the total coordinate ring of a wonderful variety abstract we study the cone of effective divisors and the total coordinate ring of wonderful varieties with applications to their automorphism group we show that the total coordinate ring of any spherical variety is obtained from that of the associated wonderful variety by a base change of invariant subrings

effective and big divisors on a projective symmetric variety abstract we describe the effective and the big cones of a projective symmetric variety moreover we give a necessary and sufficient combinatorial criterion for the bigness of a nef divisor on a projective symmetric variety when the divisor is g stable such criterion has an explicit geometric interpretation finally we describe the spherical closure of a symmetric subgroup

local rigidity of quasi regular varieties for a g variety x with an open orbit we define its boundary x as the complement of the open orbit the action sheaf s x is the subsheaf of the tangent sheaf made of vector fields tangent to x we prove for a large family of smooth spherical varieties the vanishing of the cohomology groups h i x s x for i 0 extending results of bien and brion compos math 104 1 26 1996 we apply these results to study the local rigidity of the smooth projective varieties with picard number one classified in pasquier math ann in press

spherical blow ups of grassmannians and mori dream spaces abstract in this paper we classify weak fano varieties that can be obtained by blowing up general points in prime fano varieties we also classify spherical blow ups of grassmannians in general points and we compute their effective cone these blow ups are in particular mori dream spaces furthermore we compute the stable base locus decomposition of the blow up of a grassmannian in one point and we show how it is determined by linear systems of hyperplanes containing the osculating spaces of the grassmannian at the blown up point and by the rational normal curves in the grassmannian passing through the blown up point

dynamic behaviour of beams and rectangular plates under moving loads abstract a theory concerning the dynamic response of finite elastic structures rayleigh beams and plates having arbitrary end supports and under an arbitrary number of moving masses is developed the versatile solution technique presented is based on modified generalized finite integral transforms and the modified struble s method various analytical results are presented numerical examples are given and the results compare favourably with existing ones the efficiency of the solution technique is demonstrated and discussed

on the cauchy problem of 3 d energy critical schrodinger equations with subcritical perturbations abstract we investigate the global well posedness scattering and blow up phenomena when the 3 d quintic nonlinear schrodinger equation which is energy critical is perturbed by a subcritical nonlinearity 1 u p u we find when the quintic term is defocussing then the solution is always global no matter what the sign of 1 is scattering will occur either when the perturbation is defocussing and 4 3 p 4 or when the mass of the solution is small enough and 4 3 p 4 when the quintic term is focusing we show the blow up for certain solutions

on differential operators on complete symmetric varieties of type a1 and a2 abstract in this paper we consider the algebra of global differential operators d x on wonderful compactifications x of symmetric spaces g h of type a 1 and a 2 we will first construct a global differential operator on these varieties that does not come from the infinitesimal action of the lie algebra of g we will then focus on type a 2 where we will show that d x is an algebra of finite type and that for any invertible sheaf l on x h 0 x l is either 0 or a simple left d x l module finally we will show with the help of local cohomology that this still holds for higher cohomology groups h i x l

the ring of sections of a complete symmetric variety we study the ring of sections a x of a complete symmetric variety x that is of the wonderful completion of g h where g is an adjoint semisimple group and h is the fixed subgroup for an involutorial automorphism of g we find generators for pic x we generalize the prv conjecture to complete symmetric varieties and construct a standard monomial theory for a x that is compatible with g orbit closures in x this gives a degeneration result and the rational singularityness for a x 2003 elsevier science usa all rights reserved

nonlocal elasticity theory for vibration of nanoplates classical plate theory clpt and first order shear deformation theory fsdt of plates are reformulated using the nonlocal differential constitutive relations of eringen the equations of motion of the nonlocal theories are derived navier s approach has been used to solve the governing equations for simply supported boundary conditions analytical solutions for vibration of the nanoplates such as graphene sheets are presented nonlocal theories are employed to bring out the effect of the nonlocal parameter on natural frequencies of the nanoplates the developed theory has been extended to the analysis of double layered nanoplates effect of i nonlocal parameter ii length iii height iv elastic modulus and v stiffness of winkler foundation of the plate on nondimensional vibration frequencies are investigated the theoretical development as well as numerical solutions presented herein should serve as reference for nonlocal theories of nanoplates and nanoshells

