holonomic functions of several complex variables and singularities of anisotropic ising n fold integrals focusing on examples associated with holonomic functions we try to bring new ideas on how to look at phase transitions for which the critical manifolds are not points but curves depending on a spectral variable or even fill higher dimensional submanifolds lattice statistical mechanics often provides a natural holonomic framework to perform singularity analysis with several complex variables that would in the most general mathematical framework be too complex or simply could not be defined in a learn by example approach considering several picard fuchs systems of two variables above calabi yau odes associated with double hypergeometric series we show that d finite holonomic functions are actually a good framework for finding properly the singular manifolds the singular manifolds are found to be genus zero curves we then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics the n fold integrals n corresponding to the n particle decomposition of the magnetic susceptibility of the anisotropic square ising model in this anisotropic case we revisit a set of so called nickelian singularities that turns out to be a two parameter family of elliptic curves we then find the first set of non nickelian singularities for 3 and 4 that also turns out to be rational or elliptic curves we underline the fact that these singular curves depend on the anisotropy of the ising model or equivalently that they depend on the spectral parameter of the model this has important consequences on the physical nature of the anisotropic n s which appear to be highly composite objects we address from a birational viewpoint the emergence of families of elliptic curves and that of calabi yau manifolds on such problems we also address the question of singularities of non holonomic functions with a discussion on the accumulation of these singular curves for the non holonomic anisotropic full susceptibility 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

blowing up kahler manifolds with constant scalar curvature ii in this paper we prove the existence of kahler metrics of constant scalar curvature on the blow up at finitely many points of a compact manifold that already carries a constant scalar curvature kahler metric in the case where the manifold has nontrivial holomorphic vector fields with zeros we give necessary conditions on the number and locations of the blow up points for the blow up to carry constant scalar curvature kahler metrics

electrical circuits on mesoscopic sierpinski gaskets a deterministic network of impedances on the sierpinski gasket fractal is considered with particular regard to low generation or mesoscopic systems we study the fractal set of the resonances of the circuit and present calculations on frequency dependent systems the condition of an intermediate size can lead to an oscillatory scaling of the total impedance that disappears in the asymptotic limit several examples are given

complex temperature phase diagrams for the q state potts model on self dual families of graphs and the nature of the q limit exact calculations of the potts model partition function z g q v have been presented for arbitrary q and temperature like variable v on self dual strip graphs g of the square lattice with fixed width l y and arbitrarily great length lx with two types of boundary conditions letting lx infinity the resultant free energy and complex temperature phase diagram have been computed including the locus b where the free energy is nonanalytic results are analyzed for widths l y 1 2 3 these results have been used to study the approach to the large q limit of b

scalar curvature and projective embeddings i we prove that a metric of constant scalar curvature on a polarised kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings satisfying a condition introduced by h luo this gives as a corollary the uniqueness of constant scalar curvature kahler metrics in a given rational cohomology class the proof uses results in the literature on the asymptotics of the bergman kernel the arguments are presented in a general framework involving moment maps for two different group actions

k stability and parabolic stability parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces in this paper we show that the three notions of parabolic polystability k polystability and existence of constant scalar curvature kahler metrics on the iterated blowup are equivalent for certain polarizations close to the boundary of the kahler cone

extremal metrics on blowups in this paper we provide conditions that are sufficient to guarantee the existence of extremal metrics on blowups at finitely many points of kahler manifolds which already carry an extremal metric as a particular case we construct extremal metrics on mathbb p 2 blown up k points in general position with k lt m 2

universality in fully developed turbulence we extend the numerical simulations of she et al phys rev lett 70 3251 1993 of highly turbulent flow with 15 taylor reynolds numbers re lambda 200 up to re lambda approx equals 45 000 employing a reduced wave vector set method introduced earlier to approximately solve the navier stokes equation first also for these extremely high reynolds numbers re lambda the energy spectra as well as the higher moments when scaled by the spectral intensity at the wave number kp of peak dissipation can be described by one universal function of k kp for all re lambda second the k space inertial subrange scaling exponents zeta m of this universal function are in agreement with the 1941 kolmogorov theory the better the larger re lambda is as is the re lambda dependence of kp only around kp viscous damping leads to a slight energy pileup in the spectra as in the experimental data bottleneck phenomenon

