discrete riemann surfaces and the ising model we define a new theory of discrete riemann surfaces and present its basic results the key idea is to consider not only a cellular decomposition of a surface but the union with its dual discrete holomorphy is defined by a straightforward discretisation of the cauchy riemann equation a lot of classical results in riemann theory have a discrete counterpart hodge star harmonicity hodge theorem weyl s lemma cauchy integral formula existence of holomorphic forms with prescribed holonomies giving a geometrical meaning to the construction on a riemann surface we define a notion of criticality on which we prove a continuous limit theorem we investigate its connection with criticality in the ising model we set up a dirac equation on a discrete universal spin structure and we prove that the existence of a dirac spinor is equivalent to criticality

the critical z invariant ising model via dimers the periodic case we study a large class of critical two dimensional ising models namely critical z invariant ising models on periodic graphs example of which are the classical mathbb z 2 triangular and honeycomb lattice at the critical temperature fisher j math phys 7 1776 1781 1966 introduced a correspondence between the ising model and the dimer model on a decorated graph thus setting dimer techniques as a powerful tool for understanding the ising model in this paper we give a full description of the dimer model corresponding to the critical z invariant ising model we prove that the dimer characteristic polynomial is equal up to a constant to the critical laplacian characteristic polynomial and defines a harnack curve of genus 0 we prove an explicit expression for the free energy and for the gibbs measure obtained as weak limit of boltzmann measures

regularity results for segregated configurations involving fractional laplacian abstract we study the regularity of segregated profiles arising from competition diffusion models where the diffusion process is of nonlocal type and is driven by the fractional laplacian of power s 0 1 among others our results apply to the regularity of the densities of an optimal partition problem involving the eigenvalues of the fractional laplacian more precisely we show c 0 regularity of the density where the exponent is explicit and is given by s for s 0 1 2 2 s 1 for s 1 2 1 under some additional assumptions we then show that solutions are c 0 s these results are optimal in the class of holder continuous functions thus we find a complete correspondence with known results in case of the standard laplacian

uniform bounds for strongly competing systems the optimal lipschitz case for a class of systems of semi linear elliptic equations including delta u i f i x u i beta u i sum j neq i a ij u j p quad i 1 dots k for p 2 variational type interaction or p 1 symmetric type interaction we prove that uniform l infty boundedness of the solutions implies uniform boundedness of their lipschitz norm as beta to infty that is in the limit of strong competition this extends known quasi optimal regularity results and covers the optimal case for this class of problems the proofs rest on monotonicity formulae of alt caffarelli friedman and almgren type in the variational setting and on the caffarelli jerison kenig almost monotonicity formula in the symmetric one

holder bounds and regularity of emerging free boundaries for strongly competing schrodinger equations with nontrivial grouping abstract we study interior regularity issues for systems of elliptic equations of the type u i f i x j i a i j u i u i p 1 u j p 1 set in domains r n for n 1 the paper is devoted to the derivation of c 0 estimates that are uniform in the competition parameter 0 as well as to the regularity of the limiting free boundary problem obtained for the main novelty of the problem under consideration resides in the non trivial grouping of the densities in particular we assume that the interaction parameters a i j are only non negative and thus may vanish for specific couples i j as a main consequence in the limit densities do not segregate pairwise in general but are grouped in classes which in turn form a mutually disjoint partition moreover with respect to the literature we consider more general forcing terms sign changing solutions and an arbitrary p 0 in addition we present a regularity theory of the emerging free boundary defined by the interface among different segregated groups these equations are very common in the study of bose einstein condensates and are of key importance for the analysis of optimal partition problems related to high order eigenvalues

the rh boundary value problem of the k monogenic functions in this paper we study the riemann and hilbert problems of k monogenic functions by using euler operator we transform the boundary value problem of k monogenic functions into the boundary value problems of monogenic functions then by the almansi type theorem of k monogenic functions we get the solutions of these problems

