on the hyperbolic kac moody lie algebra ha_1 1 in this paper using a homological theory of graded lie algebras and the representation theory of a we compute the root multiplicities of the hyperbolic kac moody lie algebra ha up to level 4 and deduce some interesting combinatorial identities introduction the hyperbolic kac moody lie algebras have been considered as the next natural objects of study after the affine case while the affine kac moody lie algebras have been extensively studied for their close connections to areas such as combinatorics modular forms and mathematical physics many basic questions regarding the hyperbolic case are still unresolved for example the behavior of the root multiplicities is not well understood feingold and frenkel f f and kac moody and wakimoto k m w made some progress in this area they computed the level 2 root multiplicities for the hyperbolic kac moody lie algebras ha 1 and he 1 respectively in kangl and kang2 we introduced an inductive program to study the higher level root multiplicities and the principally specialized affine characters of a certain class of lorentzian kac moody lie algebras more precisely we realize these lorentzian kac moody lie algebras as the minimal graded lie algebras l efez ln with local part v lo v where lo is an affine kac moody lie algebra v is the basic representation of lo and v is the contragredient of v thus l g i where g nez gn is the maximal graded lie algebra with local part v lo v and i ednez i is the maximal graded ideal of g which intersects the local part trivially by developing a homological theory we determined the structure of homogeneous subspaces ln g ii as modules over the affine kac moody lie algebra lo for certain higher levels the hochschild serre spectral sequences played an important role in determining the structure of i n for the hyperbolic kac moody lie algebra ha 1 we were able to determine the structure of l4 for n 1 2 5 and computed the principally specialized affine characters up to level 5 but the root multiplicities were computed up to level 3 only in this paper using the representation theory of a 1 developed received by the editors june 4 1990 and in revised form september 12 1991 1991 mathematics subject classification primary 1 7b67 1 7b65 secondary 1 7b10 this work was supported by nsf grant dms 8806371 1994 american mathematical society 0002 9947 94 1 00 25 per page

convergence of an iterative method for total variation denoising in total variation denoising one attempts to remove noise from a signal or image by solving a nonlinear minimization problem involving a total variation criterion several approaches based on this idea have recently been shown to be very effective particularly for denoising functions with discontinuities this paper analyzes the convergence of an iterative method for solving such problems the iterative method involves a lagged diffusivity approach in which a sequence of linear diffusion problems are solved global convergence in a finite dimensional setting is established and local convergence properties including rates and their dependence on various parameters are examined

a homology vanishing theorem for kac moody algebras with coefficients in the category o in this paper we study the homology of a kac moody lie algebra associated to a symmetrizable generalized cartan matrix with coefficients in a module m in the category 0 we show that all the homology groups vanish if the casimir operator acts as an automorphism on m in fact we prove a slightly more general theorem theorem 1 2 this is one of the main theorems of this paper we further study the homology of g with coefficients in arbitrary verma modules and also integrable highest weight modules wc prove proposition 1 5 that hi g m a 0 for all i o and for all the verma modules m a with highest weight e h such that i is not equal to wp p for any w e w whereas hi g m wp p w 1 ip j h this result in the case when g is a finite dimensional semi simple lie algebra is due to williams if l 1 is an integrable highest weight module with i o 2 is automatically dominant integral then we have using bgg resolution and proposition 1 5 that hi g l i 0 for all ia 0 this is the content of our theorem 1 7 in the case when g is finite dimensional the integrable highest weight modules are precisely the finite dimensional irreducible modules in this case this theorem is well known and is due to whitehead as in the finite dimensional case the situation when j 0 so that l i1 is the trivial one dimensional module is drastically different we have recently proved ku theorem 1 61 that the homology of the commutator

