log canonical thresholds of certain fano hypersurfaces we study log canonical thresholds on quartic threefolds quintic fourfolds and double spaces as an important application we show that they have kahler einstein metrics if they are general

quantum 2 spheres and big q jacobi polynomials orthogonal bases for the algebras of functions of podles quantum 2 spheres are explicitly determined in terms of bigq jacobi polynomials this gives a group theoretic interpretation of the symmetric bigq jacobi polynomials and the symmetricq hahn polynomials

a basic analog of a theorem of polya we derive a q analog of a theorem of george polya 1918 0 q 1 concer ning the zeros of basic cosine and sine transforms the results are established without further restrictions on q

identities of the rogers ramanujan bailey type abstract a multiparameter generalization of the bailey pair is defined in such a way as to include as special cases all bailey pairs considered by w n bailey in his paper identities of the rogers ramanujan type proc london math soc 2 50 1949 421 435 this leads to the derivation of a number of elegant new rogers ramanujan type identities

general logical metatheorems for functional analysis in this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in nonlinear functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters this means that the bounds are uniform for all parameters meeting these weak local boundedness conditions the results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space resp a convex subset of a normed space was assumed our results treat general classes of spaces such as metric hyperbolic cat 0 normed uniformly convex and inner product spaces and classes of functions such as nonexpansive holder lipschitz uniformly continuous bounded and weakly quasi nonexpansive ones we give several applications in the area of metric fixed point theory in particular we show that the uniformities observed in a number of recently found effective bounds by proof theoretic analysis can be seen as instances of our general logical results

fano varieties with many selfmaps abstract we study global log canonical thresholds of anticanonically embedded quasismooth weighted fano threefold hypersurfaces having terminal quotient singularities to prove the existence of a kahler einstein metric on most of them and to produce examples of fano varieties with infinite discrete groups of birational automorphisms

on quantitative versions of theorems due to f e browder and r wittmann abstract this paper is another case study in the program of logically analyzing proofs to extract new typically effective information proof mining we extract explicit uniform rates of metastability in the sense of t tao from two ineffective proofs of a classical theorem of f e browder on the convergence of approximants to fixed points of nonexpansive mappings as well as from a proof of a theorem of r wittmann which can be viewed as a nonlinear extension of the mean ergodic theorem the first rate is extracted from browder s original proof that is based on an application of weak sequential compactness in addition to a projection argument wittmann s proof follows a similar line of reasoning and we adapt our analysis of browder s proof to get a quantitative version of wittmann s theorem as well in both cases one also obtains totally elementary proofs even for the strengthened quantitative forms of these theorems that neither use weak compactness nor the existence of projections anymore in this way the present article also discusses general features of extracting effective information from proofs based on weak compactness we then extract another rate of metastability of similar nature from an alternative proof of browder s theorem essentially due to halpern that already avoids any use of weak compactness the paper is concluded by general remarks concerning the logical analysis of proofs based on weak compactness as well as a quantitative form of the so called demiclosedness principle in a subsequent paper these results will be utilized in a quantitative analysis of baillon s nonlinear ergodic theorem

the zeros of basic bessel functions the functions jv ax x and associated orthogonal polynomials abstract the modified lommel polynomials are generalized and their orthogonality relation is obtained as a byproduct we prove that the non zero roots of jv ax x are real simple and interlace with those of jv ax 1 x the q lommel polynomials are shown to play in the theory of basic bessel function the role played by the lommel polynomials in the bessel function theory the q lommel polynomials are proved to be orthogonal with respect to a purely discrete measure with bounded support this is then used to prove that the prositive zeros of x j 2 v x q are real simple and interlace with the zeros of x 1j 2 v 1 x q when 1 we also establish the complete monotonicity of 1x 1 2 j v 1 2 i x q j v 2 i x q

some logical metatheorems with applications in functional analysis in previous papers we have developed proof theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis here uniform means independence from parameters in compact spaces a recent case study in fixed point theory systematically yielded uniformity even w r t parameters in metrically bounded but noncompact subsets which had been known before only in special cases in the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all our theorems guarantee under general logical conditions such strong uniform versions of non uniform existence statements moreover they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result our metatheorems deal with general classes of spaces like metric spaces hyperbolic spaces cat 0 spaces normed linear spaces uniformly convex spaces as well as inner product spaces

on a generalization of the rogers generating function abstract we derive a generalization of the rogers generating function for the continuous q ultraspherical rogers polynomials whose coefficient is a 1 2 from that expansion we derive corresponding specialization and limit transition expansions for the continuous q hermite continuous q legendre laguerre and chebyshev polynomials of the first kind using a generalized expansion of the rogers generating function in terms of the askey wilson polynomials by ismail simeonov whose coefficient is a 7 8 we derive corresponding generalized expansions for the wilson continuous q jacobi and jacobi polynomials by comparing the coefficients of the askey wilson expansion to our continuous q ultraspherical rogers expansion we derive a new quadratic transformation for basic hypergeometric functions which relates an 7 8 to a 1 2 we also obtain several definite integral representations which correspond to the above mentioned expansions through the use of orthogonality

