the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

stationary sets for the wave equation in crystallographic domains let w be a crystallographic group in r n generated by reflections and let be the fundamental domain of w we characterize stationary sets for the wave equation in when the initial data is supported in the interior of the stationary sets are the sets of time invariant zeros of nontrivial solutions that are identically zero at t 0 lie show that for these initial data the n 1 dimensional part of the stationary sets consists of hyperplanes that are mirrors of a crystallographic group w w w this part comes from a corresponding odd symmetry of the initial data in physical language the result is that if the initial source is localized strictly inside of the crystalline then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one

noncommutative instantons and twistor transform recently n nekrasov and a schwarz proposed a modification of the adhm construction of instantons which produces instantons on a noncommutative deformation of 4 in this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces we exhibit one to one correspondences between three classes of objects framed bundles on a noncommutative 2 certain complexes of sheaves on a noncommutative 3 and the modified adhm data the modified adhm construction itself is interpreted in terms of a noncommutative version of the twistor transform we also prove that the moduli space of framed bundles on the noncommutative 2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative 2 the natural complex structures on the two moduli spaces do not coincide but are related by an so 3 rotation finally we propose a construction of instantons on a more general noncommutative 4 than the one considered by nekrasov and schwarz a q deformed 4

injectivity sets for the radon transform over circles and complete systems of radial functions abstract a necessary and sufficient characterization is given that specifies which sets of sums of translations of radial functions are dense in the set of continuous functions in the plane this problem is shown to be equivalent to inversion for the radon transform on circles centered on restricted subsets of the plane the proofs rest on the geometry of zero sets for harmonic polynomials and the microlocal analysis of this circular radon transform a characterization of nodal sets for the heat and wave equation in the plane are consequences of our theorems and questions of pinkus and ehrenpreis are answered

power convexity of a class of elliptic equations involving the hessian operator in a 3 dimensional bounded convex domain abstract in this paper we are concerned with power convexity of a class of fully nonlinear elliptic equations involving the hessian operator in a bounded and strictly convex domain in r 3 we use the constant rank theorem and deformation technique to prove that some power of the smooth admissible solution is strictly convex as a corollary we deduce that all the level sets of the solution are strictly convex

a translation principle for the four dimensional sklyanin algebras abstract in this paper we prove a translation principle for the central quotients of four dimensional sklyanin algebras which is analogous to the translation principle for semi simple lie algebras in the course of the proof we construct an elliptic analog of the sheaf of differential operators on2 1 the translation principle may be used to construct the fat points of a four dimensional non pi sklyanin algebra

convexity of solutions and brunn minkowski inequalities for hessian equations in r3 by using minkowski addition of convex functions we prove convexity and rearrangement properties of solutions to some hessian equations in r 3 and brunn minkowski and isoperimetric inequalities for related functionals

the convexity of the level sets of maximal strictly space like hypersurfaces defined on 2 dimensional space forms abstract for the maximal strictly space like hypersurface defined on 2 dimensional space forms we derive some geometrical properties including the regularity and the strict convexity and the curvature description of the level sets

convexity estimates for the solutions of a class of semi linear elliptic equations abstract in this paper we are concerned with convexity estimates for solutions of a class of semi linear elliptic equations involving the laplacian with power type nonlinearities we consider auxiliary curvature functions which attain their minimum values on the boundary and then establish lower bound convexity estimates for the solutions then we give two applications of these convexity estimates we use the deformation method to prove a theorem concerning the strictly power concavity properties of the smooth solutions to these semi linear elliptic equations finally we give a sharp lower bound estimate of the gaussian curvature for the solution surface of some specific equation by the curvatures of the domain s boundary

c1 1 solution of the dirichlet problem for degenerate k hessian equations abstract in this paper we prove the existence of the c 1 1 solution to the dirichlet problem for degenerate elliptic k hessian equations s k u f under a condition which is weaker than the condition f 1 k c 1 1

existence and convexity of local solutions to degenerate hessian equations abstract in this work we prove the existence of convex solutions to the following k hessian equation s k u k y g y u d u in the neighborhood of a point y 0 u 0 p 0 r n r r n where g c g y 0 u 0 p 0 0 k c is nonnegative near y 0 k y 0 0 and rank d y 2 k y 0 n k 1