convection diffusion equation with absorption and non decaying initial data in this paper we give the precise description of the large time behavior of the solution u of the cauchy problem tu u a u u in rn 0 u x 0 x 0in rn by using the ordinary differential equation and a linear parabolic equation here n 1 a rn 1 1 0 and is a bounded continuous function such that lp rn for some 1 p furthermore we study the large time behavior of the hot spots for the solution u

on total variation minimization and surface evolution using parametric maximum flows in a recent paper boykov et al lncs vol 3953 pp 409 422 2006 propose an approach for computing curve and surface evolution using a variational approach and the geo cuts method of boykov and kolmogorov international conference on computer vision pp 26 33 2003 we recall in this paper how this is related to well known approaches for mean curvature motion introduced by almgren et al siam journal on control and optimization 31 2 387 438 1993 and luckhaus and sturzenhecker calculus of variations and partial differential equations 3 2 253 271 1995 and show how the corresponding problems can be solved with sub pixel accuracy using parametric maximum flow techniques this provides interesting algorithms for computing crystalline curvature motion possibly with a forcing term

kac moody lie algebras spectral sequences and the witt formula this research was supported by kosef grant 98 0701 01 5 l and the young scientist award korean academy of science and technology

a convergent overlapping domain decomposition method for total variation minimization in this paper we are concerned with the analysis of convergent sequential and parallel overlapping domain decomposition methods for the minimization of functionals formed by a discrepancy term with respect to the data and a total variation constraint to our knowledge this is the first successful attempt of addressing such a strategy for the nonlinear nonadditive and nonsmooth problem of total variation minimization we provide several numerical experiments showing the successful application of the algorithm for the restoration of 1d signals and 2d images in interpolation inpainting problems respectively and in a compressed sensing problem for recovering piecewise constant medical type images from partial fourier ensembles

a nonlinear primal dual method for total variation based image restoration we present a new method for solving total variation tv minimization problems in image restoration the main idea is to remove some of the singularity caused by the nondifferentiability of the quantity nabla u in the definition of the tv norm before we apply a linearization technique such as newton s method this is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function which can be interpreted as the normal vector to the level sets of the image u our method can be viewed as a primal dual method as proposed by conn and overton a primal dual interior point method for minimizing a sum of euclidean norms preprint 1994 and andersen ph d thesis odense university denmark 1995 for the minimization of a sum of euclidean norms in addition to possessing local quadratic convergence experimental results show that the new method seems to be globally convergent

efficient schemes for total variation minimization under constraints in image processing this paper presents new fast algorithms to minimize total variation and more generally l 1 norms under a general convex constraint such problems are standards of image processing the algorithms are based on a recent advance in convex optimization proposed by yurii nesterov depending on the regularity of the data fidelity term we solve either a primal problem or a dual problem first we show that standard first order schemes allow one to get solutions of precision epsilon in o frac 1 epsilon 2 iterations at worst we propose a scheme that allows one to obtain a solution of precision epsilon in o frac 1 epsilon iterations for a general convex constraint for a strongly convex constraint we solve a dual problem with a scheme that requires o frac 1 sqrt epsilon iterations to get a solution of precision epsilon finally we perform some numerical experiments which confirm the theoretical results on various problems of image processing

subspace correction methods for total variation and ell_1 minimization this paper is concerned with the numerical minimization of energy functionals in hilbert spaces involving convex constraints coinciding with a seminorm for a subspace the optimization is realized by alternating minimizations of the functional on a sequence of orthogonal subspaces on each subspace an iterative proximity map algorithm is implemented via oblique thresholding which is the main new tool introduced in this work we provide convergence conditions for the algorithm in order to compute minimizers of the target energy analogous results are derived for a parallel variant of the algorithm applications are presented in domain decomposition methods for degenerate elliptic pdes arising in total variation minimization and in accelerated sparse recovery algorithms based on ell 1 minimization we include numerical examples which show efficient solutions to classical problems in signal and image processing