the exact location of partition function zeros a new method for statistical mechanics a new mathematical mechanism is proposed for locating exactly sections of the locus of the limiting distribution of partition function zeros for model systems in statistical mechanics illustrations and applications to the two dimensional ising model and the hard square lattice gas are given the exact location of the branch point singularity for the hard square lattice gas on the negative activity z axis is conjectured to be the negative root of the polynomial 2z6 9z5 42z4 90z3 96z2 27z 2

critical behaviour of the two dimensional potts model with a continuous number of states a finite size scaling analysis we investigate the critical behaviour of the two dimensional q state potts model using finite size scaling and transfer matrix methods for the continuous transition range 0 these results for continuous q were obtained from a transfer matrix constructed for a generalized whitney polynomial representing the potts models an effective algorithm to compute the dominant eigenvalues of this essentially nonsymmetric transfer matrix is developed

the hard hexagon partition function for complex fugacity we study the analyticity of the partition function of the hard hexagon model in the complex fugacity plane by computing zeros and transfer matrix eigenvalues for large finite size systems we find that the partition function per site computed by baxter in the thermodynamic limit for positive real values of the fugacity is not sufficient to describe the analyticity in the full complex fugacity plane we also obtain a new algebraic equation for the low density partition function per site

finite lattice extrapolations for z3 and zs models a method is presented to accelerate the convergence of finite lattice sequences to their bulk limit the calculation of highly accurate estimates of the critical parameters of the bulk system is then possible applied to the hamiltonian version of the z3 model three state potts model in 1 1 dimensions these techniques yield estimates for the exponents gamma 1 444 or 0 0001 nu 0 8333 or 0 0003 and alpha 0 33 or 0 01 for the z5 model the presence of a kosterlitz thouless transition is confirmed

extended self similarity and dissipation range dynamics of three dimensional turbulence we carry out a self consistent calculation of the structure functions in the dissipation range using the navier stokes equation combining these results with the known structures in the inertial range we actually propose crossover functions for the structure functions that take one smoothly from the inertial to the dissipation regime these crossover functions are shown to exhibit extended self similarity properties consistent with experimental findings

moduli spaces of critical riemannian metrics in dimension four abstract we obtain a compactness result for various classes of riemannian metrics in dimension four in particular our method applies to anti self dual metrics kahler metrics with constant scalar curvature and metrics with harmonic curvature with certain geometric non collapsing assumptions the moduli space can be compactified by adding metrics with orbifold like singularities similar results were obtained for einstein metrics in j amer math soc 2 3 1989 455 invent math 97 2 1989 313 invent math 101 1 1990 101 but our analysis differs substantially from the einstein case in that we do not assume any pointwise ricci curvature bound

asymptotics of the susceptibility for the ising model on the hierarchical lattices using the gks inequality griffiths and kaufman showed in 12 that the susceptibility in the ising model on the hierarchical lattices is infinite fort tc in the present work we give an analytic calculation of the dominant singularity of the susceptibility whenh 0 fort tc moreover we analytically find the spontaneous magnetization fort tc and prove that fort tc the susceptibility is finite

the algebraic construction of partition function zeros universality and algebraic cycles the author considers the one parameter lattice models of classical statistical mechanics from a simple algebraic viewpoint the way in which the limiting locus of partition function zeros emerges through a sequence of semi infinite m infinity lattice sections is considered convergence to the limiting locus cinfinity is obtained through two sequences of algebraic curves for all arbitrarily large but finite m the partition function per site is a branch of an algebraic function lambda 1 defined by an irreducible polynomial and possesses only a finite number of algebraic singular points in the limit of m to infinity infinitely many branch points accumulate on cinfinity an algebraic basis to the universality hypothesis is presented in terms of the accumulation of branch points at the critical point it is argued that for many two dimensional lattice models the critical exponents alpha and nu will be of the form alpha 2 s and nu s 1 s where s is an integer or 3

