topological properties of concept spaces full version based on the observation that the category of concept spaces with the positive information topology is equivalent to the category of countably based t 0 topological spaces we investigate further connections between the learning in the limit model of inductive inference and topology in particular we show that the texts or positive presentations of concepts in inductive inference can be viewed as special cases of the admissible representations of computable analysis we also show that several structural properties of concept spaces have well known topological equivalents in addition to topological methods we use algebraic closure operators to analyze the structure of concept spaces and we show the connection between these two approaches the goal of this paper is not only to introduce new perspectives to learning theorists but also to present the field of inductive inference in a way more accessible to domain theorists and topologists

ricci flat kahler metrics on crepant resolutions of kahler cones we prove that a crepant resolution y x of a ricci flat kahler cone x admits a complete ricci flat kahler metric asymptotic to the cone metric in every kahler class in h 2 c y mathbb r a kahler cone x bar g is a metric cone over a sasaki manifold s g i e x c s s times mathbb r 0 with bar g dr 2 r 2 g and x bar g is ricci flat precisely when s g einstein of positive scalar curvature this result contains as a subset the existence of ale ricci flat kahler metrics on crepant resolutions pi y rightarrow x mathbb c n gamma with gamma subset sl n mathbb c due to p kronheimer n 2 and d joyce n 2 we then consider the case when x c s is toric it is a result of a futaki h ono and g wang that any gorenstein toric kahler cone admits a ricci flat kahler cone metric it follows that if a toric kahler cone x c s admits a crepant resolution y x then y admits a tn invariant ricci flat kahler metric asymptotic to the cone metric x bar g in every kahler class in h 2 c y mathbb r a crepant resolution in this context is a simplicial fan refining the convex polyhedral cone defining x we then list some examples which are easy to construct using toric geometry

nonexistence for complete kahler einstein metrics on some noncompact manifolds let m be a compact kahler manifold and n be a subvariety with codimension greater than or equal to 2 we show that there are no complete kahler einstein metrics on m n as an application let e be an exceptional divisor of m then m e cannot admit any complete kahler einstein metric if blow down of e is a complex variety with only canonical or terminal singularities a similar result is shown for pairs

the moduli space of asymptotically cylindrical calabi yau manifolds we prove that the deformation theory of compactifiable asymptotically cylindrical calabi yau manifolds is unobstructed this relies on a detailed study of the dolbeault hodge theory and its description in terms of the cohomology of the compactification we also show that these calabi yau metrics admit a polyhomogeneous expansion at infinity a result that we extend to asymptotically conical calabi yau metrics as well we then study the moduli space of calabi yau deformations that fix the complex structure at infinity there is a weil petersson metric on this space which we show is kahler by proving a local families l 2 index theorem we exhibit its kahler form as a multiple of the curvature of a certain determinant line bundle

set systems by reformulating a learning process of a set system l as a game between teacher and learner we define the order type of l to be the order type of the game tree if the tree is well founded the features of the order type of l diml in symbol are 1 we can represent any well quasi order wqo for short by the set system l of the upper closed sets of the wqo such that the maximal order type of the wqo is equal to diml 2 diml is an upper bound of the mind change complexity of l diml is defined iff l has a finite elasticity fe for short where according to computational learning theory if an indexed family of recursive languages has fe then it is learnable by an algorithm from positive data regarding set systems as subspaces of cantor spaces we prove that fe of set systems is preserved by any continuous function which is monotone with respect to the set inclusion by it we prove that finite elasticity is preserved by various nondeterministic language operators kleene closure shuffle closure union product intersection the monotone continuous functions represent nondeterministic computations if a monotone continuous function has a computation tree with each node followed by at most n immediate successors and the order type of a set system l is then the direct image of l is a set system of order type at most n adic diagonal ramsey number of furthermore we provide an order type preserving contravariant embedding from the category of quasi orders and finitely branching simulations between them into the complete category of subspaces of cantor spaces and monotone continuous functions having girard s linearity between them

a geometric construction of solutions to 11d supergravity necessary and sufficient conditions are provided for a class of warped product manifolds with non vanishing flux to be supersymmetric solutions of 11d supergravity many non compact but complete solutions can be obtained in this manner including the multi membrane solution initially found by duff and stelle in a different direction an explicit 5 parameter moduli space of solutions to 11d supergravity is also constructed which can be viewed as non supersymmetric deformations of the duff stelle solution