the blow up problem for a semilinear parabolic equation with a potential abstract let be a bounded smooth domain in r n we consider the problem u t u v x u p in 0 t with dirichlet boundary conditions u 0 on 0 t and initial datum u x 0 m x where m 0 is positive and compatible with the boundary condition we give estimates for the blow up time of solutions for large values of m as a consequence of these estimates we find that for m large the blow up set concentrates near the points where p 1 v attains its maximum

on defining ideals and differential algebras of nichols algebras abstract this paper is devoted to understanding the defining ideal of a nichols algebra from the decomposition of specific elements in the group algebra of braid groups a family of primitive elements is found and algorithms are proposed to prove the main result the differential algebra of a nichols algebra is constructed moreover another point of view on serre relations is provided

blow up set of type i blowing up solutions for nonlinear parabolic systems we consider the blow up problem for systems of nonlinear parabolic inequalities and establish a criterion for the location of the blow up set our criterion enables us to obtain sufficient conditions for the boundedness of the blow up set and no boundary blow up for type i blowing up solutions

overlapping domain decomposition methods for total variation denoising alternating and parallel overlapping domain decomposition methods for the minimization of the total variation are presented their derivation is based on the predual formulation of the total variat

some blow up problems for a semilinear parabolic equation with a potential the blow up rate estimate for the solution to a semilinear parabolic equation ut u v x u p 1u in 0 t with 0 dirichlet boundary condition is obtained as an application it is shown that the asymptotic behavior of blow up time and blow up set of the problem with nonnegative initial data u x 0 m x as m goes to infinity which have been found in c cortazar m elgueta j d rossi the blow up problem for a semilinear parabolic equation with a potential preprint arxiv math ap 0607055 july 2006 is improved under some reasonable and weaker conditions compared with c cortazar m elgueta j d rossi the blow up problem for a semilinear parabolic equation with a potential preprint arxiv math ap 0607055 july 2006

unitary structure in representations of infinite dimensional groups and a convexity theorem in this paper we show that a kac moody algebra fl a associated to a symmetrizable generalized cartan matrix a carries a contravariant hermitian form which is positive definite on all root spaces we deduce that every integrable highest weight g a module l a carries a contravariant positivedefinite hermitian form this allows us to define the moment map and prove a generalization of the schur horn kosta nt heckman atiya h pressley convexity theorem the proofs are based on an identity which also gives estimates for the action of 9 a on fl a and l a we hope that the main idea behind the paper is apparent it is to use the interplay between the coadjoint and the highest weight representations we are grateful to v guillemin for an introduction to the moment map

construction of kac moody superalgebras as minimal graded lie superalgebras and weight multiplicities for kac moody superalgebras we construct a kac moody superalgebra l as the minimal graded lie superalgebra with local part v ge v where g is a smaller lie superalgebra inside l v is an irreducible highest weight g module and v is the contragredient of v we show that the weight multiplicities of irreducible highest weight modules over kac moody superalgebras of finite type and affine type more precisely kac moody superalgebras of type b 0 r b 1 0 r a 4 2r 0 a 2 2r 1 0 and c 2 r 1 are given by polynomials in the rank r the degree of these weight multiplicity polynomials are less than or equal to the depth of weights

sharp upper bound for lifespan of solutions to some critical semilinear parabolic dispersive and hyperbolic equations via a test function method abstract this paper is concerned with the blowup phenomena for initial boundary value problem 0 1 t 2 u x t u x t a x t u x t u x t p x t c 0 t u x t 0 x t c 0 t u x 0 e f x x c t u x 0 e g x x c where c is a cone like domain in r n n 2 defined as c int r r n r 0 with a connected open set in s n 1 with smooth boundary if n 1 then we only consider two cases c 0 and c r here a x is a non zero coefficient of t u which could be complex valued and space dependent is a fixed constant and e 0 is a small parameter the constants 0 1 switch the parabolicity and hyperbolicity of the problem 0 1 the result proposes an argument for sharp upper lifespan estimates of solutions to 0 1 by a test function method based on mitidieri and pokhozhaev 2001 the crucial idea is to introduce an ordinary differential inequality with a parameter as a variable

