on the birationality of complete intersections associated to nef partitions we prove that generic complete intersections associated to multiple mirror nef partitions are all birational this result solves a conjecture by batyrev and nill in 6 under some mild assumptions

on clifford double mirrors of toric complete intersections abstract we present a construction of noncommutative double mirrors to complete intersections in toric varieties this construction unifies existing sporadic examples and explains the underlying combinatorial and physical reasons for their existence

on quadrature domains and the schwarz potential abstract let t be a distribution in r n whose support is compact a domain in r n is said to be a quadrature domain with respect to t if supp t and h h t for all functions h which are harmonic and integrable over we denote this by q t hl 1 this is equivalent to saying that there is a solution u called the modified schwarz potential to the following overdetermined cauchy problem u 1 t in and u u 0 on in this paper we investigate different properties of quadrature domains we show that if t is a fixed distribution then bounded domains in q t hl1 are uniformly bounded later we prove that if the modified schwarz potential of a domain is a polynomial then is a half space we also study the uniqueness problem namely we show that if d and are in q t hl1 v and u are the msps of d and respectively and sup d u 0 then meets x v x consequently if v 0 then d 0

a discrete variational identity on semi direct sums of lie algebras the discrete variational identity under general bilinear forms on semi direct sums of lie algebras is established the constant involved in the variational identity is determined through the corresponding solution to the stationary discrete zero curvature equation an application of the resulting variational identity to a class of semi direct sums of lie algebras in the volterra lattice case furnishes hamiltonian structures for the associated integrable couplings of the volterra lattice hierarchy

variational problems with two phases and their free boundaries the problem of minimizing vu 2 q2 x 2 v dx in an appropriate class of functions v is considered here q x 0 and a2 t x2 if v 0 any minimizer u is harmonic in u 0 and vu 2 has a jump a2 x 22 across the free boundary u 0 regularity and various properties are established for the minimizer u and for the free boundary introduction in this paper we consider the problem of minimizing j v f vv 2 q2 x x2 v dx vgk where q2 x 0 a22 ifu 0 and x2 u is lower semicontinuous at v 0 it is assumed that a2 0 and a a2 a22 0 the class k consists of all functions v in l loc s2 with vu g l2 i2 such that v u on a given open subset s of 3i2 and s is a domain in r the analogous problem for functions in k v k v 0 a e was studied in 1 in that paper it was proved that any local minimizer of j v in k is lipschitz continuous and if n 2 the free boundary 8 w 0 is analytic if q x is analytic the present variational problem is motivated by applications to the now of two liquids in models of jets and cavities these applications will be studied in other forthcoming papers 5 6 the present work is aimed at extending results of 1 in particular we shall establish nondegeneracy theorems the lipschitz continuity of the solution and some properties of the free boundary for n 2 the free boundary is proved to be continuously differentiable a new and rather powerful tool introduced in this paper is the monotonicity formula lemma 5 1 asserting that for a minimizer u if u x0 0 then

derived categories of quadric fibrations and intersections of quadrics abstract we construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the clifford algebras on the base corresponding to this quadric fibration generalizing the kapranov s description of the derived category of a single quadric as an application we verify that the noncommutative algebraic variety p s 2 w b 0 where b 0 is the universal sheaf of even parts of clifford algebras is homologically projectively dual to the projective space p w in the double veronese embedding p w p s 2 w using the properties of the homological projective duality we obtain a description of the derived category of coherent sheaves on a complete intersection of any number of quadrics

integrable couplings of vector akns soliton equations by enlarging the associated matrix spectral problems two specific classes of multicomponent integrable couplings of the physically important vector akns soliton equations are constructed which can be linked to each other by a backlund transformation the resulting two hierarchies of integrable couplings possess the enlarged zero curvature representations and recursion structures and thus each system in the two hierarchies has infinitely many commuting symmetries

geometric flows for quadrature identities a new geometric flow describing the motion of quadrature surfaces is introduced this characterization enables us to study quadrature surfaces through the investigation of the flow it is proved that the flow is uniquely solvable under the geometric condition that the initial surface has positive mean curvature as a consequence a bifurcation criterion for quadrature surfaces is obtained in fact we study a generalized flow which includes the geometric flow and the hele shaw flow as special cases

properties of some balayage operators with applications to quadrature domains and moving boundary problems in this paper we define and derive some properties of a fairly wide class of balayage operators we also apply our results to obtain geometric information about quadrature domains and solutions of certain moving boundary problems e g for hele shaw flows actually it was our interest in these applications that was the starting point for the present investigations and the balayage operators were developed rather as a tool our balayage operators are denoted f fp fp r and they depend on an open set r c i and a measure p in r actually we always assume that p has a density also denoted p in l r what f does is replace a given measure p by the nearest one in the energy norm v f p which satisfies v 5 p in r this presupposes that fi has finite energy if this is not the case there is another definition which is the one we will actually use among all distributions u in irn satisfying u i 17 the newtonian potential of p in irn and au i p in r there is a largest one u vp then f p ai by definition the operator of interest in the above mentioned applications is f with r irn p 1 the definition of f y when u has finite energy can be formulated as an elliptic variational inequality or linear complementarity problem and this is one of the standard tools when dealing with free boundary problems of the above kind see l 4 however we think that the balayage point of view is quite natural in our context the reason for considering the balayage operators for more general r and p than r rn p 1 is firstly that we need them in the proof of one of our theorems theorem 4 1 secondly and perhaps more important is that we think that the more general operators have quite a lot of intrinsic interest in particular as they turn out to contain classical balayage of positive measures as a special case namely with r bounded and p 0 recently we also learnt that discrete or numerical versions of these operators have been developed and applied in geophysics during the last three decades by dimiter zidarov who calls the balayage process partial gravi equivalent mass scattering see 5 and also for example 6 the paper is organized as follows in section 2 we define and establish the basic properties of our operators at the end we also explain their relation to quadrature domains and certain moving boundary problems section 3 is devoted to one single theorem theorem 3 1 and its corollaries these results concern the f with r lrn p 1 and give rather good geometric

