the twisted xxz chain at roots of unity revisited the twisted xxz chain alias the six vertex model is investigated at roots of unity it is shown that when the twist parameter is chosen to depend on the total spin an infinite dimensional non abelian symmetry algebra can be explicitly constructed in all spin sectors this symmetry algebra can be identified with the upper or lower borel subalgebra of the sl2 loop algebra the proof uses only the intertwining property of the six vertex monodromy matrix and the familiar relations of the six vertex yang baxter algebra

conditional gauge and potential theory for the schrodinger operator on etend le theoreme de jauge conditionnelle a des operateurs plus generaux et a des domaines moins reguliers on obtient des resultats de theorie du potentiel pour l equation de schrodinger

the equivalence of certain heat kernel and green function bounds abstract we show that certain upper and lower bounds on the green function and heat kernel of a second order elliptic operator in a bounded region are equivalent and imply a number of apparently stronger bounds we also show that these bounds are equivalent to the harnack inequality except for peculiar regions and are therefore almost always valid

fibered threefolds and lang vojta s conjecture over function fields using the techniques introduced by corvaja and zannier in 2008 we solve the non split case of the geometric lang vojta conjecture for affine surfaces isomorphic to the complement of a conic and two lines in the projective plane in this situation we deal with sections of an affine threefold fibered over a curve whose boundary in the natural projective completion is a quartic bundle over the base whose fibers have three irreducible components we prove that the image of each section has bounded degree in terms of the euler characteristic of the base curve

estimates for solutions to equations of p laplace type in ahlfors regular nta domains this thesis consists of six scientific papers an introduction and a summary all six papers concern the boundary behavior of non negative solutions to partial differential equations paper i concerns solutions to certain p laplace type operators with variable coefficients suppose that u is a non negative solution that vanishes on a part of an ahlfors regular nta domain we prove among other things that the gradient du of u has non tangential limits almost everywhere on the boundary piece and that log du is a bmo function on the boundary furthermore for ahlfors regular nta domains that are uniformly n r0 approximable by lipschitz graph domains we prove a boundary harnack inequality provided that is small enough paper ii concerns solutions to a p laplace type operator with lower order terms in reifenberg flat domains we prove that the ratio of two non negative solutions vanishing on a part of the boundary is holder continuous provided that is small enough furthermore we solve the martin boundary problem provided is small enough in paper iii we prove that the boundary type riesz measure associated to an a capacitary function in a reifenberg flat domain with vanishing constant is asymptotically optimal doubling paper iv concerns the boundary behavior of solutions to certain parabolic equations of p laplace type in lipschitz cylinders among other things we prove an intrinsic carleson type estimate for the degenerate case and a weak intrinsic carleson type estimate in the singular supercritical case in paper v we are concerned with equations of p laplace type structured on hormander vector fields we prove that the boundary type riesz measure associated to a non negative solution that vanishes on a part of an x nta domain is doubling on paper vi concerns a one phase free boundary problem for linear elliptic equations of non divergence type assume that we know that the positivity set is an nta domain and that the free boundary is a graph furthermore assume that our solution is monotone in the graph direction and that the coefficients of the equation are constant in the graph direction we prove that the graph giving the free boundary is lipschitz continuous

algebraic hyperbolicity of ramified covers of mathbb g 2_m and integrap points on affine subsets of mathbb p _2 let x be a smooth affine surface x to mathbb g 2 m be a finite morphism we study the affine curves on x with bounded genus and number of points at infinity obtaining bounds for their degree in terms of euler characteristic a typical example where these bounds hold is represented by the complement of a three component curve in the projective plane of total degree at least 4 the corresponding results may be interpreted as bounding the height of integral points on x over a function field in the language of diophantine equations our results may be rephrased in terms of bounding the height of the solutions of f u v y 0 with u v y over a function field u v s units it turns out that all of this contain some cases of a strong version of a conjecture of vojta over function fields in the split case moreover our method would apply also to the nonsplit case we remark that special cases of our results in the holomorphic context were studied by m green already in the seventies and recently in greater generality by noguchi winkelmann and yamanoi however the algebraic context was left open and seems not to fall in the existing techniques