global and uniform convergence of subspace correction methods for some convex optimization problems this paper gives some global and uniform convergence estimates for a class of subspace correction based on space decomposition iterative methods applied to some unconstrained convex optimization problems some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems

blow up set for a semilinear heat equation with small diffusion abstract we consider the blow up problem for a semilinear heat equation t u u u p in 0 t u x t 0 on 0 t if u x 0 x 0 in where is a domain in r n n 1 0 p 1 and t 0 in this paper under suitable assumptions on we prove that if the family of the solutions u satisfies a uniform type i blow up estimate with respect to then the solution u blows up only near the maximum points of the initial datum for any sufficiently small 0 this is proved without any conditions on the exponent p and the domain such as n 2 p n 2 and the convexity of the domain

blow up problems for a semilinear heat equation with large diffusion abstract we consider the blow up problem of a semilinear heat equation u t d u u p in 0 t d u x t 0 on 0 t d u x 0 x 0 in where is a bounded smooth domain in r n t d 0 d 0 and p 1 we study the blow up time the location of the blow up set and the blow up profile of the blow up solution for sufficiently large d in particular we prove that for almost all initial data if d is sufficiently large then the solution blows up only near the maximum points of the orthogonal projection of the initial data from l 2 onto the second neumann eigenspace

remark on upper bound for lifespan of solutions to semilinear evolution equations in a two dimensional exterior domain abstract in this paper we consider the following initial boundary value problem with the power type nonlinearity u p with 1 p 2 in a two dimensional exterior domain 0 1 t 2 u x t u x t e i t u x t u x t p x t 0 t u x t 0 x t 0 t u x 0 e f x x t u x 0 e g x x where is given by x r 2 x 1 2 2 c and 0 1 switches the parabolicity dispersivity and hyperbolicity remark that 2 1 2 n is well known as the fujita exponent if p 2 then there exists a small global in time solution of 0 1 for some initial data small enough see ikehata 11 and if p 2 then global in time solutions cannot exist for any positive initial data see ogawa takeda 22 and lai yin 14 the result is that for given initial data f g h 0 1 l 2 satisfying f g log x l 1 with some requirement the solution blows up at finite time and moreover the upper bound for lifespan of solutions to 0 1 is given as the following double exponential type when p 2 lifespan u exp exp c e 1 the crucial idea is to use test functions which approximates the harmonic function log x satisfying dirichlet boundary condition and the technique modified from 9

the tensor hierarchy algebra we introduce an infinite dimensional lie superalgebra which is an extension of the u duality lie algebra of maximal supergravity in d dimensions for 3 d 7 the level decomposition with respect to the u duality lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor for all positive levels p we prove that these representations are always contained in those coming from the associated borcherds kac moody superalgebra and we explain why some of the latter representations are not included in the tensor hierarchy the most remarkable feature of our lie superalgebra is that it does not admit a triangular decomposition like a borcherds kac moody super algebra instead the hodge duality relations between level p and d 2 p extend to negative p relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor

semi infinite homology and 2d gravity i in 33 we studied the constraint problem for two dimensional quantum gravity in the conformal gauge in this gauge we proposed an ansatz for the gravitational sector using this ansatz we established a striking connection between the matrix models and continuum 2d gravity we also announced several results on semi infinite homology of the virasoro algebra with coefficients in a suitable class of positive energy modules in this article we will provide details of the proof of the announced results