existence and geometric properties of solutions of a free boundary problem in potential theory let 0 g h l rn n 2 be two given density functions at least one of them bounded away from zero outside a compact set and g holder continuous we prove that for any compactly supported positive measure which is sufficiently concentrated e g has sufficiently high n 1 dimensional density there exists a bounded open set rn such that the newtonian potential of the measure hln ghn 1 agrees with that of outside some regularity of is obtained as well as several results on the geometry of example if h and g are constant then for any x the inward normal ray of at x if it exists intersects the closed convex hull of typeset by ams tex 1 2 bjorn gustafsson and henrik shahgholian

semidirect sums of lie algebras and discrete integrable couplings a relation between semidirect sums of lie algebras and integrable couplings of lattice equations is established and a practicable way to construct integrable couplings is further proposed an application of the resulting general theory to the generalized toda spectral problem yields two classes of integrable couplings for the generalized toda hierarchy of lattice equations the construction of integrable couplings using semidirect sums of lie algebras provides a good source of information on complete classification of integrable lattice equations

some properties of two phase quadrature domains abstract in this paper we investigate general properties of the two phase quadrature domain which was recently introduced by emamizadeh prajapat and shahgholian the concept which is a generalization of the well known one phase domain introduces substantial difficulties with interesting features even richer than those of the one phase counterpart for given positive constants and two bounded and compactly supported measures we investigate the uniqueness of the solution of the following free boundary problem 1 u in r n n 2 u 0 in r n where it is further required that the supports of should be inside this in general may fail and give rise to non existence of solutions along the paths to various properties that we state and prove here we also present several conjectures and open problems that we believe should be true

applications of variational inequalities to the existence theorem on quadrature domains in this paper we shall study quadrature domains for the class of subharmonic functions by using the theory of variational inequalities we shall give a new proof of the existence and uniqueness theorem as an application we deal with hele shaw flows with a free boundary and show that their two weak solutions one of which was defined by the author using quadrature domains and the other was defined by gustafsson 3 using variational inequalities are identical with each other introduction in a previous paper 7 the author has defined the quadrature domains of positive measures for the class of subharmonic functions and studied their applications to complex function theory let v be a finite positive measure on the two dimensional euclidean space r2 let sl1 q be the class of subharmonic functions in an open set q which are integrable with respect to the two dimensional lebesgue measure m a nonempty open set q is called a quadrature domain of v for class sl1 if qi v is concentrated in q namely v qc 0 where qc denotes the complement of q qii jus s dv 1 a e in a connected open set w with finite area f 0 a e in wc and jf dm m w then q fm sl1 0 and there exists a minimum domain w in q fm sl1 namely q e q jm sl1 if and only if w c q and m q w o received by the editors october 5 1981 and in revised form february 23 1982 1980 mathematics subject classification primary 3 1aos 3 1bos secondary 35a15

faber krahn inequalities in sharp quantitative form the classical faber krahn inequality asserts that balls uniquely minimize the first eigenvalue of the dirichlet laplacian among sets with given volume in this paper we prove a sharp quantitative enhancement of this result thus confirming a conjecture by nadirashvili and bhattacharya weitsman more generally the result applies to every optimal poincare sobolev constant for the embeddings w 1 2 0 omega hookrightarrow l q omega

derived equivalent calabi yau threefolds from cubic fourfolds we describe pretty examples of derived equivalences and autoequivalences of calabi yau threefolds arising from pencils of cubic fourfolds the cubic fourfolds are chosen to be special so they each have an associated k3 surface thus a pencil gives rise to two different calabi yau threefolds the associated pencil of k3 surfaces and the baselocus of the original pencil the intersection of two cubic fourfolds they both have crepant resolutions which are derived equivalent

scheme of lines on a family of 2 dimensional quadrics geometry and derived category given a generic family q of 2 dimensional quadrics over a smooth 3 dimensional base y we consider the relative fano scheme m of lines of it the scheme m has a structure of a generically conic bundle m rightarrow x over a double covering x rightarrow y ramified in the degeneration locus of q rightarrow y the double covering x rightarrow y is singular in a finite number of points corresponding to the points y in y such that the quadric q y degenerates to a union of two planes the fibers of m over such points are unions of two planes intersecting in a point the main result of the paper is a construction of a semiorthogonal decomposition for the derived category of coherent sheaves on m this decomposition has three components the first is the derived category of a small resolution x of singularities of the double covering x rightarrow y the second is a twisted resolution of singularities of x given by the sheaf of even parts of clifford algebras on y and the third is generated by a completely orthogonal exceptional collection

the one phase free boundary problem for the p laplacian with non constant bernoulli boundary condition our objective here is to generalize our earlier results on the existence of classical convex solution to a free boundary problem with a bernoulli type boundary gradient condition and with the p laplacian as the governing operator the main theorems of this paper assert that the exterior and the interior free boundary problem with a bernoulli law i e with a prescribed pressure a x on the free streamline of the flow have convex solutions provided the initial domains are convex the continuous function a x is subject to certain convexity properties in our earlier results we have considered the case of constant a x in the lines of the proof of the main results we also prove the semi continuity up to the boundary of the gradient of the p capacitary potentials in convex rings with c 1 boundaries