numerical experiments on yang lee zeros it is argued that since any real space renormalisation r is a transformation under which the partition function is invariant those r which are based upon various real space approximation schemes can in principle be extended into the complex field of the parameter space in this extension the julia set of r now extended to include r which are algebraic is an approximation to the locus of the yang lee zeros finite size scaling using onsager s solution of the two dimensional ising model is used to construct r which are highly accurate in real temperatures in the critical region these are extended into the complex temperature plane and results are displayed for both the anisotropic and isotropic cases

examples of phase equilibrium in lattice models the general z4 and z5 models on the simple quadratic lattice and the pair triplet ising model on the triangular lattice are employed to illustrate an extension of the finite size scaling technique which obtains the qualitative structure of the phase equilibrium surface in terms of multiple phase coexistence

location of zeros in the complex temperature plane absence of lee yang theorem in the yang lee theory 1952 the occurrence of a phase transition at real physical values of the magnetic field is related to the behaviour of the density of zeros of the partition function in the complex magnetic field plane as shown by lee and yang these zeros fall on lines for a wide class of ferromagnetic models in extensions of the yang lee theory to the complex temperature plane it has often been assumed that the zeros of the partition function would again fall on lines rather than in areas though this happens to be true for the isotropic ising model the authors point out that there is in general no basis for this assumption they discuss the location of the zeros of the partition function of the anisotropic ising model in the complex temperature plane and show that they indeed always fall in areas

anisotropy and scaling corrections in turbulence we analyze second order turbulent velocity moments both in r and in p space finite size corrections induce dramatic differences between local r and p space scaling exponents as analytically accessible examples we focus on two popular parametrizations the batchelor parametrization for the r space structure function and a common parametrization for the energy spectrum e p is proportional to p 5 3exp p pd the spectral bottleneck energy pileup hidden in the batchelor parametrization results in an extended r space scaling range comparable to experimental ones for the same taylor reynolds number re lambda shear effects are discussed in terms of global apparent scaling correction delta zeta app re lambda to classical scaling which again depend on whether looked at in r or in p space the differences can be traced back to the subtleties of the crossovers in the velocity moments our observations emphasize the need for more experimental information on crossovers between different subranges

twice punctured hyperbolic sphere with a conical singularity and generalized elliptic integral we describe in terms of generalized elliptic integrals the hyperbolic metric of the twice punctured sphere with one conical singularity of prescribed order we also give several monotonicity properties of the metric and a couple of applications

renormalisation groups with periodic and aperiodic orbits motivated by recent results for some frustrated hierarchical models the authors analyse the consequences of renormalisation group transformations with periodic and aperiodic orbits they find that the free energy has in general an infinite number of singularities this occurs already when the transformation has a stable periodic orbit they present an idealised example and a new class of hierarchical potts models to which the theory applies

metrics with conical singularities on the sphere and sharp extensions of the theorems of landau and schottky an explicit formula for the generalized hyperbolic metric on the thrice punctured sphere mathbb p backslash z 1 z 2 z 3 with singularities of order j 1 at zj is obtained in all possible cases 1 2 3 2 the existence and uniqueness of such a metric was proved long time ago by picard j reine angew math 130 243 258 1905 and heins nagoya math j 21 1 60 1962 while explicit formulas for the cases 1 2 1 were given earlier by agard ann acad sci fenn ser a i 413 1968 and recently by anderson et al math z to appear doi 10 1007 s00209 009 0560 5 we also establish precise and explicit lower bounds for the generalized hyperbolic metric this extends work of hempel j lond math soc ii ser 20 435 445 1979 and minda complex var 8 129 144 1987 as applications sharp versions of landau and schottky type theorems for meromorphic functions are obtained