blowup stability of solutions of the nonlinear heat equation with a large life span abstract we study the cauchy problem for the nonlinear heat equation u t u u p 1 u in r n the initial data is of the form u 0 where c 0 r n is fixed and 0 we first take 1 p p f where p f is the fujita critical exponent and c 0 r n l 1 r n with nonzero mean we show that u t blows up for small extending the h fujita blowup result for sign changing solutions next we consider 1 p p s where p s is the sobolev critical exponent and x decaying as x at infinity where p 1 2 we also prove that u t blows up when is small extending a result of t lee and w ni for both cases the solution enjoys some stable blowup properties for example there is single point blowup even if is not radial

domain decomposition method for image deblurring as a fundamental problem in image processing image deblurring has still attracted a lot of research attention due to the large computational cost especially for high resolution images it becomes challenging to solve the deblurring minimization problem and the underlying partial differential equations the domain decomposition method dd as one of the most efficient algorithms for solving large scale problems had not been applied directly to image deblurring because of the global characteristic of the blur operator in this paper in order to avoid separating the blur operator we propose an algorithm for directly solving the total variational based minimization problems with dd various numerical experiments and comparisons demonstrate that the larger the image size is the more efficient the proposed method is in saving running time the parallelization has also been realized by using the parallel computing toolbox of matlab

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

isomorphism of two realizations of quantum affine algebra u_q widehat mathfrak g mathfrak l text n text we establish an explicit isomorphism between two realizations of the quantum affine algebra u q widehat mathfrak g mathfrak l text n text given previously by drinfeld and reshetikhin semenov tian shansky our result can be considered as an affine version of the isomorphism between the drinfield jimbo and the faddeev reshetikhin takhtajan constructions of the quantum algebra u q mathfrak g mathfrak l n

blow up for a semilinear parabolic equation with large diffusion on rn ii we are concerned with the cauchy problem for a semilinear heat equation p tu d u u p 1u x rn t 0 u x 0 x x rn where d 0 p 1 n 3 0 and l rn l1 rn 1 x 2dx in the paper of fujishima and ishige 2011 8 the authors of this paper studied the behavior of the blow up time and the blow up set of the solution of p as d for the case rn x dx 0 in this paper as a continuation of fujishima and ishige 2011 8 we consider the case rn x dx 0 and study the behavior of the blow up time and the blow up set of the solution of p as d the behavior in the case rn x dx 0 is completely different from the one in the case rn x dx 0

determination of the curvature of the blow up set and refined singular behavior for a semilinear heat equation we consider u x t a solution of ut u u p 1u which blows up at some time t 0 where u rn 0 t r p 1 and n 2 p n 2 under a nondegeneracy condition we show that the mere hypothesis that the blow up set s is continuous and n 1 dimensional implies that it is c2 in particular we compute the n 1 principal curvatures and directions of s moreover a much more refined blow up behavior is derived for the solution in terms of the newly exhibited geometric objects refined regularity for s and refined singular behavior of u near s are linked through a new mechanism of algebraic cancellations that we explain in detail

some forms of kac moody algebras let k be an algebraically closed field of char 0 the unique up to isomorphism three dimensional simple lie algebra s1 2 k plays an important role in the theory of the semisimple lie algebras over k for example chevalley harish chandra and serre s construction of the semisimple lie algebras g over k corresponding to a dynkin diagram 9 can be viewed as glueing together copies of sl 2 k in lf but we can not choose the distinct tds in an arbitrary way we need some compatibility conditions between the tds attached to neighbouring vertices they are described by some scalars sij and for simplicity we restrict our attention in this definition to dynkin diagrams without cycles for both definitions we also make the additional hypothesis that the

convergence rate analysis of an asynchronous space decomposition method for convex minimization we analyze the convergence rate of an asynchronous space decomposition method for constrained convex minimization in a reflexive banach space this method includes as special cases parallel domain decomposition methods and multigrid methods for solving elliptic partial differential equations in particular the method generalizes the additive schwarz domain decomposition methods to allow for asynchronous updates it also generalizes the bpx multigrid method to allow for use as solvers instead of as preconditioners possibly with asynchronous updates and is applicable to nonlinear problems applications to an overlapping domain decomposition for obstacle problems are also studied the method of this work is also closely related to relaxation methods for nonlinear network flow accordingly we specialize our convergence rate results to the above methods the asynchronous method is implementable in a multiprocessor system allowing for communication and computation delays among the processors