estimates for operator norms on weighted spaces and reverse jensen inequalities we examine the dependence on the a p norm of w of the operator norms of singular integrals maximal functions and other operators in l p w we also examine connections between some fairly general reverse jensen inequalities and the a p and rh p weight conditions

auxiliary matrices on both sides of the equator the spectra of previously constructed auxiliary matrices for the six vertex model at roots of unity are investigated for spin chains of even and odd length the two cases show remarkable differences in particular it is shown that for even roots of unity and an odd number of sites the eigenvalues contain two linear independent solutions to baxter s tq equation corresponding to the bethe ansatz equations above and below the equator in contrast one finds for even spin chains only one linear independent solution and complete strings the other main result is the proof of a previous conjecture on the degeneracies of the six vertex model at roots of unity the proof rests on the derivation of a functional equation for the auxiliary matrices which is closely related to a functional equation for the eight vertex model conjectured by fabricius and mccoy

the dirichlet problem in non smooth domains dirichlet problem for an elliptic operator 2s laj jaj with smooth coefficients aij in domains d in rm m 3 with non smooth boundaries the domains we treat are given locally in some cm coordinate system by the graph of a continuous function qs with vq5 e li for some p 1 p oo such domains are called li domains see section 1 for the precise definitions note that l domains are usually called lipschitz domains if d is an li domain and x e d we study the elliptic measure a7 associated with s and d at x thus for f a continuous function on ad the formula u x hf x f doh gives the perron wiener brelot generalized soluid

a criterion for the absolute continuity of the harmonic measure associated with an elliptic operator for some a o and for all x e q and 4 e r n we also assume a a for such operators the dirichlet problem is solvable in q if and only if it is solvable for the laplace operator according to a theorem of littman stampacchia and weinberger 1 this means that if q c r n is a sufficiently nice bounded region the unit ball b is an example and f is a given continuous function on the boundary of q then there exists a unique function u continuous on q so that l u 0 in q and u f on aq let us assume for convenience that the origin belongs to q then the mapping f c f aq u o is a positive linear functional so there exists a unique nonnegative measure w on aq such that for every f e aq

on multilinear square function and its applications to multilinear littlewood paley operators with non convolution type kernels abstract let m 2 n 1 and x r n define the multilinear square function t by t f x 0 r n m k t x y 1 y m j 1 m f j y j d y 1 d y m 2 d t t 1 2 where the kernel k satisfies a class of integral smooth conditions which is much weaker than the standard calderon zygmund kernel conditions in this paper we first established the l p 1 w 1 l p m w m l p estimate of t when each p i 1 and weak type l p 1 w 1 l p m w m l p estimate of t when there is a p i 1 where i 1 m i p p i and each w i is a nonnegative function on r n as applications of the above results we obtained the boundedness of multilinear littlewood paley operators with non convolution type kernels including multilinear g function marcinkiewicz integral and g function

homogenization of elliptic boundary value problems in lipschitz domains in this paper we study the lp boundary value problems for mathcal l u 0 in mathbb r d 1 where mathcal l rm div a nabla is a second order elliptic operator with real and symmetric coefficients assume that a is periodic in xd 1 and satisfies some minimal smoothness condition in the xd 1 variable we show that the lp neumann and regularity problems are uniquely solvable for 1 p 2 we also present a new proof of dahlberg s theorem on the lp dirichlet problem for 2 p dahlberg s original unpublished proof is given in the appendix as the periodic and smoothness conditions are imposed only on the xd 1 variable these results extend directly from mathbb r d 1 to regions above lipschitz graphs consequently by localization techniques we obtain uniform lp estimates for the dirichlet neumann and regularity problems on bounded lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization the results on the neumann and regularity problems are new even for smooth domains