gauss kronrod integration rules for cauchy principal value integrals kronrod extensions to two classes of gauss and lobatto integration rules for the evaluation of cauchy principal value integrals are derived since in one frequently occurring case the kronrod extension involves evaluating the derivative of the integrand a new extension is introduced using n 2 points which requires only values of the integrand however this new rule does not exist for all n and when it does several significant figures are lost in its use 1 introduction in this paper we shall consider kronrod extensions ke to integration rules based on gauss and lobatto points for the evaluation of cauchy principal value cpv integrals of the form fix vyx 1 since the existence theory of ke s for regular integrals insures such extensions for only certain classes of weight functions w x we shall restrict our attention here to the most important class namely w x 1 x2y x 2 where 0 p 2 in the gauss case and p 1 in the lobatto case 9 the only other relevant weight functions are the jacobi weight functions w x 1 x a l x with a i s or s y a f and w x x2 i rx2 oo r 1 5 for our case w x 1 x2 m 1 2 the corresponding orthogonal polynomials are the gegenbauer polynomials c x usually written c x which have the following normalization 11 p 174 2 w x c x cmil x dx 8 mhnil where 3 hn i ft n 2p r p n u n r p r 2p

stieltjes polynomials and related quadrature rules in this paper we present a survey of theoretical results some of them new and numerical evidence of others concerning stieltjes polynomials kronrod schemes and their generalizations applied to integrals with a classical nonnegative weight function the use of these schemes in automatic integration routines is briefly outlined

gauss kronrod quadrature formulae for weight functions of bernstein szego type ii we study the kronrod extensions of gaussian quadrature rules whose weight functions on 1 l consist of any one of the four chebyshev weights divided by an arbitrary quadratic polynomial that remains positive on 1 11 we show that in almost all cases these extended gauss kronrod quadrature rules have all the desirable properties kronrod nodes interlacing with gauss nodes all nodes contained in 1 11 and all weights positive and representable by semiexplicit formulas exceptions to these properties occur only for small values of n the number of gauss nodes namely n 3 and are carefully identified the precise degree of exactness of each of these gauss kronrod formulae is determined and shown to grow like 4n rather than 3n as is normally the case our findings are the result of a detailed analysis of the underlying orthogonal polynomials and stieltjes polynomials the paper concludes with a study of the limit case of a linear divisor polynomial in the weight function

modified optimal quadrature extensions optimal extensions of quadrature rules are of importance in the construction of automatic integrators but many sequences fail to exist in usable form the paper considers some techniques for overcoming the problem of inextensibility with a minimal effect on integrating efficiency a modification to the extension procedure proposed recently by begumisa and robinson is shown to be just a special case of the standard theory for quadrature extension some illustrative examples are included

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

hitchin thorpe inequality for noncompact einstein 4 manifolds we prove a hitchin thorpe inequality for noncompact einstein 4 manifolds with specified asymptotic geometry at infinity the asymptotic geometry at infinity is either a cusp bundle over a compact space the fibered cusps or a fiber bundle over a cone with a compact fiber the fibered boundary many noncompact einstein manifolds come with such a geometry at infinity

asymptotically conical calabi yau manifolds i this is the first of a series of articles on complete calabi yau manifolds asymptotic to riemannian cones at infinity we begin by proving general existence and uniqueness results the uniqueness part relaxes the decay condition o r n e needed in earlier work to o r e relying on some new ideas about harmonic functions we then look at two classes of examples crepant resolutions of cones this includes a new class of ricci flat small resolutions associated with flag varieties and affine deformations of cones one focus here is the question of the precise rate of decay of the metric to its tangent cone we prove that the optimal rate for the stenzel metric on t sn is 2 n n 1

on kahler einstein metrics on certain kahler manifolds with c1 m 0 on demontre qu il existe une metrique de kahler einstein sur une hypersurface de fermat a m dimensions de degre superieur a m 1

on uniqueness of tangent cones for einstein manifolds we show that for any ricci flat manifold with euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross section similarly for any noncollapsing limit of einstein manifolds with uniformly bounded einstein constants we show that local tangent cones are unique if one tangent cone has a smooth cross section

asymptotic behaviour of stieltjes polynomials for ultraspherical weight functions abstract for the ultraspherical weight functions w x 1 x 2 1 2 an asymptotic representation of the stieltjes polynomials is proved for 1 n n the main result is applied to prove positivity results for kronrod extensions of gauss and lobatto quadrature formulae

generalized associated polynomials and functions of second kind let pv v n0 pv v v 1 be a sequence of polynomials generated by a three term recurrence relation shifting the recurrence coefficients of the elements of pv v n0 we get a sequence of so called associated polynomials which play an important role in the theory of orthogonal polynomials we generalize this concept of associating for arbitrary polynomials vn n especially if vn is expanded in terms of pv v 0 n their associated polynomials are clenshaw polynomials which are used in numerical mathematics as a consequence of it we present some results from the viewpoint of associated polynomials and from the viewpoint of clenshaw polynomials analogously as for orthogonal polynomials we define functions of second kind for vn we prove some properties of them which depend on the generalized associated polynomials and functions of second kind for orthogonal polynomials