integrable roots in split graded lie algebras abstract a lie algebra g is said to be split graded if it is graded by a torsion free abelian group q in such a way that the subalgebra g 0 is abelian and the operators ad g 0 are diagonalized by the grading the elements of q 0 with g 0 are called roots and a root is said to be integrable if there are root vectors x g which are ad nilpotent and generate an s l 2 subalgebra g in this paper we study subalgebras g g generated by the subalgebras g where is a set of integrable roots for 2 these subalgebras are essentially kac moody algebras which permits us to generalize several results on root strings from kac moody algebras to split graded algebras a central result is the local finiteness theorem saying that whenever all roots of a split graded lie algebra are integrable then is locally finite if differences of roots in are not roots then is called a simple system in this case we describe the structure of the subalgebras g in their relationship to the corresponding kac moody algebras

classification of hyperbolic dynkin diagrams root lengths and weyl group orbits we give a criterion for a dynkin diagram equivalently a generalized cartan matrix to be symmetrizable this criterion is easily checked on the dynkin diagram we obtain a simple proof that the maximal rank of a dynkin diagram of compact hyperbolic type is 5 while the maximal rank of a symmetrizable dynkin diagram of compact hyperbolic type is 4 building on earlier classification results of kac kobayashi morita li and sacliolu we present the 238 hyperbolic dynkin diagrams in ranks 3 10 142 of which are symmetrizable for each symmetrizable hyperbolic generalized cartan matrix we give a symmetrization and hence the distinct lengths of real roots in the corresponding root system for each such hyperbolic root system we determine the disjoint orbits of the action of the weyl group on real roots it follows that the maximal number of disjoint weyl group orbits on real roots in a hyperbolic root system is 4

the semi infinite cohomology of affine lie algebras we study the semi infinite or brst cohomology of affine lie algebras in detail this cohomology is relevant in the brst approach to gauged wznw models our main result is to prove necessary and sufficient conditions on ghost numbers and weights for non trivial elements in the cohomology in particular we prove the existence of an infinite sequence of elements in the cohomology for non zero ghost numbers this will imply that the brst approach to the topological wznw model admits many more states than a conventional coset construction this conclusion also applies to some non topological models

on blow up at space infinity for semilinear heat equations abstract a nonnegative blowing up solution of the semilinear heat equation u t u u p with p 1 is considered when initial data u 0 satisfies lim x u 0 m 0 u 0 m and u 0 m it is shown that the solution blows up only at space infinity and that lim x u x t is the solution of the ordinary differential equation v t v p with v 0 m

solution of a nonlinear heat equation with arbitrarily given blow up points we consider the equation where i u is scalar valued and p 1 it has been proven that if u t blows up at time t the blow up points are finite in number and located in i our aim is to prove that this result is optimal that is for any given points x1 xk in i there is a solution u such that its blow up points are exactly x1 xk

single point blow up for a semilinear initial value problem abstract the initial value problem on r r is considered u t t x u xx t x u t x u t r 0 u 0 x x where 0 and is a fixed large number it is known that for some initial values the solution u t x exists only up to some finite time t and that u t as t t for the specific initial value k where 0 xx 0 r 0 k is sufficiently large it is shown that if x 0 then lim t t u t x and lim t t u x t x exist and are finite in other words blow up occurs only at the point x 0

movement of time delayed hot spots in euclidean space we investigate the shape of the solution of the cauchy problem for the damped wave equation in particular we study the existence location and number of spatial maximizers of the solution studying the shape of the solution of the damped wave equation we prepare a decomposed form of the solution into the heat part and the wave part moreover as its another application we give l p l q estimates of the solution