nilpotent orbits normality and hamiltonian group actions the theory of coadjoint orbits of lie groups is central to a number of areas in mathematics a list of such areas would include 1 group representation theory 2 symmetry related hamiltonian mechanics and attendant physical theories 3 symplectic geometry 4 moment maps and 5 geometric quantization from many points of view the most interesting cases arise when the group g in question is semisimple for semisimple g the most familiar of the orbits are orbits of semisimple elements in that case the associated representation theory is pretty much understood the borel weil bott theorem and noncompact analogs e g zuckerman functors the isotropy subgroups are reductive and the orbits are in one form or another related to flag manifolds and their natural generalizations a totally different experience is encountered with nilpotent orbits of semisimple groups here the associated representation theory unipotent representations is poorly understood and there is a loss of reductivity of isotropy subgroups to make matters worse or really more interesting orbits are no longer closed and there can be a failure of normality for orbit closures in addition simple connectivity is generally gone but more seriously there may exist no invariant polarizations the interest in nilpotent orbits of semisimple lie groups has increased sharply over the last two decades this perhaps may be attributed to the recurring experience that sophisticated aspects of semisimple group theory often lead one to these orbits e g the springer correspondence with representations of the weyl group this paper presents new results concerning the symplectic and algebraic geometry of the nilpotent orbits 0 and the symmetry groups of that geometry the starting point is the recognition made also by others that the ring r of regular functions on any g homogeneous cover m of 0 is not only a poisson algebra the case for any coadjoint orbit but that r is also naturally graded the key theme is that the same nilpotent orbit may be shared by more than one simple group the key result is the determination of all pairs of simple lie groups having a shared nilpotent orbit furthermore there is then a unique maximal such group and this group is encoded in the symplectic and algebraic

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

solitons and scattering for the cubic quintic nonlinear schrodinger equation on mathbb r 3 r 3 we consider the cubic quintic nonlinear schrodinger equation i partial t u delta u u 2u u 4u in the first part of the paper we analyze the one parameter family of ground state solitons associated to this equation with particular attention to the shape of the associated mass energy curve additionally we are able to characterize the kernel of the linearized operator about such solitons and to demonstrate that they occur as optimizers for a one parameter family of inequalities of gagliardo nirenberg type building on this work in the latter part of the paper we prove that scattering holds for solutions belonging to the region mathcal r of the mass energy plane where the virial is positive we show that this region is partially bounded by solitons also by rescalings of solitons which are not soliton solutions in their own right the discovery of rescaled solitons in this context is new and highlights an unexpected limitation of any virial based methodology

an introduction to wonderful varieties with many examples of type f4 we give an introduction to the theory of wonderful g varieties with many examples when g is simple of type f4 we present results and open problems about these varieties on their classification on their isotropy groups on morphisms between them and on their relations with the representation theory of g

self focusing on bounded domains the critical nonlinear schrodinger equation nls on bounded domains models the propagation of cw laser beams in hollow core fibers unlike the nls on unbounded domains which models propagation in bulk media the ground state waveguide solutions are stable and the condition of critical power for singularity formation is generically sharp

a classification of multiplicity free actions abstract the action of a complex algebraic groupgon an affine varietyvis said to be multiplicity free if each irreducible representation ofgoccurs at most once in the ring c v of regular functions onv this paper concerns the classical setting wherevis a vector space andgacts linearly onv the irreducible linear multiplicity free actions have been classified by v kac the focus here is on nonirreducible linear multiplicity free actions we exhibit many such actions which do not decompose as direct products of irreducible actions our main results provide a classification for all linear multiplicity free actions on a vector space