blowing up and desingularizing constant scalar curvature kahler manifolds this paper is concerned with the existence of constant scalar curvature kahler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature kahler metrics we also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature kahler metrics

complex temperature singularities in the d 2 ising model triangular and honeycomb lattices we study complex temperature singularities of the ising model on the triangular and honeycomb lattices we first discuss the complex t phases and their boundaries from exact results we determine the complex t singularities in the specific heat and magnetization for the triangular lattice we discuss the implications of the divergence of the magnetization at the point u d 1 where u d z 2 d e 4k and extend a previous study by guttmann of the susceptibility at this point with the use of differential approximants for the honeycomb lattice from an analysis of low temperature series expansions we have found evidence that the uniform and staggered susceptibilities n and n a both have divergent singularities at z d 1 z and our numerical values for the exponents are consistent with the hypothesis that the exact values are 0 d 0 a d 5 2 the critical amplitudes at this singularity were calculated using our exact results for 0 and together with numerical values for 0 from series analyses we find that the exponent relation 0 c 2 c 0 d 2 is violated at z d 1 on the honeycomb lattice the right hand side is consistent with being equal to 4 rather than 2 the connections of the critical exponents at these two singularities on the triangular and honeycomb lattice are discussed

physical meaning for mandelbrot and julia sets abstract we consider the dynamics of a kicked charged particle moving in a double well potential and a time dependent magnetic field in certain cases the stroboscopic dynamics reduces to the complex logistic map thus providing physical meaning for the mandelbrot set in other cases we obtain iterated function systems consisting of the inverse complex logistic map thus providing physical meaning for julia sets our approach can be generalized to complex mappings with a maximum of order q

renormalisation group study of a disordered model we study the potts model on a diamond hierarchical lattice with random interactions using weak disorder expansions we calculate analytically the position and the exponents of the random fixed point which appears when the specific heat exponent ap of the pure system becomes positive at ap 0 we find how the logarithmic singularity is modified by the disorder lastly we suggest that this model should present griffiths like singularities

counter examples to the generalized positive action conjecture we give examples of complete locally asymptotically flat riemannian 4 manifolds with zero scalar curvature and negative mass the generalized positive action conjecture of hawking and pope 5 is therefore false

extremal kahler metrics on projective bundles over a curve let m p e be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle e over a compact complex curve of genus 2 building on ideas of fujiki 1992 27 we prove that m admits a kahler metric of constant scalar curvature if and only if e is polystable we also address the more general existence problem of extremal kahler metrics on such bundles and prove that the splitting of e as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal kahler metrics in kahler classes sufficiently far from the boundary of the kahler cone the methods used to prove the above results apply to a wider class of manifolds called rigid toric bundles over a semisimple base which are fibrations associated to a principal torus bundle over a product of constant scalar curvature kahler manifolds with fibres isomorphic to a given toric kahler variety we discuss various ramifications of our approach to this class of manifolds

holomorphic vector bundles on sasakian threefolds and parabolic vector bundles a bijective correspondence is constructed between the parabolic vector bundles with rational parabolic weights on a riemann surface x and the holomorphic vector bundles on quasiregular sasakian threefolds over x this bijective correspondence preserves the stable objects

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

on the density of partition function temperature zeros the density of partition function temperature zeros for the two dimensional spatially anisotropic ising model in the absence of a magnetic field satisfies a linear homogeneous partial differential equation a generalised form of this equation for a non zero specific heat exponent can be derived by applying non linear scaling to the expression for the specific heat as an integral over the density of zeros

comment on the exact location of partition function zeros a new method for statistical mechanics in a recent paper wood 1985 conjectured that the exact location of the branch point singularity for the hard square lattice gas on the negative activity z axis is at z 0 119392 a careful analysis of the available series data implies that this conjecture is almost certainly false the author finds the correct value to be z 0 1193388809 or 0 000000001 with the corresponding exponent indistinguishably close to 5 6 corresponding to a zero of the singular part of the function

