some orthogonal very well poised functions recently ismail masson and suslov established a continuous orthogonality relation and some other properties of a bessel function on a q quadratic grid askey suggested that the bessel type orthogonality found in the above paper at the level really has a general character and can be extended up to the level very well poised fuctions are known as a nonterminating version of the classical askey wilson polynomials in this paper we prove askey s conjecture and discuss some properties of the orthogonal functions another type of the orthogonality relation for a very well poised function was recently found by askey rahman and suslov

exact recovery in the ising blockmodel we consider the problem associated to recovering the block structure of an ising model given independent observations on the binary hypercube this new model called the ising blockmodel is a perturbation of the mean field approximation of the ising model known as the curie weiss model the sites are partitioned into two blocks of equal size and the interaction between those of the same block is stronger than across blocks to account for more order within each block we study probabilistic statistical and computational aspects of this model in the high dimensional case when the number of sites may be much larger than the sample size

mean field theory of random site q state potts models a class of random site mean field potts models is introduced and solved exactly the bifurcation properties of the resulting mean field equations are analysed in detail particular emphasis is put on the relation between the solutions and the underlying symmetries of the model it turns out that in contrast to the ising case the introduction of randomness in the mattis potts model can change the order of the transition for q or 6 the transition becomes second order

complete analysis of phase transitions and ensemble equivalence for the curie weiss potts model using the theory of large deviations we analyze the phase transition structure of the curie weiss potts spin model which is a mean field approximation to the nearest neighbor potts model it is equivalent to the potts model on the complete graph on n vertices the analysis is carried out both for the canonical ensemble and the microcanonical ensemble besides giving explicit formulas for the microcanonical entropy and for the equilibrium macrostates with respect to the two ensembles we analyze ensemble equivalence and nonequivalence at the level of equilibrium macrostates relating these to concavity and support properties of the microcanonical entropy the curie weiss potts model is the first statistical mechanical model for which such a detailed and rigorous analysis has been carried out

geometric reid s recipe for dimer models crepant resolutions of three dimensional toric gorenstein singularities are derived equivalent to noncommutative algebras arising from consistent dimer models by choosing a special stability parameter and hence a distinguished crepant resolution y this derived equivalence generalises the fourier mukai transform relating the g hilbert scheme and the skew group algebra mathbb c x y z g for a finite abelian subgroup of mathrm sl 3 mathbb c we show that this equivalence sends the vertex simples to pure sheaves except for the zero vertex which is mapped to the dualising complex of the compact exceptional locus this generalises results of cautis logvinenko j reine angew math 636 193 236 2009 and cautis craw logvinenko j reine angew math arxiv 1205 3110 2014 to the dimer setting though our approach is different in each case we also describe some of these pure sheaves explicitly and compute the support of the remainder providing a dimer model analogue of results from logvinenko j algebra 324 2064 2087 2010

flops of g hilb and equivalences of derived categories by variation of git quotient for a finite subgroup g in sl 3 c bridgeland king and reid proved that the moduli space of g clusters is a crepant resolution of the quotient c 3 g this paper considers the moduli spaces m theta introduced by kronheimer and further studied by sardo infirri which coincide with g hilb for a particular choice of the git parameter theta for g abelian we prove that every projective crepant resolution of c 3 g is isomorphic to m theta for some parameter theta the key step is the description of git chambers in terms of the k theory of the moduli space via the appropriate fourier mukai transform we also uncover explicit equivalences between the derived categories of moduli m theta for parameters lying in adjacent git chambers

a generalized hypergeometric function ii asymptotics and d 4 symmetry in previous work we introduced and studied a function r a a bf c v hat v that generalizes the hypergeometric function in this paper we focus on a similarity transformed function mathcal e a a gamma v hat v with parameters 4 related to the couplings c 4 by a shift depending on a a we show that the function is invariant under all maps w with w in the weyl group of type d 4 choosing a a positive and bf gamma hat v real we obtain detailed information on the re v asymptotics of the function in particular we explicitly determine the leading asymptotics in terms of plane waves and the c function that implements the similarity r