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

conformal invariance of domino tiling let u be a multiply connected region in r 2 with smooth boundary let p be a polyomino in z 2 approximating u as 0 we show that for certain boundary conditions on p the height distribution on a random domino tiling dimer covering of p is conformally invariant in the limit as tends to 0 in the sense that the distribution of heights of boundary components or rather the difference of the heights from their mean values only depends on the conformal type of u the mean height is not strictly conformally invariant but transforms analytically under conformal mappings in a simple way the mean height and all the moments are explicitly evaluated

on a critical schrodinger system in r4 with steep potential wells abstract in this paper we study the following two component elliptic system u a x a 0 u u 3 v 2 u 0 in r 4 v b x b 0 v v 3 u 2 v 0 in r 4 u v h 1 r 4 h 1 r 4 where a 0 b 0 r are constants 0 and r are parameters a x b x 0 are potentials by using variational methods we prove the existence and nonexistence of general ground state solutions maybe semi trivial and ground state solutions of the above system under some further conditions on the potentials a x b x and the parameters it is worth pointing out that the cubic nonlinearities and the coupled terms of the above system are all of critical growth owing to the sobolev embedding theorem furthermore by introducing some ideas which are different from that in the literature the phenomenon of phase separation of ground state solutions of the above system is also obtained without any symmetry conditions

uniform holder estimates in a class of elliptic systems and applications to singular limits in models for diffusion flames the main result of this paper is a general holder estimate in a class of singularly perturbed elliptic systems this estimate is applied to the study of the well known burke schuman approximation in flame theory after reviewing some classical cases equidiffusional case high activation energy approximation we turn to the non equidiffusional case and to nonlinear diffusion models the limiting problems are nonlinear elliptic equations they have holder or lipschitz maximal global regularity a natural question is then whether this regularity is kept uniformly throughout the approximation process and we show that this is true in general

defect numbers of singular integral operators with carleman shift and almost periodic coefficients abstract this paper is devoted to the study of properties of the kernel and the cokernel of singular integral operators with almost periodic coefficients and a carleman shift in particular the dimensions of their kernels and cokernels are obtained this is done by considering appropriate properties of the related almost periodic elements and in special the partial indices of some of their relevant factorizations

a free boundary problem arising from segregation of populations with high competition in this work we show how to obtain a free boundary problem as the limit of a fully nonlinear elliptic system of equations that models population segregation of the gause lotka volterra type we study the regularity of the solutions in particular we prove lipschitz regularity across the free boundary the problem is motivated by the work done by caffarelli karakhanyan and fang hua lin for the linear case

conformal invariance in random cluster models i holmorphic fermions in the ising model we construct discrete holomorphic observables in the ising model at criticality and show that they have conformally covariant scaling limits as mesh of the lattice tends to zero in the sequel those observables are used to construct conformally invariant scaling limits of interfaces though ising model is often cited as a classical example of conformal invariance it seems that ours is the first paper where it is actually established

riemann hilbert problems for null solutions to iterated generalized cauchy riemann equations in axially symmetric domains we consider riemann hilbert boundary value problems for short rhbvps with variable coefficients for axially symmetric poly monogenic functions i e null solutions to iterated generalized cauchy riemann equations defined in axially symmetric domains this extends our recent results about rhbvps with variable coefficients for axially symmetric monogenic functions defined in four dimensional axially symmetric domains first we construct the almansi type decomposition theorems for poly monogenic functions of axial type then making full use of them we give the integral representation solutions to the rhbvp considered as a special case we derive solutions to the corresponding schwarz problem finally we generalize the result obtained to functions of axial type which are null solutions to perturbed iterated generalized cauchy riemann equations d k 0 k 2 k n r

toward a conformal field theory for schramm loewner evolutions we discuss the partition function point of view for chordal schramm loewner evolutions and their relationship with correlation functions in conformal field theory both are closely related to crossing probabilities and interfaces in critical models in two dimensional statistical mechanics we gather and supplement previous results with different perspectives point out remaining difficulties and suggest directions for future studies

on entire solutions of an elliptic system modeling phase separations we study the qualitative properties of a limiting elliptic system arising in phase separation for bose einstein condensates with multiple states

cuspidal representations of sl n 1 abstract in this paper we study the subcategory of cuspidal modules of the category of weight modules over the lie algebra sl n 1 our main result is a complete classification and an explicit description of indecomposable cuspidal modules

on conformally invariant cle explorations we study some conformally invariant dynamic ways to construct the conformal loop ensembles with simple loops introduced in earlier papers by sheffield and by sheffield and werner one outcome is a conformally invariant way to measure a distance of a cle4 loop to the boundary within the cle4 when one identifies all points of each loop