critical exponents of two dimensional potts and bond percolation models critical exponents of the two dimensional q state potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q results for the temperature exponent agree accurately with the conjecture of den nijs 1979 the magnetic exponent is found to behave in accordance with the conjecture of nienhuis et al 1980

analytic properties of the partition function for statistical mechanical models the author explains with examples the relationship between zeros of the partition function analyticity and degeneracy of absolute magnitude of eigenvalues of the transfer matrix for statistical mechanical models it is shown how to write down a polynomial representation of the partition function for any model with a two sheet largest eigenvalue the staggered ice model representation of the q state potts model on a sequence of semi infinite strips is solved and it is shown that for 0 q 4 the potts model result is only projected out by special boundary conditions the non potts part of the result is obtained for a 4 infinity strip and gives baxter s antiferromagnetic critical curve 1982

critical metrics on connected sums of einstein four manifolds abstract we develop a gluing procedure designed to obtain canonical metrics on connected sums of einstein four manifolds the main application is an existence result using two well known einstein manifolds as building blocks the fubini study metric on cp 2 and the product metric on s 2 s 2 using these metrics in various gluing configurations toric invariant critical metrics are found on connected sums for a specific riemannian functional which depends on the global geometry of the factors furthermore using certain quotients of s 2 s 2 as one of the gluing factors critical metrics on several non simply connected manifolds are also obtained

on the scaling of three dimensional homogeneous and isotropic turbulence abstract in this paper we investigate the scaling properties of three dimensional isotropic and homogeneous turbulence we analyze a new form of scaling extended self similarity recently introduced in the literature we found that anomalous scaling of the velocity structure functions is clearly detectable even at a moderate and low reynolds number and it extends over a much wider range of scales with respect to the inertial range

prescribing curvature on open surfaces in this article we study the problem sometimes called the berger nirenberg problem of prescribing the curvature on a riemann surface that is on an oriented surface equipped with a conformal class of riemannian metrics

phase equilibria and the scaling transformation it is argued that the scaling transformation which is currently being used to obtain numerical estimates of critical exponents cannot safely be employed without prior knowledge that the points in question are second order transition points the transformation when defined on finite or semi infinite systems will not distinguish between order disorder critical points and any other point on the coexistence surface this in turn means that the full phase diagram of the equilibrium surface can be found using the scaling transformation the three state potts model is used to illustrate this

on the isolated singularities of the solutions of the gaussian curvature equation for nonnegative curvature abstract the precise asymptotic behaviour of the solutions to the two dimensional curvature equation u k z e 2 u with e 2 u l 1 for bounded nonnegative curvature functions k z near isolated singularities is obtained

julia sets and complex singularities in hierarchical ising models we study the analytical continuation in the complex plane of free energy of the ising model on diamond like hierarchical lattices it is known 12 13 that the singularities of free energy of this model lie on the julia set of some rational endomorphismf related to the action of the migdal kadanoff renorm group we study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin off we prove that for almost all with respect to the harmonic measure geodesics the complex critical exponent is common and compute it

partition function zeros for the two dimensional ising model iii abstract the density of the complex temperature zeros of the partition function of the two dimensional ising model on completely anisotropic triangular lattices is investigated near real and complex critical points the non uniform behaviour of the density of zeros at interior and boundary critical points is studied analytically and numerically for a lattice with interactions in the ratios 3 2 1 the limiting behaviour of the density at complex critical points depends on the direction of approach in the complex plane in the neighbourhood of interior critical points one finds only a single layer of zeros but generally there are two layers or superposed sets of zeros with different distributions and different limiting densities at boundary critical points on anisotropic quadratic and on partially anisotropic triangular lattices the two layers become identical by symmetry of the partition function the divergence of the two dimensional density of zeros at real critical points is discussed briefly in relation to scaling theory

