a singular parabolic equation with logarithmic nonlinearity and l p initial data abstract we study a parabolic equation with logarithmic nonlinearity and homogeneous dirichlet condition in a smooth bounded domain we firstly prove a result about existence of positive solutions with initial data in l p spaces 1 p in order to overcome the singular logarithmic nonlinearity among other arguments we employ hardy s inequality after the life span of the obtained solutions is studied in particular we show a connection between global existence and solutions of the associated elliptic problem

the finite toda lattices connection is established between one dimensional toda lattices constructed on the basis of the systems of simple roots of classical and affine lie algebras and other integrable systems of interacting particles that connection allows us to find new lattices differing from the known ones by the interaction of particles near the ends some of the new lattices admit non abelian generalizations

q less qr decomposition in inner product spaces abstract tensor computation is intensive and difficult invariably a vital component is the truncation of tensors so as to control the memory and associated computational requirements various tensor toolboxes have been designed for such a purpose in addition to transforming tensors between different formats in this paper we propose a simple q less qr truncation technique for tensors x i with x i r n 1 n d in the simple and natural kronecker product form it generalizes the qr decomposition with column pivoting adapting the well known gram schmidt orthogonalization process the main difficulty lies in the fact that linear combinations of tensors cannot be computed or stored explicitly all computations have to be performed on the coefficients i in an arbitrary tensor v i i x i the orthonormal q factor in the qr decomposition x x 1 x p q r cannot be computed but expressed as x r 1 when required the resulting algorithm has an o p 2 d n computational complexity with n max n i some illustrative examples in the numerical solution of tensor linear equations are presented

on the hardy constant of non convex planar domains the case of the quadrilateral abstract the hardy constant of a simply connected domain r 2 is the best constant for the inequality u 2 d x c u 2 dist x 2 d x u c c after the work of ancona where the universal lower bound 1 16 was obtained there has been a substantial interest on computing or estimating the hardy constant of planar domains in this work we determine the hardy constant of an arbitrary quadrilateral in the plane in particular we show that the hardy constant is the same as that of a certain infinite sectorial region which has been studied by e b davies

bogoyavlensky volterra and birkhoff integrable systems in this paper we examine an interesting connection between the generalized volterra lattices of bogoyavlensky and a special case of an integrable system defined by sklyanin the sklyanin system happens to be one of the cases in the classification of kozlov and treshchev of birkhoff integrable hamiltonian systems using this connection we demonstrate the integrability of the system and define a new lax pair representation in addition we comment on the bi hamiltonian structure of the system

the geometric nature of the flaschka transformation we show that the flaschka map originally introduced to analyze the dynamics of the integrable toda lattice system is the inverse of a momentum map we discuss the geometrical setting of the map and apply it to the generalized toda lattice systems on semisimple lie algebras the rigid body system on toda orbits and to coadjoint orbits of semidirect products groups in addition we develop an infinite dimensional generalization for the group of area preserving diffeomorphisms of the annulus and apply it to the analysis of the dispersionless toda lattice pde and the solvable rigid body pde

generalized hardy inequality for the magnetic dirichlet forms we obtain lower bounds for the magnetic dirichlet form in dimensions d 2 for d 2 the results generalize a well known lower bound by the magnetic field strength we replace the actual magnetic field bby an non vanishing effective field which decays outside the support of bas dist x suppb 2 in the case d 3 we establish that the magnetic form is bounded from below by the magnetic field strength if one assumes that the field does not vanish and its direction is slowly varying

on nonexistence of baras goldstein type for higher order parabolic equations with singular potentials the celebrated result by baras and goldstein 1984 established that the heat equation with singular inverse square potential in a smooth bounded domain n n 3 such that 0 e u t u c x 2 u in 0 t u 0 in the supercritical range c c hardy 1 n 2 2 2 does not have a solution for any nontrivial l 1 initial data uo x 0 in or for a positive measure namely it was proved that a regular approximation of a possible solution by a sequence u n x t of classical solutions of uniformly parabolic equations with bounded truncated potentials given by v x c x 2 v n x min c x 2 n n 1 diverges and as n oo u n x t in x 0 t in the present paper we reveal the connection of this very singular evolution with a spectrum of some limiting operator the proposed approach allows us to consider more general higher order operators for which hardy s inequalities were known since rellich 1954 and initial data that are not necessarily positive in particular it is established that under some natural hypothesis the divergence result is valid for any 2mth order parabolic equation with singular potential u t m u c x 2m u in 0 t where c c h m m 1 with zero dirichlet conditions on and for a wide class of initial data in particular typically the divergence holds for any data satisfying u 0 x is continuous at x 0 and uo 0 0 similar nonexistence i e divergence as e 0 results are also derived for time dependent potentials e 2m q x e t e 2m and nonlinear reaction terms u p e 2m x 2m with p 1 applications to other linear and semilinear schrodinger and wave pdes are discussed

