodies are extended to a large class of valuations

channels of energy for the linear radial wave equation abstract exterior channel of energy estimates for the radial wave equation were first considered in three dimensions in 6 and for the 5 dimensional case in 12 in this paper we find the general form of the channel of energy estimate in all odd dimensions for the radial free wave equation this will be used in the companion paper 11 to establish the soliton resolution for equivariant wave maps in r 3 exterior to the ball b 0 1 and in all equivariance classes

approximation algorithms for the orthogonal z oriented three dimensional packing problem we present approximation algorithms for the orthogonal z oriented three dimensional packing problem tppz and analyze their asymptotic performance bound this problem consists in packing a list of rectangular boxes l b1 b2 bn into a rectangular box b l w infty orthogonally and oriented in the z axis in such a way that the height of the packing is minimized we say that a packing is oriented in the z axis when the boxes in l are allowed to be rotated by ninety degrees around the z axis this problem has some nice applications but has been less investigated than the well known variant of it denoted by tpp three dimensional orthogonal packing problem in which rotations of the boxes are not allowed the problem tpp can be reduced to tppz given an algorithm for tppz we can obtain an algorithm for tpp with the same asymptotic bound we present an algorithm for tppz called r and three other algorithms called ls bs and ss for special cases of this problem in which the instances are more restricted the algorithm ls is for the case in which all boxes in l have square bottoms bs is for the case in which the box b has a square bottom and ss is for the case in which the box b and all boxes in l have square bottoms for an algorithm wa we denote by r wa the asymptotic performance bound of wa we show that 2 5 leq r r 2 67 2 5 leq r ls leq 2 528 2 5 leq r bs leq 2 543 and 2 333 leq r ss leq 2 361 the algorithms presented here have the same complexity cal o n log n as the other known algorithms for these problems but they have better asymptotic performance bounds

asymptotic stability of solitons for subcritical generalized kdv equations we prove in this paper the asymptotic completeness of the family of solitons in the energy space for generalized korteweg de vries equations in the subcritical case this includes in particular the kdv equation and the modified kdv equation this result is obtained as a consequence of a rigidity theorem on the flow close to a soliton up to a scaling and a translation which has its own interest the proofs use some tools introduced in a previous paper to prove similar results in the case of critical generalized kdv equation

the role of the green s function in a non linear elliptic equation involving the critical sobolev exponent this paper is concerned with non linear elliptic problems of the type pe u u n 2 n 2 eu u 0 on u 0 on where is a smooth and bounded domain in rn n 4 and e 0 we show that if the ue are solutions of pe which concentrate around a point as e 0 then this point cannot be on the boundary of and is a critical point of the regular part of the green s function conversely we show that for n 5 and any non degenerate critical point x0 of the regular part of the green s function there exist solutions of pe concentrating around x0 as e 0

asymptotic stability of solitary waves we show that the family of solitary waves 1 solitons of the korteweg de vries equation partial t u u partial x u partial x 3 u 0 is asymptotically stable our methods also apply for the solitary waves of a class of generalized korteweg de vries equations partial t u partial x f u partial x 3 u 0 in particular we study the case wheref u up 1 p 1 p 1 2 3 and 3 0 withf c4 the same asymptotic stability result for kdv is also proved for the casep 2 the modified korteweg de vries equation we also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 the solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep 4 the solution is decomposed into a modulating solitary wave with time varying speedc t and phase t bound state part and an infinite dimensional perturbation radiating part the perturbation is shown to decay exponentially in time in a local sense relative to a frame moving with the solitary wave asp 4 the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

global well posedness scattering and blow up for the energy critical focusing non linear wave equation we study the energy critical focusing non linear wave equation with data in the energy space in dimensions 3 4 and 5 we prove that for cauchy data of energy smaller than the one of the static solution w which gives the best constant in the sobolev embedding the following alternative holds if the initial data has smaller norm in the homogeneous sobolev space h1 than the one of w then we have global well posedness and scattering if the norm is larger than the one of w then we have break down in finite time

dual orlicz brunn minkowski theory abstract in this paper a dual orlicz brunn minkowski theory is presented an orlicz radial sum and dual orlicz mixed volumes are introduced the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality are established the variational formula for the volume with respect to the orlicz radial sum is proved the equivalence between the dual orlicz minkowski inequality and the dual orlicz brunn minkowski inequality is demonstrated orlicz intersection bodies are defined and the orlicz busemann petty problem is posed

on dual quermassintegrals of mixed intersection bodies in this paper it is shown that a family of inequalities involving mixed intersection bodies holds the busemann intersection inequality is the first of this family all of the members of this family are strengthened versions of classical inequalities between pairs of dual quermassintegrals of a star body

lp mixed affine surface area abstract according to the three notions of mixed affine surface area l p affine surface area and l p mixed affine surface area proposed by lutwak in this article we give the concept of ith l p mixed affine surface area such that the first and second notions of lutwak are its special cases further some lutwak s results are extended associated with this concept besides applying this concept we establish an inequality for the volumes and dual quermassintegrals of a class of star bodies