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

the convexity of solution of a class hessian equation in bounded convex domain in r 3 we use the deformation methods to obtain the convexity of solution of a class hessian equation in bounded convex domain in r 3 c 2008 elsevier inc all rights reserved

nodal sets for solutions of elliptic equations on etudie sur un domaine connexe r n l ensemble zero u 1 0 d une solution u d une equation elliptique a ij d i d j u b j d j u cu 0 ou a ij b j c sont bornes et a ij est continu on demontre que la mesure de hausdorff a n 1 dimensions de u 1 0 est finie dans un voisinage d un point x 0 ou u a un ordre fini d annulation

on nodal sets for dirac and laplace operators we prove that the nodal set zero set of a solution of a generalized dirac equation on a riemannian manifold has codimension 2 at least if the underlying manifold is a surface then the nodal set is discrete we obtain a quick proof of the fact that the nodal set of an eigenfunction for the laplace beltrami operator on a riemannian manifold consists of a smooth hypersurface and a singular set of lower dimension we also see that the nodal set of a harmonic differential form on a closed manifold has codimension 2 at least a fact which is not true if the manifold is not closed examples show that all bounds are optimal

concavity estimates for a class of nonlinear elliptic equations in two dimensions we study solutions of the nonlinear elliptic equation v delta v c 2 c 1 vert dv vert 2 on a bounded domain omega in mathbb r 2 it is shown that the set m of points where the graph of the solution has negative gauss curvature always extends to the boundary unless it is empty the meethod uses an elliptic equation satisfied by an auxiliary function given by the product of the hessian determinant and a suitable power of the solutions as a consequence of the result we give a new proof for power concavity of solutions to certain semilinear boundary value problems in convex domains

convexity of solutions of semilinear elliptic equations on considere le probleme elliptique suivant u f u dans u m sur ou est un domaine convexe de r 2 et m est soit une constante reelle soit on etablit que pour une bonne fonction monotone g t pout toute solution u on a g u est strictement convexe dans

frobenius bimodules between noncommutative spaces abstract in this paper we study frobenius bimodules between noncommutative spaces quasi schemes developing some of their basic properties if x and y are spaces we study those frobenius x y bimodules x m y satisfying properties that are natural in the context of noncommutative algebraic geometry focusing in particular on cartain local conditions on m as applications we prove decomposition and gluing theorems for those frobenius bimodules which have good local properties additionally when x and y are schemes we relate frobenius x y bimodules to the sheaf x y bimodules introduced by van den bergh in j algebra 184 1996 435 490

a brunn minkowski inequality for the hessian eigenvalue in three dimensional convex domain abstract we use the deformation methods to obtain the strictly log concavity of solution of a class hessian equation in bounded convex domain in r 3 as an application we get the brunn minkowski inequality for the hessian eigenvalue and characterize the equality case in bounded strictly convex domain in r 3

the geometry of points on quantum projectivizations abstract suppose s is an affine noetherian scheme x is a separated noetherian s scheme e is a coherent o x bimodule and i t e is a graded ideal we study the geometry of the functor n of flat families of truncated b t e i point modules of length n 1 we then use the results of our study to show that if proj b is a quantum ruled surface the point modules over b are parameterized by the closed points of p x2 e when x p 1 we construct for any b point module a graded o x b bimodule resolution