zeros of linear combinations of laguerre polynomials from different sequences we study interlacing properties of the zeros of two types of linear combinations of laguerre polynomials with different parameters namely r n l n a al n a and s n l n a bl n 1 a proofs and numerical counterexamples are given in situations where the zeros of r n and s n respectively interlace or do not in general with the zeros of l k a l k a k n or n 1 the results we prove hold for continuous as well as integral shifts of the parameter a

on the asymptotic expansion of complete ricciflat kahler metrics on quasi projective manifolds in this work we describe the asymptotic behavior of complete metrics with prescribed ricci curvature on open kahler manifolds that can be compactified by the addition of a smooth and ample divisor first we construct a explicit sequence of kahler metrics with special approximating properties using those metrics as starting point we are able to work out the asymptotic behavior of the solutions given in ty1 in particular obtaining their full asymptotic expansion

an algebraic study of gauss kronrod quadrature formulae for jacobi weight functions we study gauss kronrod quadrature formulae for the jacobi weight function w d t 1 t a 1 t and its special case a f a of the gegenbauer weight function we are interested in delineating regions in the a f plane resp intervals in a for which the quadrature rule has a the interlacing property i e the gauss nodes and the kronrod nodes interlace b all nodes contained in 1 1 c all weights positive d only real nodes not necessarily satisfying a and or b we determine the respective regions numerically for n 1 1 20 4 40 in the gegenbauer case and for n 1 1 10 in the jacobi case where n is the number of gauss nodes algebraic criteria in particular the vanishing of appropriate resultants and discriminants are used to determine the boundaries of the regions identifying properties a and d the regions for properties b and c are found more directly a number of conjectures are suggested by the numerical results finally the gauss kronrod formula for the weight w al 2 is obtained from the one for the weight w ca and similarly the gauss kronrod formula with an odd number of gauss nodes for the weight function w t itlv 1 t2 c is derived from the gauss kronrod formula for the weight w a l y 2

zeros of orthogonal polynomials generated by canonical perturbations of measures abstract in this paper we analyze the behaviour of the zeros of polynomials orthogonal with respect to the uvarov perturbation of a positive borel measure d x when the measure is semiclassical then its electrostatic interpretation is given

positive quadrature formulas iii asymptotics of weights first we discuss briefly our former characterization theorem for positive interpolation quadrature formulas abbreviated qf provide an equivalent characterization in terms of jacobi matrices and give links and applications to other qf in particular to gauss kronrod quadratures and recent rediscoveries then for any polynomial t n which generates a positive qf a weight function depending on n is given with respect to which t n is orthogonal to p n 1 with the help of this result an asymptotic representation of the quadrature weights is derived in general the asymptotic behaviour is different from that of the gaussian weights only under additional conditions do the quadrature weights satisfy the so called circle law corresponding results are obtained for positive qf of radau and lobatto type

positivity of the weights of extended clenshaw curtis quadrature rules we prove that some extended clenshaw curtis quadrature rules have all weights positive we also present extended filippi rules of open type having all weights positive conjectures on the possibility of other positive quadrature rules embedded in clenshaw curtis or filippi rule are suggested

a c2 α estimate of the complex monge ampere equation abstract in this paper we prove a c 2 estimate for the solution to the complex monge ampere equation det u i j f with 0 f c under the assumption that u c 1 for some 1 which depends on n and

on a modified parabolic complex monge ampere equation with applications we study a modified parabolic complex monge ampere type equation on a complete non compact kahler manifold we prove a short time existence result and obtain basic estimates applying these results we prove that under certain assumptions on a given real and closed 1 1 form and initial kahler metric g0 on m the modified kahler ricci flow g ric has a long time solution converging to a complete kahler metric such that ric which extends the result in cao invent math 81 359 372 1985 to non compact manifolds we will also obtain a long time existence result for the kahler ricci flow which generalizes a result chau et al in can j math to appear

ricci flatness of asymptotically locally euclidean metrics in this article we study the metric property and the function theory of asymptotically locally euclidean ale kahler manifolds in particular we prove the ricci flatness under the assumption that the ricci curvature of such manifolds is either nonnegative or nonpositive the result provides a generalization of previous gap type theorems established by greene and wu mok siu and yau etc it can also be thought of as a general positive mass type result the method also proves the liouville properties of plurisubharmonic functions on such manifolds we also give a characterization of ricci flatness of an ale kahler manifold with nonnegative ricci curvature in terms of the structure of its cone at infinity