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

a harmonic algorithm for the 3d strip packing problem in the three dimensional 3d strip packing problem we are given a set of 3d rectangular items and a 3d box b the goal is to pack all the items in b such that the height of the packing is minimized we consider the most basic version of the problem where the items must be packed with their edges parallel to the edges of b and cannot be rotated building upon caprara s work for the two dimensional 2d bin packing problem we obtain an algorithm that given any epsilon 0 achieves an approximation of t infty epsilon approx1 69103 epsilon where t infty is the well known number that occurs naturally in the context of bin packing our key idea is to establish a connection between bin packing solutions for an arbitrary instance i and the strip packing solutions for the corresponding instance obtained from i by applying the harmonic transformation to certain dimensions based on this connection we also give a simple alternate proof of the t infty epsilon approximation for 2d

energy partition for the linear radial wave equation we consider the radial free wave equation in all dimensions and derive asymptotic formulas for the space partition of the energy as time goes to infinity we show that the exterior energy estimate which duyckaerts et al obtained in odd dimensions duyckaerts et al j eur math soc 13 533 599 2011 j eur math soc 2013 fails in even dimensions positive results for restricted classes of data are obtained

subword complexity and projection bodies abstract a polytope p 0 1 d and an 0 1 d induce a so called hartman sequence h p 0 1 z which is by definition 1 at the kth position if k mod 1 p and 0 otherwise k z we prove an asymptotic formula for the subword complexity of such a hartman sequence this result establishes a connection between symbolic dynamics and convex geometry if the polytope p is convex then the subword complexity of h p asymptotically equals the volume of the projection body p of p for almost all 0 1 d

inelasticity of soliton collisions for the 5d energy critical wave equation for the focusing energy critical wave equation in 5d we construct a solution showing the inelastic nature of the collision of two solitons for any choice of sign speed scaling and translation parameters except the special case of two solitons of same scaling and opposite signs beyond its own interest as one of the first rigorous studies of the collision of solitons for a non integrable model the case of the quartic gkdv equation being partially treated in martel and merle ann math 174 2 757 857 2011 invent math 183 3 563 648 2011 int math res notices 2015 3 688 739 2015 this result can be seen as part of a wider program aiming at establishing the soliton resolution conjecture for the critical wave equation this conjecture has already been proved in the 3d radial case in duyckaerts et al camb j math 1 75 144 2013 and in the general case in 3 4 and 5d along a sequence of times in duyckaerts et al geom funct anal 27 4 798 862 2017 compared with the construction of an asymptotic two soliton in martel and merle arch ration mech anal 222 3 1113 1160 2016 the study of the nature of the collision requires a more refined approximate solution of the two soliton problem and a precise determination of its space asymptotics to prove inelasticity these asymptotics are combined with the method of channels of energy from duyckaerts et al camb j math 1 75 144 2013 kenig et al geom funct anal 24 610 647 2014

on three dimensional packing the three dimensional packing problem is discussed in this paper the problem is a generalization of the one and two dimensional packing problems it is demonstrated that some basic packing strategies such as nfdh and ffdh for two dimensional packing have unbounded worst case performance ratios in the three dimensional case let r a denote the asymptotic performance bound of an approximation algorithm a an approximation algorithm g is developed and it is shown that 4 333 leqq r g leqq 4 571 the algorithm is improved to algorithm c and it is proven that r c 3 25 for the special case when all boxes have square bottoms the two algorithms are adapted to algorithms g 1 and c 1 respectively with r g 1 4 and r c 1 2 6875 for the case when both sides of the bottom of a box are no larger then 1 m two families of algorithms g m m geqq 3 and c m m geqq 2 are presented it is shown that r g m m m 2 and r c m m 1 m 1