on the sylvester equation ax x b c abstract we consider the solution of the sylvester equations ax x b c for t h and a b c n n and the related linear matrix equations axb x c axb cx d e and ax x a c solvability conditions and numerical methods are considered in terms of the generalized and periodic schur and qr decompositions we emphasize the square cases where m n but the rectangular cases will be considered

qr type factorization the yang baxter equation and an eigenvalue problem of control theory abstract it is shown that every solution to the yang baxter equation corresponds with the o r type algorithm the structure of solutions of the yang baxter equation is described and examples relevant to the algebraic riccati equation are considered

hardyʼs inequality and curvature a hardy inequality of the form f x pdx p 1p p 1 a x f x p x pdx for all f c0 r is considered for p 1 p 1 where is a domain in rnrn n 2n 2 r r is the ridge of and x x is the distance from x x to the boundary the main emphasis is on determining the dependence of a a on the geometric properties of a hardy inequality is also established for any doubly connected domain in r2r2 in terms of a uniformization of that is any conformal univalent map of onto an annulus

uniformly fat sets in this paper we study closed sets e which are locally uniformly fat with respect to a certain nonlinear riesz capacity we show that e is actually locally uniformly fat with respect to a weaker riesz capacity two applications of this result are given the first application is concerned with proving sobolev type inequalities in domains whose complements are uniformly fat the second application is concerned with the fekete points of e introduction let x x x2 x be a point in euclidean n space rn with jxl the norm of x for a 0 let i x ixi a n denote the riesz kernel of order a and put i f x f ix y f n f y dy x e r rn ic u x fr yj c n d u y x e r rn when f is a lebesgue integrable function and u a borel measure on rn here dy denotes lebesgue measure on rn if e c rn 0 1 define the a p outer riesz capacity of e by ra p e inf jfllfp ia f 1 on el where if lip is the lebesgue p norm of f we note that a 1 p 2 is the classical newtonian capacity if ap n we let rc p e inf jfjjf j f 1 on el where jc is the truncated riesz kernel defined by je x ixi a n 100 a n x 100 0 lxl 100 we shall say that a property holds a p quasi everywhere abbreviated a p q e on a set e if it holds on e except perhaps for a set of rop capacity zero next we list some properties of riesz capacities to simplify the writing we state these properties only for 0 u l p l 1 a p q e on e and p 1 a p q e on supp u c e moreover e jjli pi ujj rop e received by the editors march 11 1987 1980 mathematic3 subject cla33ification 1985 revi3ion primary 31c15 31b15 1988 american mathematical society 0002 9947 88 1 00 25 per page

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

quantum toda systems and lax pairs we describe a general method for constructing a lax pair representation of certain quantum mechanical systems that are integrable at the classical level this is then used to find conserved quantities at the quantum level for the toda systems

hardy inequalities for robin laplacians abstract in this paper we establish a hardy inequality for laplace operators with robin boundary conditions for convex domains in particular we show explicitly how the corresponding hardy weight depends on the coefficient of the robin boundary conditions we also study several extensions to non convex and unbounded domains

relativistic toda systems we present and study poincare invariant generalizations of the galilei invariant toda systems the classical nonperiodic systems are solved by means of an explicit action angle transformation

classical and quantum mechanical systems of toda lattice type i solutions to the classical periodic and non periodic toda lattice type hamiltonian systems are expressed in terms of an iwasawa type factorization of a large lie group the scattering of these systems is determined in the non periodic case for the generalized periodic toda lattices a generalization of kostant s formula is obtained using standard representations of affine lie groups

quantum cohomology of the infinite dimensional generalized flag manifolds abstract consider the infinite dimensional flag manifold lk t corresponding to the simple lie group k of rank l and with maximal torus t we show that for k of type a b or c if we endow the space h lk t r q 1 q l 1 where q1 ql 1 are multiplicative variables with an r q j bilinear product satisfying some simple properties analogous to the quantum product on qh k t then the isomorphism type of the resulting ring is determined by the integrals of motion of a certain periodic toda lattice system in exactly the same way as the isomorphism type of qh k t is determined by the integrals of motion of the non periodic toda lattice see ann math 149 1999 129 this is an infinite dimensional extension of the main result of mare relations in the quantum cohomology ring of g b preprint math dg 0210026 and at the same time a generalization of m a guest and t otofuji comm math phys 217 2001 475