mind change complexity of inferring unbounded unions of restricted pattern languages from positive data this paper shows that the mind change complexity of inferring from positive data the class of unbounded unions of languages of regular patterns with constant segment length bound is of the form w w a b assuming that the patterns are defined over a finite alphabet containing at least two elements here a and b are natural numbers and we give tight bounds on their values based on the length of the constant segments and the size of the alphabet of the pattern languages this is to the authors knowledge the first time a natural class of languages has been shown to be inferable with mind change complexity above w w the proof uses the notion of closure operators on a class of languages and also uses the order type of well partial orderings to obtain a mind change bound the inference algorithm presented can be easily applied to a wide range of classes of languages finally we show an interesting connection between proof theory and mind change complexity

some properties of zeros of sobolev type orthogonal polynomials abstract for polynomials orthogonal with respect to a discrete sobolev product we prove that for each n q n has at least n m zeros on the convex hull of the support of the measure where m denotes the number of terms in the discrete part interlacing properties of zeros are also described

kahler einstein metrics on complex surfaces with c 1 0 various estimates of the lower bound of the holomorphic invariant m defined in t are given here by using branched coverings potential estimates and lelong numbers of positive d closed 1 1 currents of certain type etc these estimates are then applied to produce kahler einstein metrics on complex surfaces withc1 0 in particular we prove that there are kahler einstein structures withc1 0 on any manifold of differential type cp 2 overline ncp 2 3 leqq n leqq 8

complex monge ampere equations on quasi projective varieties we introduce generalized monge ampere capacities and use these to study complex monge ampere equations whose right hand side is smooth outside a divisor we prove in many cases that there exists a unique normalized solution which is smooth outside the divisor our results still hold if the divisor is replaced by any closed subset

topological strings and integrable hierarchies we consider the topological b model on local calabi yau geometries we show how one can solve for the amplitudes by using w algebra symmetries which encodes the symmetries of holomorphic diffeomorphisms of the calabi yau in the highly effective fermionic brane formulation this leads to a free fermion description of the amplitudes furthermore we argue that topological strings on calabi yau geometries provide a unifying picture connecting non critical super strings integrable hierarchies and various matrix models in particular we show how the ordinary matrix model the double scaling limit of matrix models and kontsevich like matrix model are all related and arise from studying branes in specific local calabi yau three folds we also show how a model topological string on p 1 and local toric threefolds and in particular the topological vertex can be realized and solved as b model topological string amplitudes on a calabi yau manifold

linear combinations of orthogonal polynomials generating positive quadrature formulas let pk x x k e n0 be the polynomials orthogonal on 1 1 with respect to the positive measure da we give sufficient conditions on the real numbers p j 0 m such that the linear combination of orthogonal polynomials yflo jpn j has n simple zeros in 1 1 1 and that the interpolatory quadrature formula whose nodes are the zeros of yfj osjpn j has positive weights

interlacing of zeros of shifted sequences of one parameter orthogonal polynomials we study the interlacing property of zeros of laguerre polynomials of adjacent degree where the free parameters differ by an integer and of the same degree where the free parameter is shifted continuously similar interlacing results are proven for the positive zeros of gegenbauer polynomials

the complex hessian equations with gradient terms on hermitian manifolds abstract in this paper we consider the complex hessian equation involving additional gradient terms on both left and right hand sides we derive the second order estimates for plurisubharmonic solutions to this equation on compact hermitian manifolds such an estimate may be helpful to study the existence and regularity for the solution of the equation

mind change efficient learning this paper studies efficient learning with respect to mind changes our starting point is the idea that a learner that is efficient with respect to mind changes minimizes mind changes not only globally in the entire learning problem but also locally in subproblems after receiving some evidence formalizing this idea leads to the notion of strong mind change optimality we characterize the structure of language classes that can be identified with at most mind changes by some learner not necessarily effective a language class l is identifiable with mind changes iff the accumulation order of l is at most accumulation order is a classic concept from point set topology we show that accumulation order is related to other established notions of structural complexity such as thickness and intrinsic complexity to aid the construction of learning algorithms we show that the characteristic property of strongly mind change optimal learners is that they output conjectures languages with maximal accumulation order we illustrate the theory by describing strongly mind change optimal learners for various problems such as identifying linear subspaces one variable patterns and fixed length patterns

a family of gauss kronrod quadrature formulae etude d une famille de formules de quadrature de gauss kronrad formules explicites des poids