the jones polynomial and certain separable frobenius extensions 1 introductionthe jones knot polynomial was a significant advance in the mid 1980 sfor the theory of knot invariants as well as an intriguing object ofcross disciplinary research within days of a jones seminar talk at pennstate the homfly polynomial in two variables was discovered it combinedthe power of discrimination among knots and indeed oriented links ofboth the jones polynomial and the much older alexander conway polyno mial the homfly polynomial was derived by ocneanu as the trace oncertain hecke algebras this was succeeded by the kaufmann polynomialand other link invariants from partition functions in statistical mechanics then came the exhaustive works by drinfeld 10 turaev 36 vassilievwx wxwx37 and others which produced the so called quantum group hopf algebra invariants and vassiliev invariants that include all the previous knot polynomials and enough new ones to possibly classify knots theinterested reader should first consult 21 and 3 wx wxjones originally discovered his knot polynomials through his work ontype

subspaces of non commutative spaces this paper concerns the closed points closed subspaces open subspaces weakly closed and weakly open subspaces and effective divisors on a non commutative space

the injective spectrum of a noncommutative space abstract for a noncommutative space x we study inj x the set of isomorphism classes of indecomposable injective x modules in particular we look at how this set suitably topologized can be viewed as an underlying spectrum for x as applications we discuss noncommutative notions of irreducibility and integrality and a way of associating an integral subspace of x to each element of inj x which behaves like a weak point

integral non commutative spaces abstract a non commutative space x is a grothendieck category mod x we say x is integral if there is an indecomposable injective x module e x such that its endomorphism ring is a division ring and every x module is a subquotient of a direct sum of copies of e x a noetherian scheme is integral in this sense if and only if it is integral in the usual sense we show that several classes of non commutative spaces are integral we also define the function field and generic point of an integral space and show that these notions behave as one might expect

frobenius extensions of subalgebras of hopf algebras we consider when extensions s r of subalgebras of a hopf algebra are frobenius that is frobenius of the second kind given a hopf algebra h we show that when s r are hopf algebras in the yetter drinfeld category for h the extension is frobenius provided r is finite over s and the extension of biproducts s h r h is cleft more generally we give conditions for an extension to be frobenius in particular we study extensions of integral type and consider when the frobenius property is inherited by the subalgebras of coinvariants we apply our results to extensions of enveloping algebras of lie coloralgebras thus extending a result of bell and farnsteiner for lie superalgebras 0 introduction in this paper we consider when various extensions s r of subalgebras of a hopf algebra are frobenius such extensions generalize the usual notion of frobenius extensions by having the module action on one side twisted by an automorphism of s it was already known that any extension of finite dimensional hopf algebras is frobenius a result of the third author sch 92 as is any finite extension u k u l of enveloping algebras of lie superalgebras a result of bell and farnsteiner bf note that u l is not an ordinary hopf algebra but rather a hopf algebra in the category of z2 graded modules these results were an important motivation for this paper and raised the question as to when an extension of hopf algebras in a category was frobenius one of the main results of this paper is that an extension s r of hopf algebras of finite index in the yetter drinfeld category hyd for a given hopf algebra h is frobenius provided that the associated extension of hopf algebras s h r h the biproducts of s and r with h has a normal basis theorem 5 6 this will happen whenever r and h are finite dimensional or when r h is pointed as an application we are able to generalize the bf result to lie coloralgebras if u k u l is a finite extension of enveloping algebras of lie coloralgebras then it is frobenius moreover we give an explicit description of the automorphism of u k and of the frobenius homomorphism f u l u k corollary 6 3 along the way we prove a number of other results about frobenius extensions and conditions that ensure that an extension is frobenius in section 1 we give a short direct proof of the fact that an extension b a of finite dimensional hopf algebras is always frobenius moreover we give explicit received by the editors december 10 1995 1991 mathematics subject classification primary 16w30 secondary 17b35 17b37 c 1997 american mathematical society

range descriptions for the spherical mean radon transform abstract the transform considered in the paper averages a function supported in a ball in r n over all spheres centered at the boundary of the ball this radon type transform arises in several contemporary applications e g in thermoacoustic tomography and sonar and radar imaging range descriptions for such transforms are important in all these areas for instance when dealing with incomplete data error correction and other issues four different types of complete range descriptions are provided some of which also suggest inversion procedures necessity of three of these appropriately formulated conditions holds also in general domains while the complete discussion of the case of general domains would require another publication

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m