the solution of the kato problem for divergence form elliptic operators with gaussian heat kernel bounds which in turnhas applications to the perturbation theory for certain classes of hyperbolicequations see 16 we remark that 1 3 is equivalent to the oppositeinequality for the square root of the adjoint operator l in 13 14 kato conjectured that an abstract version of 1 3 might hold for regularly accretive operators see 14 18 for the details a counter example to this abstract conjecture was obtained by mcintosh 17 whothen reformulated the conjecture in the following form bearing in mind thatkato s interest in the problem had been motivated by the special case ofelliptic di erential operators conjecture 1 4 the estimate 1 3 holds for lde ned as in 1 2 for anyl

multilinear littlewood paley estimates with applications to partial differential equations abstract we obtain a collection of multilinear littlewood paley estimates which we then apply to two problems in partial differential equations the first problem is the estimation of the square root of an elliptic operator in divergence form and the second is the estimation of solutions to the cauchy problem for nondivergence form parabolic equations

eigenvectors in the superintegrable model i sl2 generators in order to calculate correlation functions of the chiral potts model one only needs to study the eigenvectors of the superintegrable model here we start this study by looking for eigenvectors of the transfer matrix of the periodic 2 tq model which commutes with the chiral potts transfer matrix we show that the degeneracy of the eigenspace of 2 tq in the q 0 sector is 2r with r n 1 l n when the size of the transfer matrix l is a multiple of n we introduce chiral potts model operators different from the more commonly used generators of quantum group from these we can form the generators of a loop algebra for this algebra we then use the roots of the drinfeld polynomial to give new explicit expressions for the generators representing the loop algebra as the direct sum of r copies of the simple algebra

regular xxz bethe states at roots of unity as highest weight vectors of the sl2 loop algebra we show that every regular bethe ansatz eigenvector of the xxz spin chain at roots of unity is a highest weight vector of the sl2 loop algebra for some restricted sectors with respect to eigenvalues of the total spin operator sz and evaluate explicitly the highest weight in terms of the bethe roots we also discuss whether a given regular bethe state in the sectors generates an irreducible representation or not in fact we present such a regular bethe state in the inhomogeneous case that generates a reducible weyl module here we call a solution of the bethe ansatz equations which is given by a set of distinct and finite rapidities regular bethe roots we call a nonzero bethe ansatz eigenvector with regular bethe roots a regular bethe state

new developments in the eight vertex model ii chains of odd length we study the transfer matrix of the 8 vertex model with an odd number of lattice sites n for systems at the root of unity points mk l with m odd the transfer matrix is known to satisfy the famous tq equation where q is a specifically known matrix we demonstrate that the location of the zeroes of this q matrix is qualitatively different from the case of evenn and in particular they satisfy a previously unknown equation which is more general than what is often called bethe s equation for the case of even m where no q matrix is known we demonstrate that there are many states which are not obtained from the formalism of the sos model but which do satisfy the tq equation the ground state for the particular case of 2k 3 and n odd is investigated in detail

the dirichlet problem for higher order equations in composition form the present paper commences the study of higher order differential equations in composition form specifically we consider the equation lu divb adiva u 0 where a and b are elliptic matrices with complex valued bounded measurable coefficients and a is an accretive function elliptic operators of this type naturally arise for instance via a pull back of the bilaplacian 2 from a lipschitz domain to the upper half space more generally this form is preserved under a lipschitz change of variables contrary to the case of divergence form fourth order differential equations we establish well posedness of the dirichlet problem for the equation lu 0 with boundary data in l2 and with optimal estimates in terms of nontangential maximal functions and square functions

bethe s equation is incomplete for the xxz model at roots of unity we demonstrate for the six vertex and xxz model parameterized by q q 1 2 1 that when q2n 1 for integer n 2 the bethe s ansatz equations determine only the eigenvectors which are the highest weights of the infinite dimensional sl2 loop algebra symmetry group of the model therefore in this case the bethe s ansatz equations are incomplete and further conditions need to be imposed in order to completely specify the wave function we discuss how the evaluation parameters of the finite dimensional representations of the sl2 loop algebra can be used to complete this specification