the critical z invariant ising model via dimers locality property we study a large class of critical two dimensional ising models namely critical z invariant ising models fisher j math phys 7 1776 1781 1966 introduced a correspondence between the ising model and the dimer model on a decorated graph thus setting dimer techniques as a powerful tool for understanding the ising model in this paper we give a full description of the dimer model corresponding to the critical z invariant ising model consisting of explicit expressions which only depend on the local geometry of the underlying isoradial graph our main result is an explicit local formula for the inverse kasteleyn matrix in the spirit of kenyon invent math 150 2 409 439 2002 as a contour integral of the discrete exponential function of mercat discrete period matrices and related topics 2002 and kenyon invent math 150 2 409 439 2002 multiplied by a local function using results of boutillier and de tiliere prob theor rel fields 147 3 4 379 413 2010 and techniques of de tiliere prob th rel fields 137 3 4 487 518 2007 and kenyon invent math 150 2 409 439 2002 this yields an explicit local formula for a natural gibbs measure and a local formula for the free energy as a corollary we recover baxter s formula for the free energy of the critical z invariant ising model baxter in exactly solved models in statistical mechanics academic press london 1982 and thus a new proof of it the latter is equal up to a constant to the logarithm of the normalized determinant of the laplacian obtained in kenyon invent math 150 2 409 439 2002

a matrix hilbert transform in hermitean clifford analysis abstract orthogonal clifford analysis is a higher dimensional function theory offering both a generalization of complex analysis in the plane and a refinement of classical harmonic analysis during the last years hermitean clifford analysis has emerged as a new and successful branch of it offering yet a refinement of the orthogonal case recently in f brackx b de knock h de schepper d pena pena f sommen submitted for publication a hermitean cauchy integral was constructed in the framework of circulant 2 2 matrix functions in the present paper a new hermitean hilbert transform is introduced arising naturally as part of the non tangential boundary limits of that hermitean cauchy integral the resulting matrix operator is shown to satisfy properly adapted analogues of the characteristic properties of the hilbert transform in classical analysis and orthogonal clifford analysis

an optimal partition problem related to nonlinear eigenvalues we extend to the case of many competing densities the results of the paper ann inst h poincare 6 2002 more precisely we are concerned with an optimal partition problem in n dimensional domains related to the method of nonlinear eigenvalues introduced by nehari acta math 105 1961 we prove existence of the minimal partition and some extremality conditions moreover in the case of two dimensional domains we give an asymptotic formula near the multiple intersection points finally we show some connections between the variational problem and the behavior of competing species systems with large interaction

root fernando kac subalgebras of finite type abstract let g be a finite dimensional lie algebra and m be a g module the fernando kac subalgebra of g associated to m is the subset g m g of all elements g g which act locally finitely on m a subalgebra l g for which there exists an irreducible module m with g m l is called a fernando kac subalgebra of g a fernando kac subalgebra of g is of finite type if in addition m can be chosen to have finite jordan holder l multiplicities under the assumption that g is simple i penkov has conjectured an explicit combinatorial criterion describing all fernando kac subalgebras of finite type which contain a cartan subalgebra in the present paper we prove this conjecture for g e 8

boundary value problems with higher order lipschitz boundary data for polymonogenic functions in fractal domains in this note we consider certain jump problem for poly monogenic functions in fractal domains with higher order lipschitz boundary data this is accomplished by using a higher order teodorescu operator which replaces the expected surface integral also we give out the uniqueness of solutions basing the work on the method of removable singularities for monogenic functions making use of a dolzhenko type theorem

extensions of the conformal representations for orthogonal lie algebras abstract the conformal transformations with respect to the metric defining o n c give rise to an inhomogeneous polynomial representation of o n 2 c using shen s technique of mixed product we generalize the above representation to an inhomogeneous representation of o n 2 c on the tensor space of any finite dimensional irreducible o n c module with the polynomial space where a hidden central transformation is involved moreover we find a condition on the constant value taken by the central transformation such that the generalized representation is irreducible in our approach pieri s formulas invariant operators and the idea of kostant s characteristic identities play key roles the result could be useful in understanding higher dimensional conformal field theory with the constant value taken by the central transformation as the central charge our representations virtually provide natural extensions of the conformal transformations on a riemannian manifold to its vector bundles

conformal invariance in the fk representation of the quantum ising model and convergence of the interface to the mathrm sle _ 16 3 we study the interface in the fk representation of the 1d quantum ising model and show that in the limit it converges to the mathrm sle 16 3 curve

convolution singular integrals on lipschitz surfaces we prove the lp boundedness of convolution singular integral op erators on a lipschitz surface g x eo x e r 1 x e rn where g is a lipschitz function which satisfies ilvgli we have embedded r i in the clifford algebra r n with identity eo and are considering convolution with right monogenic functions 0 which satisfy 10 x i 0 so x xo x e r lxol where p a w provided there exists an l function q satisfying 0o r 0 r xern then the related convolution singular integral operator t u x lim o f yei x y n y u y dsy 0 en x u x e o jl is bounded on lp for 1 tronics macquarie university nsw 2109 australia e mail address chun li chun macadam mpce mq edu au e mail address alan mcintosh alan macadam mpce mq edu au stephen semmes department of mathematics rice university houston texas 77251 e mail address semmes rice edu this content downloaded from 157 55 39 143 on wed 15 jun 2016 04 40 33 utc all use subject to http about jstor org terms