a generalized hypergeometric function satisfying four analytic difference equations of askey wilson type the hypergeometric function 2 f 1 can be written in terms of a contour integral involving gamma functions we generalize this barnes representation by using a certain generalized gamma function as a building block in this way we obtain a new 2 f 1 generalization with various symmetry features we determine the analyticity properties of the r function in all of its eight arguments and show that it is a joint eigenfunction of four distinct askey wilson type difference operators two acting on v and two on the askey wilson polynomials can be obtained by a suitable discretization of v or

artin group actions on derived categories of threefolds motivated by the enhanced gauge symmetry phenomenon of the physics literature and mirror symmetry this paper constructs an action of an artin group on the derived category of coherent sheaves of a smooth quasiprojective threefold containing a configuration of ruled surfaces described by a finite type dynkin diagram the action ex tends over deformations of the threefold via a compatible action of the corresponding re flection group on the base of its deformation space all finite type dynkin diagrams are realized

limit theorems for the bipartite potts model we consider the bipartite potts model in which interaction strengths depend only on the groups the particles belong to we first rigorously compute the exact value of the free energy then put forward two scaling limit theorems for the joint distribution of two empirical vectors which serve as order parameters it turns out that the limit distribution can be gaussian or the exponential distribution of higher order under different conditions

the weil petersson geometry of the moduli space of su n 3 calabi yau manifolds i the weil petersson metric is defined on the moduli space of calabi yau manifolds the curvature of this weil petersson metrics is computed and its potential is explicitely defined it is proved that the moduli space of calabi yau manifolds is unobstructed see tian

the tangent bundle of a calabi yau manifold deformations and restriction to rational curves the tangent bundle x of a calabi yau threefoldx is the only known example of a stable bundle with non trivial restriction to any rational curve onx by deforming the direct sum of x and the trivial line bundle one can try to obtain new examples we use algebro geometric techniques to derive results in this direction the relation to the finiteness of rational curves on calabi yau threefolds is discussed

the associated askey wilson polynomials we derive some contiguous relations for very well poised 8 7 series and use them to construct two linearly independent solutions of the three term recurrence relation of the associated askey wilson polynomials we then use these solutions to find explicit representations of two families of associated askey wilson polynomials we identify the corresponding continued fractions as quotients of two very well poised 807 series and find the weight functions

deformation theory of singular symplectic n folds by a symplectic manifold or a symplectic n fold we mean a compact kaehler manifold of even dimension n with a non degenerate holomorphic 2form i e is a nowhere vanishing n form this notion is generalized to a variety with singularities we call x a projective symplectic variety if x is a normal projective variety with rational gorenstein singularities and if the regular locus u of x admits a non degenerate holomorphic 2 form a symplectic variety will play an important role together with a singular calabi yau variety in the generalized bogomolov decomposition conjecture now that essentially a few examples of symplectic manifolds are discovered it seems an important task to seek new symplectic manifolds by deforming symplectic varieties in this paper we shall study a projective symplectic variety from a view point of deformation theory if x has a resolution x x such that x is a symplectic manifold we say that x has a symplectic resolution our first results are concerned with a birational contraction map of a symplectic manifold

a generalized hypergeometric function iii associated hilbert space transform for generic parameters a a c 0 2 4 we associate a hilbert space transform to the relativistic hypergeometric function r a a bf c v hat v studied in previous papers restricting the couplings c to a certain polytope we show that the renormalized r function kernel gives rise to an isometry from the even subspace of l 2 mathbb r hat w hat v d hat v to the even subspace of l2 w v dv where hat w hat v and w v are positive and even weight functions we prove that the orthogonal complement of the range of this isometry is spanned by n pairwise orthogonal functions the latter are in essence askey wilson polynomials arising from the r function by choosing hat v i kappa n with kappa 0 ldots kappa n 1 distinct negative numbers the two commuting analytic difference operators acting on the variable v for which r is a joint eigenfunction give rise to two commuting self adjoint hamiltonians on the even subspace of l2 w v dv we explicitly determine the relation of the time dependent scattering theory for these dynamics to their joint spectral transform