completing bethe s equations at roots of unity in a previous paper we demonstrated that bethe s equations are not sufficient to specify the eigenvectors of the xxz model at roots of unity for states where the hamiltonian has degenerate eigenvalues we here find the equations which will complete the specification of the eigenvectors in these degenerate cases and present evidence that the sl 2 loop algebra symmetry is sufficiently powerful to determine that the highest weight of each irreducible representation is given by bethe s ansatz

l2 solvability of boundary value problems for divergence form parabolic equations with complex coefficients abstract we consider parabolic operators of the form t l l div a x t in r n 2 x t x x n 1 t r n r r x n 1 0 n 1 we assume that a is a n 1 n 1 dimensional matrix which is bounded measurable uniformly elliptic and complex and we assume in addition that the entries of a are independent of the spatial coordinate x n 1 as well as of the time coordinate t for such operators we prove that the boundedness and invertibility of the corresponding layer potential operators are stable on l 2 r n 1 c l 2 r n 2 c under complex l perturbations of the coefficient matrix subsequently using this general result we establish solvability of the dirichlet neumann and regularity problems for t l by way of layer potentials and with data in l 2 assuming that the coefficient matrix is a small complex perturbation of either a constant matrix or of a real and symmetric matrix

csos models descending from chiral potts models degeneracy of the eigenspace and loop algebra monodromy matrices of the model are known to satisfy a yang baxter equation with a six vertex r matrix as the intertwiner the commutation relations of the elements of the monodromy matrices are completely determined by this r matrix we show the reason why in the superintegrable case the eigenspace is degenerate but not in the general case we then show that the eigenspaces of special csos models descending from the chiral potts model are also degenerate the existence of an quantum loop algebra or subalgebra in these models is established by showing that the serre relations hold for the generators the highest weight polynomial or the drinfeld polynomial of the representation is obtained by using the method of baxter for the superintegrable case as a byproduct the eigenvalues of all such csos models are given explicitly

the theory of weights and the dirichlet problem for elliptic equations in this work we consider questions from the area of partial differential equations and the related questions from harmonic analysis specifically we will study the dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients roughly speaking we consider the following question suppose that we are given two operators lo and li and that we have good estimates for the dirichlet problem for lo good in this context could mean several different things for instance it might mean that there exists p 1 p oo so that the dirichlet problem is solvable in a suitable sense with lp data what are the optimal conditions on the difference between the coefficients of lo and l1 which guarantee that the dirichlet problem for li also has good estimates this question has been extensively studied recently cf f j k r f and d4 our work uses in a crucial way the results in r f the main results in d4 and r f are proved using an ingenious differential inequality for a family of harmonic measures introduced in d4 another of our goals in this article is to give new direct proofs of the results in d4 and r f without the use of this differential inequality theorems 2 4 2 5 and 2 18 these direct approaches have already had important applications k p2 an interesting class of examples for the theorems quoted above arises when one pulls back the laplacian via a quasiconformal mapping of the plane into itself specializing our theorem to this case suggests a new characterization of the class of am weights introduced in connection with several problems in harmonic analysis by muckenhoupt mul and coifman c fefferman co f we shall present in theorem 2 3 a criterion in terms of carleson measures which is very close to log w e bmo and is necessary and sufficient for w e ad