uniform holder estimate for singularly perturbed parabolic systems of bose einstein condensates and competing species abstract we prove the uniform holder continuity of solutions for two classes of singularly perturbed parabolic systems these systems arise in bose einstein condensates and in competing models in population dynamics the proof relies upon the blow up technique and the monotonicity formulas by almgren and alt caffarelli and friedman

bose fluids and positive solutions to weakly coupled systems with critical growth in dimension two abstract we prove using variational methods the existence in dimension two of positive vector ground states solutions for the bose einstein type systems u 1 u 1 u e u 2 1 v e u v 1 in v 2 v 2 v e v 2 1 u e u v 1 in u v h 0 1 where is a bounded smooth domain 1 2 1 the first eigenvalue of h 0 1 1 2 0 and is either positive small or large or negative small the nonlinear interaction between two bose fluids is assumed to be of critical exponential type in the sense of j moser for small solutions the system is asymptotically equivalent to the corresponding one in higher dimensions with power like nonlinearities

asymptotic estimates for the spatial segregation of competitive systems abstract for a class of population models of competitive type we study the asymptotic behavior of the positive solutions as the competition rate tends to infinity we show that the limiting problem is a remarkable system of differential inequalities which defines the functional class s in 2 by exploiting the regularity theory recently developed in conti et al indiana univ math j to appear for the elements of functional classes of the form s we provide some qualitative and regularity property of the limiting configurations besides for the case of two competing species we obtain a full description of the limiting states and we prove some quantitative estimates for the rate of convergence finally we prove some new liouville type results which allow to have uniform regularity estimates of the solutions

irreducible completely pointed modules of quantum groups of type a abstract we give a classification of all irreducible completely pointed u q sl n 1 modules over a characteristic zero field in which q is not a root of unity this generalizes the classification result of benkart britten and lemire in the non quantum case we also show that any infinite dimensional irreducible completely pointed u q sl n 1 can be obtained from some irreducible completely pointed module over the quantized weyl algebra a n 1 q

finite weight modules over twisted affine lie superalgebras abstract this work provides the first step toward the classification of irreducible finite weight modules over twisted affine lie superalgebras we divide the class of such modules into two subclasses called hybrid and tight we reduce the classification of hybrid irreducible finite weight modules to the classification of cuspidal modules of finite dimensional cuspidal lie superalgebras which is discussed in a work of dimitrov mathieu and penkov

a natural constraint approach to normalized solutions of nonlinear schrodinger equations and systems abstract the paper deals with the existence of normalized solutions to the system u 1 u 1 u 3 u v 2 in r 3 v 2 v 2 v 3 u 2 v in r 3 r 3 u 2 a 1 2 and r 3 v 2 a 2 2 for any 1 2 a 1 a 2 0 and 0 prescribed we present a new approach that is based on the introduction of a natural constraint associated to the problem we also show that as phase separation occurs for the solutions that we find our method can be adapted to scalar nonlinear schrodinger equations with normalization constraint and leads to alternative and simplified proofs to some results already available in the literature

monotonicity and 1 dimensional symmetry for solutions of an elliptic system arising in bose einstein condensation we study monotonicity and 1 dimensional symmetry for positive solutions with algebraic growth of the following elliptic system left begin array ll delta u u upsilon 2 quad rm in mathbb r n delta upsilon u 2 upsilon quad rm in mathbb r n end array right for every dimension n geqq 2 in particular we prove a gibbons type conjecture proposed by berestycki et al

half dirichlet problem for matrix functions on the unit ball in hermitian clifford analysis abstract the simultaneous null solutions of the two complex hermitian dirac operators are focused on in hermitian clifford analysis where the hermitian cauchy integral was constructed and will play an important role in the framework of circulant 2 2 matrix functions under this setting we will present the half dirichlet problem for circulant 2 2 matrix functions on the unit ball of even dimensional euclidean space we will give the unique solution to it merely by using the hermitian cauchy transformation get the solution to the dirichlet problem on the unit ball for circulant 2 2 matrix functions and the solution to the classical dirichlet problem as the special case derive a decomposition of the poisson kernel for matrix laplace operator and further obtain the decomposition theorems of solution space to the dirichlet problem for circulant 2 2 matrix functions