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

positions of convex bodies associated to extremal problems and isotropic measures abstract we show that there are close relations between extremal problems in dual brunn minkowski theory and isotropic type properties for some borel measures on the sphere the methods we use allow us to obtain similar results in the context of firey brunn minkowski theory we also study reverse inequalities for dual mixed volumes which are related with classical positions such as l position or isotropic position

inequalities for mixed projection bodies mixed projection bodies are related to ordinary projection bodies zonoids in the same way that mixed volumes are related to ordinary volume analogs of the classical inequalities from the brunn minkowski theory such as the minkowski brunn minkowski and aleksandrov fenchel inequalities are developed for projection and mixed projection bodies

moment entropy inequalities it is shown that the product of the renyi entropies of two independent random vectors provides a sharp lower bound for the expected value of the moments of the inner product of the random vectors this new inequality contains important geometry such as extensions of one of the fundamental affine isoperimetric inequalities the blaschke santalo inequality

existence of blow up solutions in the energy space for the critical generalized kdv equation from these conservation laws h appears as an energy space so that it is a natural space in which to study the solutions note that p 2 is a special case for equation 2 indeed from the integrability theory see lax 14 we have for suitable u0 u0 and its derivatives with fast decay at infinity an infinite number of conservation laws the general question is to understand the dynamics induced by such equations local existence in time of solutions of 2 in the energy space is now well understood see kato 10 ginibre and tsutsumi 8 for the h theory s 32 kenig ponce and vega 11 for the l theory in the case of equation 1 and sharp h theory for 2 and bourgain 3 and 4 for the periodic case

anisotropic fractional perimeters the anisotropic fractional s perimeter with respect to a convex body k in rn is shown to converge as s 1 to the anisotropic perimeter with respect to the moment body of k for anisotropic fractional s seminorms on bv rn the corresponding result is established generalizing results of bourgain brezis mironescu and davila minimizers of the anisotropic fractional isoperimetric inequality with respect to k are shown to converge to the moment body of k as s 1 anisotropic fractional sobolev inequalities are established for a borel set e r and 0 s 1 the fractional s perimeter of e is given by ps e

parametric on line algorithms for packing rectangles and boxes abstract we present approximation algorithms for the following problems the two dimensional bin packing 2bp the three dimensional packing problem tpp and the container packing problem 3bp we consider the special case in which the items to be packed are small and must be packed on line we say an item is small if each of its dimension is at most 1 m of the respective dimension of the recipient where m is an integer greater than 1 these problems are denoted by 2bpm tppm and 3bpm and the performance of the algorithms are given in terms of the parameter m to our knowledge the parametric on line algorithms we present here have the best so far achieved asymptotic performance bounds for these problems for 2bpm and tppm we present algorithms with performance bound close to m 2 m 1 m 1 2 and for 3bpm we describe an algorithm with performance bound close to m 3 m 2 m2 1 m 1 2 for m 2 respectively m 3 these bounds are 2 112 and 3 112 respectively 1 73 and 2 285 the results on 2bpm and 3bpm extend the results on multidimensional on line packing presented by coppersmith and raghavan oper res lett 8 1 1989 17 the result on tppm improves the bound m 1 m 1 due to li and cheng siam j comput 19 1990 847 all these algorithms can be implemented to run in o nlogn time where n is the number of items to be packed

equation of motion for interacting pulses we develop a systematic method of deriving the equation of motion for interacting fronts or pulses in one dimension the theory is applicable to both dissipative and dispersive systems in the case of the time dependent ginzburg landau equation which is a typical example of a dissipative system the front equation obtained is the same as has been obtained previously the pulse interaction is also derived for the korteweg char21 de vries equation emphasizing the difference between the cases with and without dissipative terms

extended affine surface area during the twentieth century many of the notions of classical differential geometry have been extended so as to be defined on the boundaries of arbitrary convex bodies without smoothness requirements for example the projection measures quermassintegrals are the natural extensions of the integrals of mean curvature see santa16 27 pp 215 2321 for a discussion the surface area measures of aleksandrov fenchel jessen are extensions of the indefinite integrals of the elementary symmetric functions of the principal radii of curvature while federer s curvature measures when restricted to convex hypersurfaces are extensions of the indefinite integrals of the elementary symmetric functions of the principal curvatures see schneider 30 pp 27 331 one of the important concepts of classical differential geometry which has been difficult to extend is afline surface area this has often led to difficulties which when circumvented have been done so only by considerable effort and ingenuity for example suppose we are given a measure on the unit sphere which satisfies the hypothesis of the minkowski problem the solution of the minkowski problem guarantees the existence of a convex body whose surface area measure is the given measure from the given measure we can find the surface area of the body the minkowski mixed volume inequality can be used to obtain various upper bounds for the volume of the body however we cannot even inquire about the affine surface area of the body unless we can somehow convince ourselves that the convex body is sufficiently smooth results regarding aftine surface area have a surprising number of applications for example a number of geometric inequalities which appear to have no connection with affine surface area are in fact consequences of inequalities involving afline surface area see for example petty 25 and also 151 since polytopes have zero afftne surface area it seems quite surprising that the notion of afline surface area arises frequently in the area of polytope approximation see the survey of gruber 9 recently leichtweiss 11 121 presented a definition of extended afftne