auxiliary matrices for the six vertex model at qn 1 and a geometric interpretation of its symmetries the construction of auxiliary matrices for the six vertex model at a root of unity is investigated from a quantum group theoretic point of view employing the concept of intertwiners associated with the quantum loop algebra uq 2 at qn 1 a three parameter family of auxiliary matrices is constructed the elements of this family satisfy a functional relation with the transfer matrix allowing one to solve the eigenvalue problem of the model and to derive the bethe ansatz equations this functional relation is obtained from the decomposition of a tensor product of evaluation representations and involves auxiliary matrices with different parameters because of this dependence on additional parameters the auxiliary matrices break in general the finite symmetries of the six vertex model such as spin reversal or spin conservation more importantly they also lift the extra degeneracies of the transfer matrix due to the loop symmetry present at rational coupling values the extra parameters in the auxiliary matrices are shown to be directly related to the elements in the enlarged centre z of the algebra uq 2 at qn 1 this connection provides a geometric interpretation of the enhanced symmetry of the six vertex model at rational coupling the parameters labelling the auxiliary matrices can be interpreted as coordinates on a hypersurface spec z 4 which remains invariant under the action of an infinite dimensional group g of analytic transformations called the quantum coadjoint action

trace operator and the dirichlet problem for elliptic equations on arbitrary bounded open sets abstract we consider the dirichlet problem and the weak dirichlet problem on a general possibly nonregular bounded domain for elliptic linear equation with uniformly elliptic divergence form operator we investigate carefully the relationship between weak integral probabilistic soft and the perron wiener brelot solutions of the problem we also study the relationship between solutions of the dirichlet problem and weak dirichlet problem to this end we extend the usual notion of the trace operator and consider the dirichlet problem with martin boundary

the neumann problem for elliptic equations with non smooth coefficients in this paper we study the neumann and regularity problems for divergence form symmetric elliptic operators l div av where a aifix is a matrix of bounded measurable functions and ellipticity is the existence of a 0 such that 2 114 2 ais ir 21412 for all i r we assume that l is defined in the unit ball b but the methods of proof are sufficiently general to allow b to be a bounded starlike lipschitz domain the neumann problem lu 0 in b avun q f

construction of some missing eigenvectors of the xyz spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix we discuss an algebraic method for constructing eigenvectors of the transfer matrix of the eight vertex model at the discrete coupling parameters we consider the algebraic bethe ansatz of the elliptic quantum group e sl2 for the case where the parameter satisfies 2n m1 m2 for arbitrary integers n m1 and m2 when m1 or m2 is odd the eigenvectors thus obtained have not been discussed previously furthermore we construct a family of degenerate eigenvectors of the xyz spin chain some of which are shown to be related to the sl2 loop algebra symmetry of the xxz spin chain we show that the dimension of some degenerate eigenspace of the xyz spin chain on l sites is given by n2l n if l n is an even integer the construction of eigenvectors of the transfer matrices of some related interaction round a face models is also discussed

estimates for 8706 μ 1 and calderon s theorem on the cauchy integral etude du caractere borne de l operateur integral de cauchy a l aide d une transformation quasi conforme

necessary and sufficient conditions for absolute continuity of elliptic harmonic measure our purpose in this paper is to give a necessary and sufficient condition on the modulus of continuity of the coefficients of an elliptic operator in divergence form in order that the corresponding poisson kernel for the dirichlet problem exist this condition on the coefficients is global continuity together with the additional property that the modulus of continuity along some nontangential direction at each boundary point be bounded uniformly in these points and directions by a function 7 t satisfying the dini type condition

generalizations of siegel s and picard s theorems we prove new theorems that are higher dimensional generalizations of the classical theorems of siegel on integral points on affine curves and of picard on holomorphic maps from c to affine curves these include results on integral points over varying number fields of bounded degree and results on kobayashi hyperbolicity we give a number of new conjectures describing from our point of view how we expect siegel s and picard s theorems to optimally generalize to higher dimensions in this article we prove new theorems that are higher dimensional generaliza tions of the classical theorems of siegel on integral points on affine curves and of picard on holomorphic maps from c to affine curves in section 2 we will give the statements of siegel s and picard s theorems and we will recall how these two theorems from such seemingly different areas of mathematics are related we will then proceed to give a number of new conjectures describing from our point of view how we expect siegel s and picard s theorems to optimally generalize to higher dimensions these include conjectures on integral points over varying number fields of bounded degree and conjectures addressing hyperbolic questions these conjectures appear to be fundamentally new we will then summarize our progress on these conjectures we have been able to get results in all dimensions with best possible results in many cases for surfaces