construction of a solution for the two component radial gross pitaevskii system with a large coupling parameter abstract we consider strongly coupled competitive elliptic systems that arise in the study of two component bose einstein condensates as the coupling parameter tends to infinity solutions that remain uniformly bounded are known to converge to a segregated limiting profile with the difference of its components satisfying a limit scalar pde in the case of radial symmetry under natural non degeneracy assumptions on a solution of the limit problem we establish by a perturbation argument its persistence as a solution to the elliptic system

infinite conformal symmetry of critical fluctuations in two dimensions we study the massless quantum field theories describing the critical points in two dimensional statistical systems these theories are invariant with respect to the infinite dimensional group of conformal analytic transformations it is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of the virasoro algebra exactly solvable theories associated with degenerate representations are analized in these theories the anomalous dimensions are known exactly and the correlation functions satisfy the system of linear differential equations

almost monotonicity formulas for elliptic and parabolic operators with variable coefficients in this paper we extend the results of caffarelli jerison and kenig ann of math 2 155 2002 and caffarelli and kenig amer j math 120 1998 by establishing an almost monotonicity estimate for pairs of continuous functions satisfying u 0 lu 1 u u 0 in an infinite strip global version or a finite parabolic cylinder localized version where l is a uniformly parabolic operator lu l a b c u div a x s u b x s u c x s u s u with double dini continuous a and uniformly bounded b and c we also prove the elliptic counterpart of this estimate this closes the gap between the known conditions in the literature both in the elliptic and parabolic case imposed on u in order to obtain an almost monotonicity estimate at the end of the paper we demonstrate how to use this new almost monotonicity formula to prove the optimal c 1 1 regularity in a fairly general class of quasi linear obstacle type free boundary problems

liouville theorems and 1 dimensional symmetry for solutions of an elliptic system modelling phase separation abstract we consider solutions of the competitive elliptic system s u i j i u i u j 2 in r n u i 0 in r n i 1 k we are concerned with the classification of entire solutions according to their growth rate the prototype of our main results is the following there exists a function k n n increasing in k such that if u 1 u k is a solution of s and u 1 x u k x c 1 x d for every x r n then d this means that the number of components k of the solution imposes a lower bound increasing in k on the minimal growth of the solution itself if n 2 the expression of is explicit and optimal while in higher dimension it can be characterised in terms of an optimal partition problem we discuss the sharpness of our results and as a further step for every n 2 we can prove the 1 dimensional symmetry of the solutions of s satisfying suitable assumptions extending known results which are available for k 2 the proofs rest upon a blow down analysis and on some monotonicity formulae

rhombic embeddings of planar quad graphs given a finite or infinite planar graph all of whose faces have degree 4 we study embeddings in the plane in which all edges have length 1 that is in which every face is a rhombus we give a necessary and sucient condition for the existence of such an embedding as well as a description of the set of all such embeddings r etant donne un graphe planaire fini ou infini dont toutes les faces sont de degre 4 onetudie ses plongements dans le plan dont toutes les aretes sont de longueur 1 c estdire dont toutes les faces sont des losanges on donne une condition necessaire et susante pour l existence d un tel plongement et on decrit l ensemble de ces plongements

on modules of bounded multiplicities for the symplectic algebras simple infinite dimensional highest weight modules having bounded weight multipicities are classified as submodules of a tensor product also it is shown that a simple torsion free module of finite degree tensored with a finite dimensional module is completely reducible

the limit equation for the gross pitaevskii equations and s terraciniʼs conjecture abstract we establish the limit system for the gross pitaevskii equations when the segregation phenomenon appears and shows this limit is the one arising from the competing systems in population dynamics this covers and verifies a conjecture of s terracini et al both in the parabolic case and the elliptic case

modular representations of the special linear groups with small weight multiplicities abstract we classify irreducible representations of the special linear groups in positive characteristic with small weight multiplicities with respect to the group rank and give estimates for the maximal weight multiplicities for the natural embeddings of the classical groups inductive systems of representations with totally bounded weight multiplicities are classified an analogue of the steinberg tensor product theorem for arbitrary indecomposable inductive systems for such embeddings is proved

integrability as a consequence of discrete holomorphicity loop models it has recently been established that imposing the condition of discrete holomorphicity on a lattice parafermionic observable leads to the critical boltzmann weights in a number of lattice models remarkably the solutions of these linear equations also solve the yang baxter equations we extend this analysis for the zn model by explicitly considering the condition of discrete holomorphicity on two and three adjacent rhombi for two rhombi this leads to a quadratic equation in the boltzmann weights and for three rhombi a cubic equation the two rhombus equation implies the inversion relations the star triangle relation follows from the three rhombus equation we also show that these weights are self dual as a consequence of discrete holomorphicity this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