integrability versus non integrability hard hexagons and hard squares compared in this paper we compare the integrable hard hexagon model with the nonintegrable hard squares model by means of partition function roots and transfer matrix eigenvalues we consider partition functions for toroidal cylindrical and free free boundary conditions up to sizes 40 40 and transfer matrices up to 30 sites for all boundary conditions the hard squares roots are seen to lie in a bounded area of the complex fugacity plane along with the universal hard core line segment on the negative real fugacity axis the density of roots on this line segment matches the derivative of the phase difference between the eigenvalues of largest and equal moduli and exhibits much greater structure than the corresponding density of hard hexagons we also study the special point z 1 of hard squares where all eigenvalues have unit modulus and we give several conjectures for the value at z 1 of the partition functions

algebraic approximations to the locus of partition function zeros applications of the mathematical formalism put forward in the previous paper see ibid p 3471 1987 are made to a number of well known lattice models in statistical mechanics sequences of algebraic curves are obtained from the branches of an algebraic function lambda 1 where lambda 1 at real temperatures is the branch of lambda 1 which is the partition function per site of an m infinity lattice section these sequences are viewed as approximations to parts of the limiting locus cinfinity of partition function zeros approximations to critical points can be obtained in a natural extension of these curves beyond the branch points of lambda 1

zeros of the partition function for the two dimensional ising model the zeros of the partition function for the zero field two dimensional ising model in the complex temperature plane are represented in terms of the eigenvalues of the transfer matrix the boundaries of the limiting distribution have a particular simple representation in terms of the correlation range along the two principal axes of the quadratic lattice examples of the distribution in the complex plane of z exp 2k are given for the anisotropic model when the horizontal and vertical interactions are in integer ratio

blowing up extremal kahler manifolds ii this is a continuation of the work of arezzo pacard singer and the author on blowups of extremal kahler manifolds we prove the conjecture stated in szekelyhidi duke math j 161 8 1411 1453 2012 and we relate this result to the k stability of blown up manifolds as an application we prove that if a kahler manifold m of dimension 2 admits a constant scalar curvature csck metric then the blowup of m at a point admits a csck metric if and only if it is k stable as long as the exceptional divisor is sufficiently small

moduli spaces of weighted pointed stable curves abstract a weighted pointed curve consists of a nodal curve and a sequence of marked smooth points each assigned a number between zero and one a subset of the marked points may coincide if the sum of the corresponding weights is no greater than one we construct moduli spaces for these objects using methods of the log minimal model program and describe the induced birational morphisms between moduli spaces as the weights are varied in the genus zero case we explain the connection to geometric invariant theory quotients of points in the projective line and to compactifications of moduli spaces studied by kapranov keel and losev manin

the construction of ale spaces as hyper kahler quotients on decrit la construction d une famille particuliere de 4 varietes hyper kahler les espaces asymptotiquement localement euclidiens ale on decrit une 4 variete de riemann avec juste une extremite qui a l infini ressemble au quotient r 4 de l espace euclidien r 4 par un groupe fini d identifications

chromatic polynomials of large triangular lattices evaluating the q colourings of a lattice is equivalent to solving the q state zero temperature antiferromagnetic potts model this has recently been done exactly for an infinite triangular lattice with q real here the results are extended to the full complex q plane giving the limiting distribution of the zeros of the chromatic polynomial the results are compared with finite lattice calculations and the occurrence of isolated real zeros converging on the beraha numbers is noted

q state potts models in hamiltonian field theory for q 4 in 1 1 dimensions finite lattice sequence extrapolation methods are applied to the q state potts models in 1 1 d hamiltonian field theory thermal exponents nu 0 71 or 0 02 alpha 0 53 or 0 02 are obtained for the four state model methods are discussed for treating the first order transitions for q 4 and a sequence extrapolation is used to estimate the latent heat exact analytic formulae are derived for the ground state energy the mass gap and the latent heat at the transition point these are compared with the numerical results