limit matrices for the toda flow and periodic flags for loop groups sato s theory of infinite dimensional grassmannians has been applied to explain the geometry of the k p equation s djkm it has been used as a tool to study blow up behaviors and to regularize the solutions near the blow up a vm2 the point is that realizing the k p flow as a holomorphic flow of planes enables one to follow what happens to the limiting planes as the equation in the original bad coordinates blows up the blow up behaviors are characterized by the various strata the orbit of planes visits in the grassmannian in this paper such ideas are applied to the n periodic toda flow on periodic jacobi matrices which translates into a flow on the space of n periodic flags of planes in the grassmannians indeed here the n periodic toda flow amounts to n coupled kp equations with special interactions between time flows u t how such matrices blow up has been studied in fi f1 ha a vm1 for arbitrary lie algebras and kac moody lie algebras whereas this paper focusses on regularizing the flow near the blow up locus that is on finding the boundary of isospectral sets if n periodic jacobi matrices

sharp hardy inequalities on the solid torus abstract in this paper we establish the classical hardy inequality in the solid torus and some variants of it the general idea is to use the fact that sobolev embeddings can be improved in the presence of symmetries in all cases using techniques that exploit the symmetry presented by the solid torus we calculate the displayed best constants and we prove that they are the same as the standard hardy best constants which appear in convex domains although the solid torus is not convex

algorithms for numerical analysis in high dimensions nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension the separated representation introduced previously allows many operations to be performed with scaling that is formally linear in the dimension in this paper we further develop this representation by i discussing the variety of mechanisms that allow it to be surprisingly efficient ii addressing the issue of conditioning iii presenting algorithms for solving linear systems within this framework and iv demonstrating methods for dealing with antisymmetric functions as arise in the multiparticle schrodinger equation in quantum mechanics numerical examples are given

symmetries conserved quantities and hierarchies for some lattice systems with soliton structure basic invariants such as conserved quantities symmetries mastersymmetries and recursion operators are explicitly constructed for the following nonlinear lattice systems the modified korteweg de vries lattice the ablowitz ladik lattice the brusci ragnisco lattice the ragnisco tu lattice and some cases of the class of integrable systems introduced by bogoyavlensky the algorithmic basis for obtaining these quantities is described and the interrelation between the underlying mastersymmetry approach and the lax pair analysis is discussed by explicit presentation of the higher order members of the corresponding hierarchies new completely integrable lattice flows are found for all systems multi hamiltonian formulations are given

a unified approach to improved l p hardy inequalities with best constants we present a unified approach to improved l p hardy inequalities in r n we consider hardy potentials that involve either the distance from a point or the distance from the boundary or even the intermediate case where the distance is taken from a surface of codimension 1 k n in our main result we add to the right hand side of the classical hardy inequality a weighted l p norm with optimal weight and best constant we also prove nonhomogeneous improved hardy inequalities where the right hand side involves weighted l q norms q p

a geometrical version of hardy s inequality we prove a version of hardy s type inequality in a domain omega subset of r n which involves the distance to the boundary and the volume of omega in particular we obtain a result which gives a positive answer to a question asked by h brezis and m marcus

integrable hamiltonians with exponential potential abstract we construct a large class of integrable hamiltonian systems with n degrees of freedom this class naturally extends the nonperiodic hamiltonians of toda lattice type

a wpn theory of parabolic equations with unbounded leading coefficients on non smooth domains we study parabolic partial differential equations with unbounded second first and zero order coefficients on non smooth domains allowing hardy inequality existence and uniqueness results are given in weighted sobolev spaces and holder estimates of the solutions are also obtained the number of derivatives of the solutions can be any nonnegative real number in particular it can be fractional

unitarity of the sov transform for the toda chain the quantum separation of variables method consists in mapping the original hilbert space where a spectral problem is formulated onto one where the spectral problem takes a simpler separated form in order to realise such a program one should construct the map explicitly and then show that it is unitary in the present paper we develop a technique which allows one to prove the unitarity of this map in the case of the quantum toda chain our proof solely builds on objects and relations naturally arising in the framework of the so called quantum inverse scattering method hence with minor modifications it should be readily transposable to other quantum integrable models solvable by the quantum separation of variables method as such it provides an important alternative to the proof of the map s unitarity based on the group theoretical interpretation of the quantum toda chain which is absent for more complex quantum integrable models