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

addition formulas for q bessel functions the two dimensional euclidean algebra provides an algebraic setting for q bessel functions using a two variable realization of this algebra q analogs of the lommel and graf addition formulas for the q bessel functions are obtained

on a conjecture of tian we study tian s alpha invariant in comparison with the alpha 1 invariant for pairs s d h consisting of a smooth surface s d of degree d in the projective three dimensional space and a hyperplane section h a conjecture of tian asserts that alpha s d h alpha 1 s d h we show that this is indeed true for d 4 the result is well known for d leqslant 3 and we show that alpha s d h alpha 1 s d h for d geqslant 8 provided that s d is general enough we also construct examples of s d for d 6 and d 7 for which tian s conjecture fails we provide a candidate counterexample for s 5

positivity preserving transformations for q binomial coefficients several new transformations for q binomial coefficients are found which have the special feature that the kernel is a polynomial with nonnegative coefficients by studying the group like properties of these positivity preserving transformations as well as their connection with the bailey lemma many new summation and transformation formulas for basic hypergeometric series are found the new q binomial transformations are also applied to obtain multisum rogers ramanujan identities to find new representations for the rogers szego polynomials and to make some progress on bressoud s generalized borwein conjecture for the original borwein conjecture we formulate a refinement based on new triple sum representations of the borwein polynomials

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

an integral representation and some transformation properties of q bessel functions a poisson type integral representation for jackson s q bessel function is obtained by using askey and wilson s q beta integral and nassrallah and rahman s integral formula for an 8 7 series this representation along with some transformtion formulas for basic hypergeometric series help express the q bessel functions as a 3 2 series in base q a 2 2 series in base q2 and a 2 1 series in base qq12 where 0 q 1 as an application q analogues are found for gegenbauer s degenerate addition formulas for bessel functions

quadratic transformation formulas for basic hypergeometric series starting with some of the known transformation formulas for well poised 2 1 and very well poised 8 7 basic hypergeometric series we obtain q analogues of 36 quadratic transformation formulas given in 2 11 of higher transcendental functions vol 1 edited by erdelyi et al we also derive some new quadratic transformation formulas that give rise to identities connecting very well poised but unbalanced 10 9 series in base q with very well poised and balanced 12 11 series in base q 2 a rogers ramanujan type identity is also found as a limiting case

norm convergence of nilpotent ergodic averages we show that multiple polynomial ergodic averages arising from nilpotent groups of measure preserving transformations of a probability space always converge in the l 2 norm

critical exponents from nonlinear functional equations for partially directed cluster models we present a method for the derivation of the generating function and computation of critical exponents for several cluster models staircase bar graph and directed column convex polygons as well as partially directed self avoiding walks starting with nonlinear functional equations for the generating function by linearizing these equations we first give a derivation of the generating functions the nonlinear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz alternatively taking the continuum limit leads to nonlinear differential equations from which one can extract the scaling function we find that all the above models are in the same universality class with exponents y 1 2 7 1 3 and b 2 3 all models have as their scaling function the logarithmic derivative of the airy function

automatic generation of hypergeometric identities by the beta integral method in this article hypergeometric identities or transformations for p 1fp series and for kampe de feriet series of unit arguments are derived systematically from known transformations of hypergeometric series and products of hypergeometric series respectively using the beta integral method in an automated manner based on the mathematica package hyp as a result we obtain some known and some identities which seem to not have been recorded before in literature

a quantitative nonlinear strong ergodic theorem for hilbert spaces abstract we give a quantitative version of a strong nonlinear ergodic theorem for a class of possibly even discontinuous selfmappings of an arbitrary subset of a hilbert space due to r wittmann and outline how the existence of uniform bounds in such quantitative formulations of ergodic theorems can be proved by means of a general logical metatheorem in particular these bounds depend neither on the operator nor on the initial point furthermore we extract such uniform bounds in our quantitative formulation of wittmann s theorem implicitly using the proof theoretic techniques on which the metatheorem is based however we present our result and its proof in analytic terms without any reference to logic as such our bounds turn out to involve nested iterations of relatively low computational complexity while in theory these kind of iterations ought to be expected so far this seems to be the first occurrence of such a nested use observed in practice

an a _2 bailey lemma and rogers ramanujan type identities author s andrews george e schilling anne warnaar s ole abstract using new q functions recently introduced by hatayama et al and by two of the authors we obtain an a 2 version of the classical bailey lemma we apply our result which is distinct from the a 2 bailey lemma of milne and lilly to derive rogers ramanujan type identities for characters of the w 3 algebra