rigidity and stability of einstein metrics for quadratic curvature functionals we investigate rigidity and stability properties of critical points of quadratic curvature functionals on the space of riemannian metrics we show it is possible to gauge the euler lagrange equations in a self adjoint fashion to become elliptic fredholm theory may then be used to describe local properties of the moduli space of critical metrics we show a number of compact examples are infinitesimally rigid and consequently are isolated critical points in the space of unit volume riemannian metrics we then give examples of critical metrics which are strict local minimizers up to diffeomorphism and scaling a corollary is a local reverse bishop s inequality for such metrics in particular any metric g in a c neighborhood of the round metric s gs satisfying ric g ric gs has volume v ol g v ol gs with equality holding if and only if g is isometric to gs

variation of cone metrics on riemann surfaces we discuss the variational properties of the unique conical metric of constant curvature 1 associated to a compact riemann surface together with a weighted divisor 2005 elsevier inc all rights reserved

smoothing singular constant scalar curvature kahler surfaces and minimal lagrangians abstract given a complex surface x with singularities of class t and no nontrivial holomorphic vector field endowed with a kahler class 0 we consider smoothings m t t where t is a kahler class on m t degenerating to 0 under an hypothesis of nondegeneracy of the smoothing at each singular point we prove that if x admits a constant scalar curvature kahler metric in 0 then m t admits a constant scalar curvature kahler metric in t for small t in addition we construct small lagrangian stationary spheres which represent lagrangian vanishing cycles when t is small

conformal duality and compact complex surfaces on etudie les structures demi conformement plates sur une variete de riemann a 4 dimensions m g avec l hypothese supplementaire que m est complexe

stability of vector bundles and extremal metrics on donne des exemples de surfaces reglees m qui n ont pas de champs holomorphes vectoriels non triviaux et qui n admettent pas de metrique de kahler extremale dans une classe de kahler specifique donnee

extended universality in moderate reynolds number flows is the kolmogorov viscous scale this form seems to be obeyed on a range of scales that is larger than power law scaling it is shown here that this extended universality stems from the structure of the navier stokes equations and from the property of the locality of interactions the approach discussed here allows us to estimate the range of validity of the universal form in addition we examine the possi bility that the observed deviations from the classical values of

re examination of yang lee zeros of the anisotropic ising models on square triangular and honeycomb lattices we have determined the zeros of the partition functions of the anisotropic ising models on square triangular and honeycomb lattices in the absence of a magnetic field with arbitrary combinations of interactions it is found that the zeros are generally distributed over areas however the zeros near the positive real axis are distributed on the unit circle in a complex plane with the zero density g which leads to logarithmic singularity of the free energy our results generalize fisher s results for isotropic cases

isolated singularities of affine special kahler metrics in two dimensions we prove that there are just two types of isolated singularities of special kahler metrics in real dimension two provided the associated holomorphic cubic form does not have essential singularities we also construct examples of such metrics

multiple phase coexistence and the scaling transformation it is shown that a simple extension of the finite size scaling method in the theory of critical phenomena can yield a sequence of approximants to any point subset of the interaction parameter space where multiple phase coexistence is possible the method is illustrated by an application to the three four and five state potts models and even in its lowest order of approximation is able to distinguish the difference in the zero field critical behaviour between the five state and the three and four state models

on the boundaries and density of partition function temperature zeros for the two dimensional ising model explicit expressions which determine the boundaries of the regions containing temperature zeros of the partition function of the two dimensional ising model on an anisotropic triangular lattice are obtained along with a general formula for the density of zeros valid everywhere in regions containing zeros

calculating dynamic structure factors with the real space renormalisation group the authors develop exact methods based on the real space renormalisation group for calculating dynamic structure factors of heisenberg ferromagnets the method is applied to a pure chain a sierpinski gasket and a quasi one dimensional model of a dilute chain the result for the sierpinski gasket exhibits a rich structure because of the requirement of discrete scale invariance by the use of the distribution scaling the method is generalised to treat a randomly diluted chain