stability of blow up profile and lower bounds for blow up rate for the critical generalized kdv equation the generalized korteweg de vries equations are a class of hamiltonian systems in infinite dimension derived from the kdv equation where the quadratic term is replaced by a higher order power term these equations have two conservation laws in the energy space h 1 l 2 norm and energy we consider in this paper the critical generalized kdv equation which corresponds to the smallest power of the nonlinearity such that the two conservation laws do not imply a bound in h 1 uniform in time for all h 1 solutions and thus global existence from 15 there do exist for this equation solutions u t such that u t h1 as t t where t we call them blow up solutions the question is to describe in a qualitative way how blow up occurs for solutions with l 2 mass close to the minimal mass allowing blow up and with decay in l 2 at the right we prove after rescaling and translation which leave invariant the l 2 norm that the solution converges to a universal profile locally in space at the blow up time t from the nature of this profile we improve the standard lower bound on the blow up rate for finite time blow up solutions

on polars of mixed projection bodies abstract recently lutwak established general minkowski inequality brunn minkowski inequality and aleksandrov fenchel inequality for mixed projection bodies in this paper following lutwak we established their polar forms as applications we prove some interrelated results

asymptotic n soliton like solutions of the subcritical and critical generalized korteweg de vries equations we consider the generalized korteweg de vries equations ut uxx upx 0t x r inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in the subcritical and critical cases p 2 3 4 or 5 let r j t x qc j x c j t x j where j 1 n be n soliton solutions of this equation with corresponding speeds 0 c 1 c 2 c n in this paper we construct a solution u t of the generalized korteweg de vries equation such that inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i this solution behaves asymptotically as t as the sum of n solitons without loss of mass by dispersion this is an exceptional behavior indeed being given the parameters c j 1 j n x j 1 j n we prove uniqueness of such a solution in the integrable cases p 2 and 3 such solutions are explicitly known and their properties were extensively studied in the literature they are called n soliton solutions therefore the existence result is new only for the nonintegrable cases the uniqueness result is new for all cases

lyapunov stability of ground states of nonlinear dispersive evolution equations on presente une nouvelle demonstration de la stabilite orbitale des solitons d etat fondamental de l equation de schrodinger non lineaire pour une grande classe de non linearites on demontre la stabilite de l onde solitaire pour l equation de korteweg de vries generalisee

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

the steiner formula for minkowski valuations a steiner type formula for continuous translation invariant minkowski valuations is established in combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations this new steiner type formula is used to obtain a family of brunn minkowski type inequalities for rigid motion intertwining minkowski valuations

the cos λ and sin λ transforms as intertwining operators between generalized principal series representations of sl n 1 k abstract in this article we connect topics from convex and integral geometry with well known topics in representation theory of semisimple lie groups by showing that the cos and sin transforms on the grassmann manifolds gr p k su n 1 k s u p k u n 1 p k are standard intertwining operators between certain generalized principal series representations induced from a maximal parabolic subgroup p p of sl n 1 k the index p indicates the dependence of the parabolic on p the general results of knapp and stein 1971 1980 20 21 and vogan and wallach 1990 44 then show that both transforms have meromorphic extension to c and are invertible for generic c furthermore known methods from representation theory combined with a selberg type integral allow us to determine the k spectrum of those operators

on universality of blow up profile for l 2 critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation iut u u 4 nu with initial condition u0 h1 existence of such solutions is known but the complete blow up dynamic is not understood so far for a specific set of initial data finite time blow up with a universal sharp upper bound on the blow up rate has been proved in 22 23

the three dimensional bin packing problem the problem addressed in this paper is that of orthogonally packing a given set of rectangular shaped items into the minimum number of three dimensional rectangular bins the problem is strongly np hard and extremely difficult to solve in practice lower bounds are discussed and it is proved that the asymptotic worst case performance ratio of the continuous lower bound is an exact algorithm for filling a single bin is developed leading to the definition of an exact branch and bound algorithm for the three dimensional bin packing problem which also incorporates original approximation algorithms extensive computational results involving instances with up to 90 items are presented it is shown that many instances can be solved to optimality within a reasonable time limit