on surfaces of general type with maximal albanese dimension abstractgiven a minimal surface s equipped with a generically nite map to an abelianvariety and c s a rational or an elliptic curve we show that the canonical degreeof c is bounded by four times the self intersection of the canonical divisor of s asa corollary we obtain the niteness of rational and elliptic curves with an optimaluniform bound on their canonical degrees on any surface of general type with twolinearly independent regular one forms 1 1 introduction the object of this paper is to give an e ective bound on the canonical degrees of rationaland elliptic curves on a minimal surface of general type with generically nite albanese maplinearly in terms of the canonical volume of the surface here the canonical degree is thedegree with respect to the canonical polarization of the surface and the self intersection ofthis polarization is the canonical volume we will work in the complex analytic setting so that all varieties are complex analytic wewill assume the rudiments from the the theory of classi cation of complex projective surfacesas found for example in 3 we call a smooth projective surface to be of maximal albanesedimension if its albanese map is generically nite n b this terminology is di erent fromthat introduced in 21 we recall that any smooth projective surface s of maximal albanesedimension admits a morphism to a minimal one whose canonical divisor pulls back to thepositive part p of the zariski decomposition of the canonical divisor k of s and the canonicalvolume of s is by de nition vol k p

divergence form operators in reifenberg flat domains we study the boundary regularity of solutions of elliptic operators in divergence form with c 0 coefficients or operators which are small perturbations of the laplacian in non smooth domains we show that as in the case of the laplacian there exists a close relationship between the regularity of the corresponding elliptic measure and the geometry of the domain

greatest common divisors and vojta s conjecture for blowups of algebraic tori we give results and inequalities bounding the greatest common divisor of multivariable polynomials evaluated at s unit arguments generalizing to an arbitrary number of variables results of bugeaud corvaja zannier hernandez luca and corvaja zannier in closely related results and in line with observations of silverman we prove special cases of vojta s conjecture for blowups of toric varieties as an application we classify when terms from simple linear recurrence sequences can have a large greatest common divisor in an appropriate sense the primary tool used in the proofs is schmidt s subspace theorem from diophantine approximation

the a property of elliptic measures for operators with coefficients supported in whitney type cubes abstract it is proved the preservation of the a property with respect to surface measure of harmonic measure associated to second order divergence form operators with bounded measurable coefficients when the discrepancy between the main coefficients of the operators is supported on whitney type rectangles of the unit ball of r n in particular neither the discrepancy between the coefficients satisfies a carleson measure condition nor the coefficients of the operators involved are radially independent as in previous well known works as an application of the perturbation result we provide a class of operators with coefficients supported on whitney type rectangles for which the a property holds

functional calculus of dirac operators and complex perturbations of neumann and dirichlet problems we prove that the neumann dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in l 2 for small complex l infty perturbations of a coefficient matrix which is either real symmetric of block form or constant all matrices are assumed to be independent of the transversal coordinate we solve the neumann dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order dirac type operator we establish quadratic estimates for this dirac operator which implies that the associated hardy projection operators are bounded and depend continuously on the coefficient matrix we also prove that certain transmission problems for k forms are well posed for small perturbations of block matrices

hausdorff measure estimates and lipschitz regularity in inhomogeneous nonlinear free boundary problems in this paper we prove a hausdorff measure estimate for the free boundaries of subsolutions of fully nonlinear and quasilinear equations of the type f d 2u x geqq f x and rm div a x nabla u geqq mu where f in l q q n and is a signed radon measure with some appropriate growth condition gradient estimates for nonnegative harmonic functions with bounded normal derivatives along the boundary obtained by caffarelli and salsa geometric approach to free boundary problems 2005 are extended to the context of inhomogeneous problems involving fully nonlinear and p laplace equations as an application lipschitz regularity is obtained for one phase solutions of inhomogeneous nonlinear free boundary problems

