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

complete lagrange family for the cube in finite element interpolations abstract noting a vagueness in the serendipity definition the complete lagrange family for the cube and the mixed complete lagrange family are introduced the unisolvent lagrange interpolations over rectangular finite elements are then presented definitively in hierarchy ranking expressions so that the polynominal completeness is realized with appropriately necessary minimum nodes the corresponding interpolation bases in the conventional form can easily be derived through the basis transformation without matrix inversion

bounded depth polynomial size circuits for symmetric functions abstract let f f 1 f 2 be a family of symmetric boolean functions where f n has n boolean variables for each n 1 let f n be the minimum number of variables of f n that each have to be set to constant values so that the resulting function is a constant function we show that the growth rate of f n completely determines whether or not the family f is good that is can be realized by a family of constant depth polynomial size circuits with unbounded fan in furthermore if f n log n k for some k then the family f is good however if f n n for some 0 then the family is not good

finite monoids and the fine structure of nc 1 recently a new connection was discovered between the parallel complexity class nc 1 and the theory of finite automata in the work of barrington on bounded width branching programs there nonuniform nc 1 was characterized as those languages recognized by a certain nonuniform version of a dfa here we extend this characterization to show that the internal structures of nc 1 and the class of automata are closely related in particular using therien s classification of finite monoids we give new characterizations of the classes ac 0 depth k ac 0 and acc the last being the ac 0 closure of the mod q functions for all constant q we settle some of the open questions in 3 give a new proof that the dot depth hierarchy of algebraic automata theory is infinite 8 and offer a new framework for understanding the internal structure of nc 1

a logic for constant depth circuits consider a family of boolean circuits c c2 c constructed by some uniform effective procedure operating on input n such a procedure provides a concise representation of a family of parallel algorithms for computing boolean values a formula of first order logic may also be viewed as a concise representation of a family of parallel algorithms for evaluating boolean functions the parallelism is implicit in the quantification a formula gx q x is true if and only if each of the formulas q a is true and all these formulas can be checked simultaneously and universes of different sizes give rise to boolean functions with different numbers of inputs the boolean values of the formula s predicates on various combinations of elements of the universe this note presents an extended first order logic designed to be exactly equivalent in expressiveness to polynomialsize constant depth unbounded fan in circuits constructed by turing machines of bounded computational complexity 1984 academic press inc

finite semigroup varieties defined by programs abstract we study the regular languages recognized by polynomial length programs over finite semigroups belonging to product varieties v li where v is a variety of finite monoids and li is the variety of finite locally trivial semigroups in the case where the semigroup variety has a particular closure property with respect to programs we are able to give precise characterizations of these regular languages as a corollary we obtain new proofs of the results of barrington compton straubing and therien characterizing the regular languages in certain circuit complexity classes

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

regular languages defined with generalized quantifiers we study an extension of first order logic obtained by adjoining quantifiers that count with respect to an integer modulus it is shown that the languages definable in this framework are precisely the regular languages whose syntactic monoids contain only solvable groups we obtain an analogous result for regular languages and establish some connections with complexity theory for fixed depth families of circuits

general complete lagrange interpolations with applications to three dimensional finite element analysis abstract in this paper we have first derived the interpolation polynomials for the general serendipity solid elements of rectangular shape which allow arbitrarily placed nodes along the edges we have then presented a method to determine the interpolation functions for the general complete lagrange elements which allow arbitrarily placed nodes explicit expressions for interpolation functions of the serendipity and complete lagrange families with uniform spacing of nodes over the element domain are derived for elements of orders 4 10 we have also modified the shape functions of complete lagrange family to correctly interpolate the complete polynomials in the global space for angular distortions of quadrilaterals over the six planar facets of linear solid hexahedron elements

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

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

programs over semigroups of dot depth one abstract the notion of a p variety arises in the algebraic approach to boolean circuit complexity it has great significance since many known and conjectured lower bounds on circuits are equivalent to the assertion that certain classes of semigroups form p varieties in this paper we prove that semigroups of dot depth one form a p variety this example has the following implication if a boolean combination of 1 formulas using arbitrary numerical predicates defines a regular language one can then find an equivalent 1 formula all of whose numerical predicates are regular

general complete lagrange family for the cube in finite element interpolations abstract in this paper we have first derived the interpolation polynomials for the general serendipity elements which allow arbitrarily placed nodes along the edges we have then presented a method to determine the interpolation functions for the general complete lagrange elements which allow arbitrarily placed nodes explicit expressions for interpolation functions of the serendipity and complete lagrange family elements which allow uniform spacing of nodes over the element domain are derived for elements of orders 4 10 we have also modified the shape functions of complete lagrange family so that they can correctly interpolate the complete polynomial in the global space for angular distortions

classification of finite monoids the language approach abstract the concept of a variety of congruences is introduced and related to variety of languages and variety of monoids a systematic construction of increasingly complex varieties of congruences is presented this construction is powerful enough to generate all monoids containing solvable groups some hierarchies occurring through this process are shown to correspond to well known hierarchies of monoids thus indicating that our construction is natural from an algebraic point of view some problems that remain open are also discussed

effects of element distortions on the performance of isoparametric elements we discuss the effects of element distortions on the performance of displacement based isoparametric quadrilateral finite elements suitable example problems for both the lagrangian and the serendipity types of elements are used to show numerically the effects of element distortions the effects of angular and curved edge distortions are evaluated analytically in terms of the abilities of the elements to represent exactly various polynomial displacement fields we then solve a plane stress problem adaptively to demonstrate the effect of element distortions on the accuracy of the stress distribution the conclusion is that the lagrangian elements are not affected by angular distortions and are therefore the more reliable elements for general use

languages defined with modular counting quantifiers we prove that a regular language defined by a boolean combination of generalized 1 sentences built using modular counting quantifiers can be defined by a boolean combination of 1 sentences in which only regular numerical predicates appear the same statement with 1 replaced by first order is equivalent to the conjecture that the nonuniform circuit complexity class acc is strictly contained in nc1 the argument introduces some new techniques based on a combination of semigroup theory and ramsey theory which may shed some light on the general case

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

non uniform automata over groups abstract a new model non uniform deterministic finite automata nudfa s over general finite monoids has recently been developed as a strong link between the theory of finite automata and low level parallel complexity achievements of this model include the proof that width 5 branching programs recognize exactly the languages in non uniform nc1 nudfa characterizations of several important subclasses of nc1 and a new proof of the old result that the dot dephth hierarchy is infinite using m sipser s 1983 in proceedings 15th acm symposium on the theory of computing association for computing machinery new york pp 61 69 work on constant depth circuits here we extend this theory to nudfa s over solvable groups nudfa s over non solvable groups have the maximum possible computing power we characterize the power of nudfa s over nilpotent groups and prove some optimal lower bounds for nudfa s over certain groups which are solvable but not nilpotent most of these results appeared in preliminary form in d a barrington and d therien 1987 in automata languages and programming 14th international colloquium springer verlag berlin pp 163 173

an improved isoparametric transformation for finite element analysis the standard isoparametric transformation for finite element analysis is shown to be only a special case of a more general transformation which allows for flexibility in locating side and interior element nodes side nodes positioned at the same relative distance from corner nodes in both local and global space and interior nodes positioned such that they do not affect the jacobian of the co ordinate transformation provide improved accuracy over nodes positioned locally without regard for the global configuration this concept is demonstrated in the context of two dimensional quadratic quadrilateral isoparametrics but is also applicable to higher orders and dimensions in the test cases examined the new formulation provides a reduction in error of up to three orders of magnitude and also provides accurate solutions for nodal placements which are not allowed using the standard transformations