slow blow up solutions for the h 1 mathbb r 3 critical focusing semilinear wave equation given 1 2 and 0 arbitrary we prove the existence of energy solutions of ttu u u5 0 0 1 in r3 1 which blow up exactly at r t 0 as t 0 these solutions are radial and of the form u t 1 2w t r r t inside the cone r t where t t 1 w r 1 r2 3 1 2 is the stationary solution of 0 1 and is a radiation term with r t x t 2 t x t 2 x t 6 dx 0 t 0 outside of the light cone there is the energy bound r t u x t 2 ut x t 2 u x t 6 dx 0 the regularity of u increases with as in our accompanying article on wave maps 10 the argument is based on a renormalization method for the soliton profile w r

projection problems for symmetric polytopes abstract let k r n be a convex body a compact convex subset with non empty interior k its projection body finding the least upper bound as k ranges over the class of origin symmetric convex bodies of the affine invariant ratio v k v k n 1 being called schneider s projection problem is a well known open problem in the convex geometry to study this problem lutwak yang and zhang recently introduced a new affine invariant functional for convex polytopes in r n for origin symmetric convex polytopes they posed a conjecture for the new functional u p in this paper we give an affirmative answer to the conjecture in r n thereby obtain a modified version of schneider s projection problem

a new affine invariant for polytopes and schneider s projection problem new affine invariant functionals for convex polytopes are introduced some sharp affine isoperimetric inequalities are established for the new functionals these new inequalities lead to fairly strong volume estimates for projection bodies two of the new affine isoperimetric inequalities are extensions of ball s reverse isoperimetric inequalities if k is a convex body i e a compact convex subset with nonempty interior in euclidean n space rn then on the unit sphere sn l its support function h k sn 1 r is defined for u e sn l by h k u max u y y e k where u y denotes the standard inner product of u and y the projection body hk of k can be defined as the convex body whose support function for u e sn 1 is given by h ihk u vol n1 klu1 where voln l denotes n 1 dimensional volume and klul denotes the image of the orthogonal projection of k onto the codimension 1 subspace orthogonal to u an important unsolved problem regarding projection bodies is schneider s projection problem what is the least upper bound as k ranges over the class of origin symmetric convex bodies in r of the affine invariant ratio v hnk v k n1 1 n1 where v is used to abbreviate vol see s1 s2 sw and le schneider s1 conjectured that this ratio is maximized by parallelotopes in s1 schneider also presented applications of such results in stochastic geometry however a counterexample was produced in br to show that this is not the case we will present a modified version of schneider s conjecture that has an affirmative answer in addition we will obtain an inequality that gives an upper bound for the affine ratio while our upper bound is not sharp for any n nevertheless it is asymptotically optimal to be more specific in this paper we introduce a new centro affine functional u defined on the class of polytopes which is closely related to the volume functional v while in general u k v k if k is a random polytope with many faces then u k is very close to v k we shall prove the following variation of schneider s projection conjecture received by the editors february 26 2000 1991 mathematics subject classification primary 52a40

general l_p affine isoperimetric inequalities sharp l p affine isoperimetric inequalities are established for the entire class of l p projection bodies and the entire class of l p centroid bodies these new inequalities strengthen the l p petty projection and the l p busemann petty centroid inequality

classification of invariant valuations on the quaternionic plane abstract we describe the orbit space of the action of the group sp 2 sp 1 on the real grassmann manifolds gr k h 2 in terms of certain quaternionic matrices of moore rank not larger than 2 we then give a complete classification of valuations on the quaternionic plane h 2 which are invariant under the action of the group sp 2 sp 1

lp affine isoperimetric inequalities the lp analogues of the petty projection inequality and the busemannpetty centroid inequality are established an affine isoperimetric inequality compares two functionals associated with convex or more general bodies where the ratio of the functionals is invariant under non degenerate linear transformations these isoperimetric inequalities are more powerful than their better known euclidean relatives

a global optimization algorithm for the three dimensional packing problem abstract this paper introduces a geometrical model which reduces the general three dimensional packing problem to a finite enumeration scheme a very efficient algorithm is derived several tests prove the validity of the algorithm in particular a numerical estimate of the asymtotic performance bound is given

inelastic interaction of nearly equal solitons for the quartic gkdv equation this paper describes the interaction of two solitons with nearly equal speeds for the quartic gkdv equation partial tu partial x partial x 2u u 4 0 quad t x in mathbb r 0 1 we call soliton a solution of 0 1 of the form u t x qc x ct y0 where c 0 y0 and q c q c 4 cq c since 0 1 is not an integrable model the general question of the collision of two given solitons q c 1 x c 1 t q c 2 x c 2 t with c1 c2 is an open problem