automorphisms root systems and compactifications of homogeneous varieties let g be a complex semisimple group and let h c g be the group of fixed points of an involutive automorphism of g then x g h is called a symmetric variety in cp de concini and procesi have constructed an equivariant compactification x which has a number of remarkable properties some of them being i the boundary is the union of divisors d1 dr ii there are exactly 2r orbits their closures are the intersections di1 n ndi even schematically in particular there is only one closed orbit iii in case g is of adjoint type all orbit closures are smooth it is called the wonderful embedding of x or a complete symmetric variety and is the foundation for most deeper results about x independently luna and vust developed in lv a general theory of equivariant compactifications of homogeneous varieties under a connected reductive group g in particular they realized the reason which makes symmetric varieties behave so nicely a borel subgroup b has an open dense orbit in g h varieties with this property are called spherical luna and vust were able to describe all equivariant compactifications of them in terms of combinatorial data very similar to torus embeddings which are actually a special case they obtained in particular that every spherical embedding has only finitely many orbits nevertheless the reason for the existence of a compactification with properties i iii remained mysterious then brion and pauer established a relation with the automorphism group they proved in bp a spherical variety x g h possesses an equivariant compactification with exactly one closed orbit if and only if autg x ng h h is finite in this case there is a unique one which dominates all others the wonderful compactification x they also showed that the orbits of x correspond to the faces of a strictly convex polyhedral cone z then properties i and ii above are equivalent to z being simplicial this fact is much deeper and was proved by brion in brl in fact he showed much more let f be the set of characters of b which are the characters of a rational b eigenfunction on x this is a finitely generated free abelian group then the cone z is a subset of the real vector space hom f r brion showed that there is a finite reflection group wx acting on f such that z is one of its weyl chambers in case of a symmetric variety wx is its little weyl group

a natural constraint approach to normalized solutions of nonlinear schrodinger equations and systems abstract the paper deals with the existence of normalized solutions to the system u 1 u 1 u 3 u v 2 in r 3 v 2 v 2 v 3 u 2 v in r 3 r 3 u 2 a 1 2 and r 3 v 2 a 2 2 for any 1 2 a 1 a 2 0 and 0 prescribed we present a new approach that is based on the introduction of a natural constraint associated to the problem we also show that as phase separation occurs for the solutions that we find our method can be adapted to scalar nonlinear schrodinger equations with normalization constraint and leads to alternative and simplified proofs to some results already available in the literature

the cox ring of a spherical embedding abstract let g be a connected reductive group and g h a spherical homogeneous space we show that the ideal of relations between a natural set of generators of the cox ring of a g embedding of g h can be obtained by homogenizing certain equations which depend only on the homogeneous space using this result we describe some examples of spherical homogeneous spaces such that the cox ring of any of their g embeddings is defined by one equation

the dynamics of the 3d radial nls with the combined terms in this paper we show the scattering and blow up result of the radial solution with the energy below the threshold for the nonlinear schrodinger equation nls with the combined terms iu t delta u u 4 u u 2 u qquad qquad qquad qquad rm cnls in the energy space h 1 mathbb r 3 the threshold is given by the ground state w for the energy critical nls iu t u u 4 u this problem was proposed by tao visan and zhang in comm pdes 32 1281 1343 2007 the main difficulty is lack of the scaling invariance illuminated by ibrahim et al in analysis pde 4 3 405 460 2011 we need to give the new radial profile decomposition with the scaling parameter then apply it to the scattering theory our result shows that the defocusing dot h 1 subcritical perturbation u 2 u does not affect the determination of the threshold of the scattering solution of cnls in the energy space

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

global well posedness and scattering for nonlinear schrodinger equations with combined nonlinearities in the radial case we consider the cauchy problem for the nonlinear schrodinger equation with combined nonlinearities one of which is defocusing mass critical and the other is focusing energy critical or energy subcritical the threshold is given by means of variational argument we establish the profile decomposition in h1 rd and then utilize the concentration compactness method to show the global wellposedness and scattering versus blowup in h1 rd below the threshold for radial data when d 4

global spherical tropicalization via toric embeddings the first steps in defining tropicalization for spherical varieties have been taken in the last few years there are two parts to this theory tropicalizing subvarieties of homogeneous spaces and tropicalizing their closures in spherical embeddings in this paper we obtain a new description of spherical tropicalization that is equivalent to the other theories this works by embedding in a toric variety tropicalizing there and then applying a particular piecewise projection map we use this theory to prove that taking closures commutes with the spherical tropicalization operation

normalized ground states for the nls equation with combined nonlinearities abstract we study existence and properties of ground states for the nonlinear schrodinger equation with combined power nonlinearities u u u q 2 u u p 2 u in r n n 1 having prescribed mass r n u 2 a 2 under different assumptions on q p a 0 and r we prove several existence and stability instability results in particular we consider cases when 2 q 2 4 n p 2 q p i e the two nonlinearities have different character with respect to the l 2 critical exponent these cases present substantial differences with respect to purely subcritical or supercritical situations which were already studied in the literature we also give new criteria for global existence and finite time blow up in the associated dispersive equation