some functions that generalize the askey wilson polynomials we determine all biinfinite tridiagonal matrices for which some family of eigenfunctions are also eigenfunctions of a second order q difference operator the solution is described in terms of an arbitrary solution of a q analogue of gauss hypergeometric equation depending on five free parameters and extends the four dimensional family of solutions given by the askey wilson polynomials there is some evidence that this bispectral problem for an arbitrary order q difference operator is intimately related with some q deformation of the toda lattice hierarchy and its virasoro symmetries when tridiagonal matrices are replaced by the schroedinger operator and q 1 this statement holds with toda replaced by kdv in this context this paper determines the analogs of the bessel and airy potentials

constraining the kahler moduli in the heterotic standard model phenomenological implications of the volume of the calabi yau threefolds on the hidden and observable m theory boundaries together with slope stability of their corresponding vector bundles constrain the set of kahler moduli which give rise to realistic compactifications of the strongly coupled heterotic string when vector bundles are constructed using extensions we provide simple rules to determine lower and upper bounds to the region of the kahler moduli space where such compactifications can exist we show how small these regions can be working out in full detail the case of the recently proposed heterotic standard model more explicitly we exhibit kahler classes in these regions for which the visible vector bundle is stable on the other hand there is no polarization for which the hidden bundle is stable

scaling limits for multi species statistical mechanics mean field models we study the limiting thermodynamic behavior of the normalized sums of spins in multi species curie weiss models we find sufficient conditions for the limiting random vector to be gaussian or to have an exponential distribution of higher order and compute the covariance matrix in terms of model parameters

on 2 4 complete intersection threefolds that contain an enriques surface we study nodal complete intersection threefolds of type 2 4 in mathbb p 5 which contain an enriques surface in its fano embedding we completely determine calabi yau birational models of a generic such threefold these models have hodge numbers h 11 2 h 12 32 we also describe calabi yau varieties with hodge numbers h 11 h 12 equal to 2 26 23 5 and 31 1 the last two pairs of hodge numbers are to the best of our knowledge new

sectional curvatures of kahler moduli if x is a compact kahler manifold of dimension n we let open image in new window denote the cone of kahler classes and open image in new window the level set given by classes d with d n 1 this space is naturally a riemannian manifold and is isometric to the manifold open image in new window of kahler forms with n some fixed volume form equipped with the hodge metric as studied previously by huybrechts we study these spaces further in particular their geodesics and sectional curvatures conjecturally at least for calabi yau manifolds and probably rather more generally these sectional curvatures should be bounded between open image in new window and zero we find simple formulae for the sectional curvatures and prove both the bounds hold for various classes of varieties developing along the way a mirror to the weil petersson theory of complex moduli in the case of threefolds with h1 1 3 we produce an explicit formula for this curvature in terms of the invariants of the cubic form this enables us to check the bounds by computer for a wide range of examples finally we explore the implications of the non positivity of these curvatures

model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data we consider the problem of estimating the parameters of a gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse our approach is to solve a maximum likelihood problem with an added l1 norm penalty term the problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes we present two new algorithms for solving problems with at least a thousand nodes in the gaussian case our first algorithm uses block coordinate descent and can be interpreted as recursive l1 norm penalized regression our second algorithm based on nesterov s first order method yields a complexity estimate with a better dependence on problem size than existing interior point methods using a log determinant relaxation of the log partition function wainwright and jordan 2006 we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case we test our algorithms on synthetic data as well as on gene expression and senate voting records data