cuspidal sln modules and deformations of certain brauer tree algebras we show that the algebras describing blocks of the category of cuspidal weight resp generalized weight sln modules are one parameter resp multi parameter deformations of certain brauer tree algebras we explicitly determine these deformations both graded and ungraded the algebras we deform also appear as special centralizer subalgebras of temperley lieb algebras or as generalized khovanov algebras they show up in the context of highest weight representations of the virasoro algebra in the context of rational representations of the general linear group and schur algebras and in the study of the milnor fiber of kleinian singularities

long range properties of spanning trees we compute some large scale properties of the uniform spanning tree process on bounded regions in z2 in particular we compute the distribution of the meeting point of the branches of the tree issued from three boundary points we also compute the crossing probabilities of branches of the tree on rectangular and annular regions as well as the winding number of the branches of the tree

lie algebra modules with finite dimensional weight spaces i let g denote a reductive lie algebra over an algebraically closed field of characteristic zero and let 4 denote a cartan subalgebra of g in this paper we study finitely generated g modules that decompose into direct sums of finite dimensional 4 weight spaces we show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules those we call torsion free modules we also show that if g is a simple lie algebra that admits a torsion free module then g is of type a or c

the scaling limit of the energy correlations in non integrable ising models we obtain an explicit expression for the multipoint energy correlations of a non solvable two dimensional ising models with nearest neighbor ferromagnetic interactions plus a weak finite range interaction of strength in a scaling limit in which we send the lattice spacing to zero and the temperature to the critical one our analysis is based on an exact mapping of the model into an interacting lattice fermionic theory which generalizes the one originally used by schultz mattis and lieb for the nearest neighbor ising model the interacting model is then analyzed by a multiscale method first proposed by pinson and spencer if the lattice spacing is finite then the correlations cannot be computed in closed form rather they are expressed in terms of infinite convergent power series in in the scaling limit these infinite expansions radically simplify and reduce to the limiting energy correlations of the integrable ising model up to a finite renormalization of the parameters explicit bounds on the speed of convergence to the scaling limit are derived

on the plemelj privalov theorem in clifford analysis involving higher order lipschitz classes abstract the main purpose of this work is to prove that the higher order lipschitz classes behave invariant under the action of a singular integral operator which arise naturally in polymonogenic clifford algebra valued function theory

variational problems with long range interaction we consider a class of variational problems for densities that repel each other at a distance typical examples are given by the dirichlet functional and the rayleigh functional begin aligned d mathbf u sum i 1 k int omega nabla u i 2 quad text or quad r mathbf u sum i 1 k frac int omega nabla u i 2 int omega u i 2 end aligned minimized in the class of h 1 omega mathbb r k functions attaining some boundary conditions on and subjected to the constraint begin aligned text dist u i 0 u j 0 ge 1 quad forall i neq j end aligned for these problems we investigate the optimal regularity of the solutions prove a free boundary condition and derive some preliminary results characterizing the free boundary partial sum i 1 k u i 0

uniform holder bounds for nonlinear schrodinger systems with strong competition for the positive solutions of the gross pitaevskii system u u 1 u 3 u v 2 v v 2 v 3 u 2 v we prove that l boundedness implies c 0 boundedness for every 0 1 uniformly as moreover we prove that the limiting profile as is lipschitz continuous the proof relies upon the blowup technique and the monotonicity formulae by almgren and alt caffarelli and friedman this system arises in the hartree fock approximation theory for binary mixtures of bose einstein condensates in different hyperfine states extensions to systems with k 2 densities are given

dominos and the gaussian free field we define a scaling limit of the height function on the domino tiling model dimer model on simply connected regions in z 2 and show that it is the massless free field a gaussian process with independent coefficients when expanded in the eigenbasis of the laplacian

critical ising on the square lattice mixes in polynomial time the ising model is widely regarded as the most studied model of spin systems in statistical physics the focus of this paper is its dynamic stochastic version the glauber dynamics introduced in 1963 and by now the most popular means of sampling the ising measure intensive study throughout the last three decades has yielded a rigorous understanding of the spectral gap of the dynamics on mathbb z 2 everywhere except at criticality while the critical behavior of the ising model has long been the focus for physicists mathematicians have only recently developed an understanding of its critical geometry with the advent of sle cle and new tools to study conformally invariant systems