moving load analysis of arbitrarily shaped plates using the b3 spline finite strip method the b3 spline finite strip method is reviewed and its application is extended to the moving load analysis of arbitrarily shaped plates rectangular skew and circular sector plates are traversed by constant force moving loads it is shown that this method gives results comparing favorably with results from the literature

new and old results on spherical varieties via moduli theory abstract given a connected reductive algebraic group g and a finitely generated monoid of dominant weights of g in 2005 alexeev and brion constructed a moduli scheme m for multiplicity free affine g varieties with weight monoid this scheme is equipped with an action of an adjoint torus t ad and has a distinguished t ad fixed point x 0 in this paper we obtain a complete description of the t ad module structure in the tangent space of m at x 0 for the case where is saturated using this description we prove that the root monoid of any affine spherical g variety is free as another application we obtain new proofs of uniqueness results for affine spherical varieties and spherical homogeneous spaces first proved by losev in 2009 furthermore we obtain a new proof of alexeev and brion s finiteness result for multiplicity free affine g varieties with a prescribed weight monoid at last we prove that for saturated all the irreducible components of m equipped with their reduced subscheme structure are affine spaces

symmetric decreasing rearrangement can be discontinuous suppose f xl x2 0 is a continuously differentiable function supported in the unit disk in the plane its symmetric decreasing rearrangement is the rotationally invariant function f xl x2 whose level sets are circles enclosing the same area as the level sets of f such rearrangement preserves lp norms but decreases convex gradient integrals e g p p 1 p 0 j 1 2 3 is a sequence of infinitely differentiable functions also supported in the unit disk which converge uniformly together with first derivatives to f the symmetzed functions also converge uniformly the real question is about convergence of the derivatives of the symmetrized functions we announce that the derivatives of the symmetrized functions need not converge strongly e g it can happen that fj f p 0 for every p we further characterize exactly those f s for which convergence is assured and for which it can fail

a new rearrangement inequality and its application for l 2 constraint minimizing problems in this paper we introduce a new type rearrangement inequality based on the steiner rearrangement to show orbital stability of standing wave solutions with respect to a nonlinear schrodinger equation h 1 precompactness of minimizing sequence of l 2 constraint minimizing problem is very important usually by using the concentration compactness principle and some scaling argument h 1 precompactness is obtained to prove h 1 precompactness without scaling arguments we introduce the rearrangement the steiner rearrangement is defined as a map from h 1 to h 1 whereas our rearrangement cdot star cdot is defined as a map from h 1 times h 1 to h 1 by using the rearrangement we show a strict inequality vert nabla u star v vert l 2 2 vert nabla u vert l 2 2 vert nabla v vert l 2 2 under simple assumptions

uniqueness property for spherical homogeneous spaces let g be a connected reductive group recall that a homogeneous g space x is called spherical if a borel subgroup b g has an open orbit on x to x one assigns certain combinatorial invariants the weight lattice the valuation cone and the set of b stable prime divisors we prove that two spherical homogeneous spaces with the same combinatorial invariants are equivariantly isomorphic further we recover the group of g equivariant automorphisms of x from these invariants

multiple normalized solutions for quasi linear schrodinger equations abstract in this paper we prove the existence of two solutions having a prescribed l 2 norm for a quasi linear schrodinger equation one of these solutions is a mountain pass solution relative to a constraint and the other one a minimum either local or global to overcome the lack of differentiability of the associated functional we rely on a perturbation method developed in 25

homogeneous coordinates for algebraic varieties abstract we associate to every divisorial e g smooth variety x with only constant invertible global functions and finitely generated picard group a pic x graded homogeneous coordinate ring this generalizes the usual homogeneous coordinate ring of the projective space and constructions of cox and kajiwara for smooth and divisorial toric varieties we show that the homogeneous coordinate ring defines in fact a fully faithful functor for normal complex varieties x with only constant global functions we even obtain an equivalence of categories finally the homogeneous coordinate ring of a locally factorial complete irreducible variety with free finitely generated picard group turns out to be a krull ring admitting unique factorization