korn inequality on irregular domains abstract in this paper we study the weighted korn inequality on some irregular domains e g s john domains and domains satisfying quasihyperbolic boundary conditions examples regarding sharpness of the korn inequality on these domains are presented moreover we show that korn inequalities imply certain poincare inequality

estimates for the optimal constants in multipolar hardy inequalities for schrodinger and dirac operators by expanding squares we prove several hardy inequalities with two critical singularities and constants which explicitly depend upon the distance between the two singularities these inequalities involve the l2 norm such results are generalized to an arbitrary number of singularities and compared with standard results given by the ims method the generalized version of hardy inequalities with several singularities is equivalent to some spectral information on a schrodinger operator involving a potential with several inverse square singularities we also give a generalized hardy inequality for dirac operators in the case of a potential having several singularities of coulomb type which are critical for dirac operators

the toda lattice dynkin diagrams singularities and abelian varieties the periodic l 1 particle toda lattices coresponding to extended dynkin diagrams have l 1 polynomial constants of the motion namely as many as there are dots in the dynkin diagram for most values of the constants their intersection defines as l dimensional affine invariant manifold which completes into a complex algebraic torus abelian variety by glueing on a divisor entirely specified by the extended dynkin diagram therefore the global geometry of the complex invariant tori such as polarization dimension of certain linear systems divisor equivalences etc is entirely given by the extended dynkin diagram in particular an explicit geometric description for the case of three particle toda lattices is given which describes the divisors their singularities and how the various divisors intersect

weighted hardy inequalities abstract for bounded lipschitz domains d in r n it is known that if 1 p then for all 0 0 where 0 p 1 0 there is a constant c with d u x p dist x d p d x c d u x p dist x d d x for all u c 0 d we show that if d is merely assumed to be a bounded domain in r n that satisfies a whitney cube counting condition with exponent and has plump complement then the same inequality holds with 0 now taken to be p n n p n p 2 n further we extend the known results see h brezis m marcus hardy s inequalities revisited dedicated to ennio de giorgi ann scuola norm sup pisa cl sci 4 25 1997 1998 217 237 m hoffmann ostenhof t hoffmann ostenhof a laptev a geometrical version of hardy s inequality j funct anal 189 2002 537 548 j tidblom a geometrical version of hardy s inequality for w 1 p proc amer math soc 132 2004 2265 2271 concerning the improved hardy inequality d u x p dist x d p d x d p n d u x p d x c d u x p d x c c n p by showing that the class of domains for which the inequality holds is larger than that of all bounded convex domains

four algorithms for the the efficient computation of truncated pivoted qr approximations to a sparse matrix in this paper we propose four algorithms to compute truncated pivoted qr approximations to a sparse matrix three are based on the gram schmidt algorithm and the other on householder triangularization all four algorithms leave the original matrix unchanged and the only additional storage requirements are arrays to contain the factorization itself thus the algorithms are particularly suited to determining low rank approximations to a sparse matrix

hardy inequalities for p laplacians with robin boundary conditions in this paper we study the best constant in a hardy inequality for the p laplace operator on convex domains with robin boundary conditions we show in particular that the best constant equals p

a hardy inequality in the half space paper 1 a geometrical version of hardy s inequality for w 0 1 p d the aim of this article is to prove a hardy type inequality concerning functions in w 0 1 p d for some domain d in r n involving the volume of d and the distance to the boundary of d the inequality is a generalization of a previously proved inequality by m and t hoffmann ostenhof and a laptev which dealt with the special case p 2 paper 2 a hardy inequality in the half space here we prove a hardy type inequality in the half space which generalize an inequality originally proved by v maz ya to the so called l p case this inequality had previously been conjectured by the mentioned author we will also improve the constant appearing in front of the reminder term in the original inequality which is the first improved hardy inequality appearing in the litterature paper 3 hardy type inequalities for many particle systems in this article we prove some results about the constants appearing in hardy inequalities related to many particle systems we show that the problem of estimating the best constants there is related to some interesting questions from geometrical combinatorics the asymptotical behaviour when the number of particles approaches infinity of a certain quantity directly related to this is also investigated paper 4 various results in the theory of hardy inequalities and personal thoughts in this article we give some further results concerning improved hardy inequalities in half spaces and other conic domains also some examples of applications of improved hardy inequalities in the theory of viscous incompressible flow will be given