convergence of finite size scaling renormalisation techniques abstract the rate of convergence of the transfer matrix finite size scaling or phenomenological renormalisation method is studied it is shown both heuristically and numerically that the convergence of estimates for exponents etc is governed asymptotically by the leading irrelevant variable scaling exponent the more rapid apparent convergence rates observed in many practical calculations for two dimensional lattices of widths up to ten lattice spacings are attributed to cancellation between various correction terms

boundary conditions and the free energy for statistical mechanical models the author shows that the choice of boundary conditions can significantly affect the calculation of the free energy for discrete statistical mechanical models on finite or semi infinite lattices when the exponentiated coupling is extended into the complex plane he shows however that the boundary conditions most consistent with boundary configuration independence in the direction in which a lattice is infinite give a unique analytic function for the free energy this unique function is the free energy most appropriate for extrapolation to the full thermodynamic limit the author shows that such boundary conditions are orthogonal to all but one degenerate ferromagnetic ground states at zero temperature

einstein metrics and complex singularities this paper is concerned with the construction of special metrics on non compact 4 manifolds which arise as resolutions of complex orbifold singularities our study is close in spirit to the construction of the hyperkahler gravitational instantons but we focus on a different class of singularities we show that any resolution x of an isolated cyclic quotient singularity admits a complete scalar flat kahler metric which is hyperkahler if and only if k x is trivial and that if k x is strictly nef then x also admits a complete non kahler self dual einstein metric of negative scalar curvature in particular complete self dual einstein metrics are constructed on simply connected non compact 4 manifolds with arbitrary second betti number

a new real space renormalisation method and its julia set the authors introduce a new real space renormalisation method which is valid not only on the real axis but also everywhere in the complex plane of temperatures the method requires the knowledge of the partition function on small lattices and can in principle be improved systematically they test this method on the two dimensional ising model they find that it gives a satisfactory picture for the set of zeros of the partition function when restricted to the real axis the estimates for the renormalisation transformation the critical point and the thermal exponent are also very good

special kahler manifolds we give an intrinsic definition of the special geometry which arises in global n 2 supersymmetry in four dimensions the base of an algebraic integrable system exhibits this geometry and with an integrality hypothesis any special kahler manifold is so related to an integrable system the cotangent bundle of a special kahler manifold carries a hyperkahler metric we also define special geometry in supergravity in terms of the special geometry in global supersymmetry

prescribing curvature on compact surfaces with conical singularities we study the berger nirenberg problem on surfaces with conical singularities i e we discuss conditions under which a function on a riemann surface is the gaussian curvature of some conformal metric with a prescribed set of singularities of conical types

non minimal scalar flat kahler surfaces and parabolic stability a new construction is presented of scalar flat kahler metrics on non minimal ruled surfaces the method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank 2 parabolically stable holomorphic bundles this rather general construction is shown also to give new examples of low genus in particular it is shown that mathbb cp 2 blown up at 10 suitably chosen points admits a scalar flat kahler metric this answers a question raised by claude lebrun in 1986 in connection with the classification of compact self dual 4 manifolds

exact finite lattice results for zn symmetric spin models the author obtains the exact finite lattice partition functions for a sequence of two dimensional zn symmetric spin models interpolating between the two phase ferromagnetic potts model and the three phase n 4 vector model the author presents these results through the distribution of zeros of the partition function in the complex coupling variable the author finds that the phase structure for both two and three phase models is manifested in this presentation in the potts model case the zeros approaching the ferromagnetic phase transition point are restricted to a circular locus although well away from the ferromagnetic region this restriction does not apply as the interpolation proceeds the simple ferromagnetic locus is modified revealing a three phase structure

twist geometry of the c map we discuss the geometry of the c map from projective special kahler to quaternionic kahler manifolds using the twist construction to provide a global approach to hitchin s description as found by alexandrov et al and alekseevsky et al this is related to the quaternionic flip of haydys we prove uniqueness statements for several steps of the construction in particular we show that given a hyperkahler manifold with a rotating symmetry there is essentially only a one parameter degree of freedom in constructing a quaternionic kahler manifold of the same dimension we demonstrate how examples on group manifolds arise from this picture