assessment of nanotube structures under a moving nanoparticle using nonlocal beam theories abstract dynamic analysis of nanotube structures under excitation of a moving nanoparticle is carried out using nonlocal continuum theory of eringen to this end the nanotube structure is modeled by an equivalent continuum structure ecs according to the nonlocal euler bernoulli timoshenko and higher order beam theories the nondimensional equations of motion of the nonlocal beams acted upon by a moving nanoparticle are then established analytical solutions of the problem are presented for simply supported boundary conditions the explicit expressions of the critical velocities of the nonlocal beams are derived furthermore the capabilities of various nonlocal beam models in predicting the dynamic deflection of the ecs are examined through various numerical simulations the role of the scale effect parameter the slenderness ratio of the ecs and velocity of the moving nanoparticle on the time history of deflection as well as the dynamic amplitude factor of the nonlocal beams are scrutinized in some detail the results show the importance of using nonlocal shear deformable beam theories particularly for very stocky nanotube structures acted upon by a moving nanoparticle with low velocity

existence and instability of standing waves with prescribed norm for a class of schrodinger poisson equations in this paper we study the existence and the instability of standing waves with prescribed l norm for a class of schrodinger poisson slater equations in r 0 1 i t x 1 p 2 0 when p 10 3 6 to obtain such solutions we look to critical points of the energy functional f u 1 2 ou 2l2 r3 1 4

primitive wonderful varieties we complete the classification of wonderful varieties initiated by d luna we review the results that reduce the problem to the family of primitive varieties and report the references where some of them have already been studied finally we analyze the rest case by case

manin s conjecture for certain spherical threefolds abstract we prove manin s conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for a spherical threefold with canonical singularities and two infinite families of spherical threefolds with log terminal singularities moreover we show that one of these families does not satisfy a conjecture of batyrev and tschinkel on the leading constant in the asymptotic formula our proofs are based on the universal torsor method using brion s description of cox rings of spherical varieties

small scale effect on the vibration of thin nanoplates subjected to a moving nanoparticle via nonlocal continuum theory the vibration of elastic thin nanoplates traversed by a moving nanoparticle involving coulomb friction is investigated using the nonlocal continuum theory of eringen the eigen function technique and the laplace transform method are employed to solve the governing equations of the nanoplate the explicit expressions of the in plane and transverse displacements are obtained when the moving nanoparticle traverses the nanoplate on an arbitrary straight line in a special case the obtained results are also compared with those of other researchers and a reasonably good agreement is achieved the effects of small scale parameters and velocity of the moving nanoparticle on the dynamic response as well as the dynamic amplitude factors dafs of the in plane and transverse displacements are then explored in some detail the results indicate that the magnitude of daf of the transverse displacement of the nanoplate i e daf w increases with the first small scale effect parameter irrespective of the values of the second small scale effect parameter and the velocity of the moving nanoparticle as the first small scale effect parameter grows the maximum values of dafw as a function of the moving nanoparticle velocity increase and generally occur in the lower levels of the moving nanoparticle velocity

prediction capabilities of classical and shear deformable beam models excited by a moving mass abstract in this paper a comprehensive assessment of design parameters for various beam theories subjected to a moving mass is investigated under different boundary conditions the design parameters are adopted as the maximum dynamic deflection and bending moment of the beam to this end discrete equations of motion for classical euler bernoulli timoshenko and higher order beams under a moving mass are derived based on hamilton s principle the reproducing kernel particle method rkpm and extended newmark method are utilized for spatial and time discretization of the problem correspondingly the design parameter spectra in terms of the beam slenderness mass weight and velocity of the moving mass are introduced for the mentioned beam theories as well as various boundary conditions the results indicate the existence of a critical beam slenderness mostly as a function of beam boundary condition in which for slenderness lower than this so called critical one the application of euler bernoulli or even timoshenko beam theories would underestimate the real dynamic response of the system moreover there would be a roughly linear relation between the weight of the moving mass and the design parameters for a certain value of the moving mass velocity in most cases of boundary conditions

wonderful subgroups of reductive groups and spherical systems abstract let g be a semisimple complex algebraic group and h g a wonderful subgroup we prove several results relating the subgroup h to the properties of a combinatorial invariant s of g h called its spherical system it is also possible to consider a spherical system s as a datum defined by purely combinatorial axioms and under certain circumstances our results prove the existence of a wonderful subgroup h associated with s as a byproduct we reduce for any group g the proof of the classification of wonderful g varieties known as the luna conjecture to its verification on a small family of cases called primitive