the energy density in the planar ising model we study the critical ising model on the square lattice in bounded simply connected domains with and free boundary conditions we relate the energy density of the model to a discrete fermionic spinor and compute its scaling limit by discrete complex analysis methods as a consequence we obtain a simple exact formula for the scaling limit of the energy field one point function in terms of the hyperbolic metric this confirms the predictions originating in physics but also provides a higher precision

universality in the 2d ising model and conformal invariance of fermionic observables it is widely believed that the celebrated 2d ising model at criticality has a universal and conformally invariant scaling limit which is used in deriving many of its properties however no mathematical proof has ever been given and even physics arguments support a priori weaker mobius invariance we introduce discrete holomorphic fermions for the 2d ising model at criticality on a large family of planar graphs we show that on bounded domains with appropriate boundary conditions those have universal and conformally invariant scaling limits thus proving the universality and conformal invariance conjectures

a classification of simple lie modules having a 1 dimensional weight space let l denote a simple lie algebra over the complex numbers in this paper we classify and construct all simple l modules which may be infinite dimensional but have at least one 1 dimensional weight space this completes the study begun earlier by the authors for the case of l a the approach presented here relies heavily on the results of suren fernando whose dissertation dealt with simple weight modules and their weight systems 0 introduction let l be a finite dimensional simple lie algebra over the complex field c having a cartan subalgebra h and denote by c l the centralizer of h in the universal enveloping algebra u of l if x h c is a weight function of a 1 dimensional weight space mx in a simple l module m then q c l c defined by q c v cv for v e mx and c e c l is an algebra homomorphism called a mass function of m clearly x restricted to h is equal to x conversely given any algebra homomorphism q c l c one can construct a unique simple l module which admits x as a mass function 4 10 in 4 the authors determined all algebra homomorphisms q c l c for the simple lie algebras l of type an and using these classified all pointed an modules where we call a module pointed if it is simple and has at least one 1 dimensional weight space in this paper we complete the classification of all pointed l modules for arbitrary simple lie algebras the collection of all pointed l modules clearly includes the highest weight l modules and is included in the collection of all harish chandra l modules relative to the cartan subalgebra h i e simple l modules having a weight space decomposition and finite dimensional weight spaces relative to h the latter inclusion is strict since there exist examples of harish chandra a2 modules in which every weight space is two dimensional in the special case of a1 modules every harish chandra a1 module is pointed our approach to this problem makes heavy use of the results from fernando s thesis 8 in particular from fernando s results we see that in place of determining all algebra homomorphisms q c l c it suffices to find only those algebra homomorphisms x which are associated with so called pointed torsion free l received by the editors may 23 1984 this paper was presented at the 1984 canadian mathematical society s summer seminar on lie algebras and related topics held at the university of windsor 1980 mathematics subject classification primary 17b10

the laplacian and dirac operators on critical planar graphs on a periodic planar graph whose edge weights satisfy a certain simple geometric condition the discrete laplacian and dirac operators have the property that their determinants and inverses only depend on the local geometry of the graph we obtain explicit expressions for the logarithms of the normalized determinants as well as the inverses of these operators we relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron in the associated dimer and spanning tree models for which the determinants of the dirac operator and the laplacian respectively play the role of the partition function this allows us to compute the entropy and correlations in terms of the local geometry in addition we define a continuous family of special discrete holomorphic functions which via convolutions gives a general process for constructing discrete holomorphic functions and discrete harmonic functions on critical planar graphs

on phase separation models asymptotics and qualitative properties in this paper we study bound state solutions of a class of two component nonlinear elliptic systems with a large parameter tending to infinity the large parameter giving strong intercomponent repulsion induces phase separation and forms segregated nodal domains divided by an interface to obtain the profile of bound state solutions near the interface we prove the uniform lipschitz continuity of bound state solutions when the spatial dimension is n 1 furthermore we show that the limiting nonlinear elliptic system that arises has unbounded solutions with symmetry and monotonicity these unbounded solutions are useful for rigorously deriving the asymptotic expansion of the minimizing energy which is consistent with the hypothesis of du and zhang discontin dynam sys 2012 when the spatial dimension is n 2 we establish the de giorgi type conjecture for the blow up nonlinear elliptic system under suitable conditions at infinity on bound state solutions these results naturally lead us to formulate de giorgi type conjectures for these types of systems in higher dimensions