differentiability of solutions to second order elliptic equations via dynamical systems for a second order elliptic equation in divergence form we investigate conditions on the coefficients which imply that all solutions are lipschitz continuous or differentiable at a given point we assume the coefficients have modulus of continuity satisfying the square dini condition and obtain additional conditions that examples show are sharp our results extend those of previous authors who assume the modulus of continuity satisfies the dini condition our method involves the study of asymptotic properties of solutions to a dynamical system that is derived from the coefficients of the elliptic equation

the l p dirichlet problem and nondivergence harmonic measure we consider the dirichlet problem lu 0 in d u g on d for two second order elliptic operators l k u n i j 1 a i j k x ij u x k 0 1 in a bounded lipschitz domain d r n the coefficients a i j k belong to the space of bounded mean oscillation bmo with a suitable small bmo modulus we assume that l 0 is regular in l p d d for some p 1 p that is nu lp c g lp for all continuous boundary data g here is the surface measure on d and nu is the nontangential maximal operator the aim of this paper is to establish sufficient conditions on the difference of the coefficients e i j x a i j 1 x a i j 0 x that will assure the perturbed operator l 1 to be regular in l q d d for some q 1 q

weighted maximal regularity estimates and solvability of non smooth elliptic systems i we develop new solvability methods for divergence form second order real and complex elliptic systems above lipschitz graphs with l 2 boundary data the coefficients a may depend on all

new developments in the eight vertex model we demonstrate that the q matrix introduced in baxter s 1972 solution of the eight vertex model has some eigenvectors which are not eigenvectors of the spin reflection operator and conjecture a new functional equation for q v which both contains the bethe equation that gives the eigenvalues of the transfer matrix and computes the degeneracies of these eigenvalues

completeness of the bethe ansatz for the six and eight vertex models we discuss some of the difficulties that have been mentioned in the literature in connection with the bethe ansatz for the six vertex model and xxz chain and for the eight vertex model in particular we discuss the beyond the equator infinite momenta and exact complete string problems we show how they can be overcome and conclude that the coordinate bethe ansatz does indeed give a complete set of states as expected

the regularity problem for second order elliptic operators with complex valued bounded measurable coefficients the present paper establishes a certain duality between the dirichlet and regularity problems for elliptic operators with t independent complex bounded measurable coefficients t being the transversal direction to the boundary to be precise we show that the dirichlet boundary value problem is solvable in l p subject to the square function and non tangential maximal function estimates if and only if the corresponding regularity problem is solvable in l p moreover the solutions admit layer potential representations in particular we prove that for any elliptic operator with t independent real possibly non symmetric coefficients there exists a p 1 such that the regularity problem is well posed in l p

on fewnomials integral points and a toric version of bertini s theorem an old conjecture of erd s and renyi proved by schinzel predicted a bound for the number of terms of a polynomial g x x when its square g x has a given number of terms further conjectures and results arose but some fundamental questions remained open in this paper with methods which appear to be new we achieve a final result in this direction for completely general algebraic equations f x g x 0 where f x y is monic of arbitrary degree in y and has boundedly many terms in x we prove that the number of terms of such a g x is necessarily bounded this includes the previous results as extremely special cases we shall interpret polynomials with boundedly many terms as the restrictions to 1 parameter subgroups or cosets of regular functions of bounded degree on a given torus glm such a viewpoint shall lead to some best possible corollaries in the context of finite covers of glm concerning the structure of their integral points over function fields in the spirit of conjectures of vojta and a bertini type irreducibility theorem above algebraic multiplicative cosets a further natural reading occurs in non standard arithmetic where our result translates into an algebraic and integral closedness statement inside the ring of non standard polynomials