effective metastability of halpern iterates in cat 0 spaces abstract this paper provides an effective uniform rate of metastability in the sense of tao on the strong convergence of halpern iterations of nonexpansive mappings in cat 0 spaces the extraction of this rate from an ineffective proof due to saejung is an instance of the general proof mining program which uses tools from mathematical logic to uncover hidden computational content from proofs this methodology is applied here for the first time to a proof that uses banach limits and hence makes a substantial reference to the axiom of choice

multiplier ideal sheaves and kahler einstein metrics of positive scalar curvature we present a method for proving the existence of kiihler einstein metrics of positive scalar curvature on certain compact complex manifolds and use the method to produce a large class of examples of compact kiihler einstein manifolds of positive scalar curvature suppose that m is a compact complex manifold of positive first chern class as is well known the existence of a kiihler einstein metric on m is equivalent to the existence of a solution to a certain complex monge ampere equation on m to solve this complex mongeampere equation by the method of continuity one needs only to establish the appropriate zeroth order a priori estimate suppose now that m does not admit a kiihler einstein metric so that the zeroth order a priori estimate fails to hold from this lack of an estimate we extract various global algebro geometric properties of m by introducing a coherent sheaf of ideals j on m called the multiplier ideal sheaf which carefully measures the extent to which the estimate fails the sheaf y is analogous to the subelliptic multiplier ideal sheaf that j j kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d neumann problem now j is a global algebro geometric object on m and it so happens that j satisfies a number of highly nontrivial global algebro geometric conditions including a cohomology vanishing theorem in particular the complex analytic subspace v c m cut out by j is nonempty connected and has arithmetic genus zero if v is zero dimensional then it is a single reduced point while if v is one dimensional then its support is a tree of smooth rational curves the logarithmic geometric genus of m v always vanishes these considerations place nontrivial global algebro geometric restric

quantitative results for halpern iterations of nonexpansive mappings abstract we give a rate of metastability for halpern s iteration relative to a rate of metastability for the resolvent for nonexpansive mappings in uniformly smooth banach spaces extracted from a proof due to xu in hilbert space the latter is known so we get an explicit rate of metastability we also extract a rate of asymptotic regularity for general normed spaces such rates have already been extracted by kohlenbach and leustean for different and incomparable conditions on n the proof analyzed in this paper is also more effective than the proofs treated by kohlenbach and leustean in that it does not use banach limits or weak compactness which makes the extraction particularly efficient moreover we also give an equivalent axiomatization of uniformly smooth banach spaces this paper is part of an ongoing case study of proof mining in nonlinear fixed point theory

tribasic integrals and identities of rogers ramanujan type some integrals involving three bases are evaluated as infinite products using complex analysis many special cases of these integrals may be evaluated in another way to find infinite sum representations for these infinite products the resulting identities are identities of rogers ramanujan type some integer partition interpretations of these identities are given generalizations of the rogers ramanujan type identities involving polynomials are given again as corollaries of integral evaluations

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

local stability of ergodic averages we consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages it is not difficult to construct an example of a computable lebesgue measure preserving transformation of 0 1 and a characteristic function f xa such that the ergodic averages a n f do not converge to a computable element of l 2 0 1 in particular there is no computable bound on the rate of convergence for that sequence on the other hand we show that for any nonexpansive linear operator t on a separable hilbert space and any element f it is possible to compute a bound on the rate of convergence of a n f from t f and the norm f of the limit in particular if t is the koopman operator arising from a computable ergodic measure preserving transformation of a probability space x and f is any computable element of l 2 x then there is a computable bound on the rate of convergence of the sequence anf the mean ergodic theorem is equivalent to the assertion that for every function k n and every e 0 there is an n with the property that the ergodic averages a m f are stable to within e on the interval n k n even in situations where the sequence a n f does not have a computable limit one can give explicit bounds on such n in terms of k and f e this tells us how far one has to search to find an n so that the ergodic averages are locally stable on a large interval we use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem and we show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to bishop and ivanov finally we explain how our positive results can be viewed as an application of a body of general proof theoretic methods falling under the heading of proof mining

a product formula for the continuous q jacobi polynomials abstract using a set of transformations on a series form product formula for the q wilson polynomials pn x a b c d q and the nassrallah rahman integral representation of an 8 7 series a product formula in the integral form for the continuous q jacobi polynomials pn x b bq c cq q2 with a positive kernel is obtained in the case 0 b

on the q analogues of some transformations of nearly poised hypergeometric series a number of transformation formulas for very well poised basic hypergeometric series have been obtained which in the limit q 1 approach the known transformation formulas for nearly poised ordinary hypergeometric series