discrete complex analysis on isoradial graphs abstract we study discrete complex analysis and potential theory on a large family of planar graphs the so called isoradial ones along with discrete analogues of several classical results we prove uniform convergence of discrete harmonic measures green s functions and poisson kernels to their continuous counterparts among other applications the results can be used to establish universality of the critical ising and other lattice models

irreducible representations of _ with a 1 dimensional weight space in this paper we classify all irreducible linear representations of the simple lie algebra an which admit a one dimensional weight space with respect to some cartan subalgebra h of a we first show that the problem is equivalent to determining all algebra homomorphisms from the centralizer of the cartan subalgebra h in the universal enveloping algebra of an to the base field we construct all such algebra homomorphisms and provide conditions under which two such algebra homomorphisms provide inequivalent irreducible representations of an

kac ward formula and its extension to order disorder correlators through a graph zeta function a streamlined derivation of the kac ward formula for the planar ising model s partition function is presented and applied in relating the kernel of the kac ward matrices inverse with the correlation functions of the ising model s order disorder correlation functions a shortcut for both is facilitated by the bowen lanford graph zeta function relation the kac ward relation is also extended here to produce a family of non planar interactions on mathbb z 2 for which the partition function and the order disorder correlators are solvable at special values of the coupling parameters temperature

singularly perturbed elliptic systems and multi valued harmonic functions with free boundaries here we study the asymptotic limits of solutions of some singularly perturbed elliptic systems the limiting problems involve multiple valued harmonic functions or in general harmonic maps to singular spaces and free interfaces between supports of various components of the maps the main results of the paper are the uniform lipschitz regularity of solutions as well as the regularity of free interfaces

mixed boundary value problems with a shift for a pair of metaanalytic and analytic functions in this article we reconsider the mixed boundary value problem on the unit circle for a pair of metaanalytic and analytic functions as in du and wang 2008 9 by adopting appropriate transformations we convert the problem into two independent boundary value problems for analytic functions we then obtain expressions of solution and condition of solvability for the mixed boundary value problem the forms of the solutions and the condition of solvability here are rather dissimilar to those in du and wang 2008 9 but the equivalence is established at the end of this article

z graded oscillator representations of symplectic lie algebras abstract in this paper we study two parameter z graded oscillator representations of symplectic lie algebras obtained from the natural oscillator representation by partially swapping differential operators and multiplication operators we prove that the homogeneous subspaces with nonzero degree or unequal parameters are infinite dimensional irreducible weight submodules with finite dimensional weight subspaces when the two parameters are equal the homogeneous subspace of degree zero is exactly a direct sum of two infinite dimensional irreducible weight submodules with finite dimensional weight subspaces in generic case these irreducible modules are non unitary modules without highest weight vectors our results are extensions of howe s oscillator construction of infinite dimensional multiplicity free irreducible representations for special linear lie algebras

some new monotonicity theorems with applications to free boundary problems the monotonicity theorem of alt caffarelli and friedman acf lemma 5 1 plays a central role in the existence and regularity theory of two phase free boundary problems it has been extended for example to variable coefficient operators c3 lemma 1 and to eigenvalue problems fl in this paper we reformulate the theorem so that it applies to inhomogeneous equations in which the right hand side of the equation need not vanish at the free boundary of particular interest is the case in which au takes two different constant values on the set where u 0 and on the set where u 0 the prandtl batchelor

weight modules for current algebras for any finite dimensional simple lie algebra g and commutative associative algebra s of finite type we give a complete classification of the simple weight modules of g s with bounded weight multiplicities the classification of simple weight modules is widely seen as a difficult problem in a spectacular tour de force olivier mathieu gave a complete description of such modules for finite dimensional simple lie algebras ma the crucial part of mathieu s classifi cation was understanding the simple admissible highest weight modules in the present paper we consider this question in the context of generalised current algebras l g k s where g is a finite dimensional simple lie algebra over k and s is a commutative associative and unital k algebra which is of finite type these algebras include the finite dimensional simple lie algebras g and the loop algebras g c c t t 1 of affine kac moody theory our methods work not only for the classification of simple admissible highest weight modules but also describe the simple objects in the much larger category of admissible modules weight modules whose weight multiplicities are uniformly bounded we hope that this paper will contribute to solving the harder problem of classifying all simple weight modules for these algebras the category of admissible modules includes the finite dimensional simple modules all of which are known to be evaluation modules see the introductions of l1 cfk or the more general results in nss lp l2 for details what is surprising is that this larger category of generally infinite dimensional modules can also be described in terms of evaluations we now summarize the content of this paper funding from the natural sciences and engineering research council of canada is gratefully ac knowledged