resolution of singularities of threefolds in positive characteristic ii abstract together with vincent cossart olivier piltant resolution of singularities of threefolds in positive characteristic i reduction to local uniformization on artin schreier and purely inseparable coverings j algebra 320 3 2008 1051 1082 this article gives a complete proof of desingularization of quasiprojective varieties of dimensional 3 on fields which are differentially finite over perfect fields

quantization and dynamisation of trace poisson brackets the quantization problem for the trace bracket algebra derived from double poisson brackets is discussed we obtain a generalization of the boundary ybe or so called abcd algebra for the quantization of quadratic trace brackets a dynamical deformation is proposed on the lines of gervais neveu felder dynamical quantum algebras

rational recursion operators for integrable differential difference equations in this paper we introduce the concept of prehamiltonian pairs of difference operators demonstrate their connections with nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential difference equations we begin with a rigorous setup of the problem in terms of the skew field of rational pseudo difference operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational pseudo difference operators in particular we give a criteria for a rational operator to be weakly nonlocal a difference operator is called prehamiltonian if its image is a lie subalgebra with respect to the lie bracket on the difference field two prehamiltonian operators form a prehamiltonian pair if any linear combination of them is prehamiltonian then we show that a prehamiltonian pair naturally leads to a nijenhuis operator and a nijenhuis operator can be represented in terms of a prehamiltonian pair this provides a systematic method to check whether a rational operator is nijenhuis as an application we construct a prehamiltonian pair and thus a nijenhuis recursion operator for the differential difference equation recently discovered by adler and postnikov the nijenhuis operator obtained is not weakly nonlocal we prove that it generates an infinite hierarchy of local commuting symmetries we also illustrate our theory on the well known examples including the toda the ablowitz ladik and the kaup newell differential difference equations

discrete integrable systems and deformations of associative algebras interrelations between discrete deformations of the structure constants for associative algebras and discrete integrable systems are reviewed theory of deformations for associative algebras is presented closed left ideal generated by the elements representing the multiplication table plays a central role in this theory deformations of the structure constants are generated by the deformation driving algebra and governed by the central system of equations it is demonstrated that many discrete equations such as discrete boussinesq equation discrete wdvv equation discrete schwarzian kp and bkp equations discrete hirota miwa equations for kp and bkp hierarchies are particular realizations of the central system an interaction between the theories of discrete integrable systems and discrete deformations of associative algebras is reciprocal and fruitful an interpretation of the menelaus relation discrete schwarzian kp equation discrete hirota miwa equation for kp hierarchy consistency around the cube as the associativity conditions and the concept of gauge equivalence for instance between the menelaus and kp configurations are particular examples

logistic regression analysis of sample survey data summary standard chi squared x2 or likelihood ratio g2 test statistics for logistic regression analysis involving a binary response variable are adjusted to take account of the survey design these adjustments are based on certain generalized design effects logistic regression diagnostics to detect any outlying cell proportions in the table and influential points in the factor space are also developed taking account of the survey design finally the results are used to analyse some data from the october 1980 canadian labour force survey

integrable odes on associative algebras in this paper we give definitions of basic concepts such as symmetries first integrals hamiltonian and recursion operators suitable for ordinary differential equations on associative algebras and in particular for matrix differential equations we choose existence of hierarchies of first integrals and or symmetries as a criterion for integrability and justify it by examples using our componentless approach we have solved a number of classification problems for integrable equations on free associative algebras also in the simplest case we have listed all possible hamiltonian operators of low order

geometric proofs of theorems of ax kochen and ersov we give an algebraic geometric proof of the theorem of ax and kochen on p adic diophantine equations in many variables unlike ax kochen s proof ours does not use any notions from mathematical logic and is based on weak toroidalization of morphisms we also show how this geometric approach yields new proofs of the ax kochen er v s ov transfer principle for local fields and of quantifier elimination theorems of basarab and pas

counterexamples to local monomialization in positive characteristic a local monomialization of an algebraic map between varieties along a valuation is a sequence of blowups of nonsingular subvarieties along the valuation in the domain and target of the map such that the resulting map can be expressed by monomial inclusion in a suitable sense in this paper we give examples showing that a local monomialization does not always exist over fields of positive characteristic it was shown in earlier work of the author that a local monomialization always exists over fields of characteristic zero

noncommutative extensions of elliptic integrable euler arnold tops and painleve vi equation in this paper we suggest generalizations of elliptic integrable tops to matrix valued variables our consideration is based on the r matrix description which provides lax pairs in terms of quantum and classical r matrices first we prove that for relativistic and non relativistic tops such lax pairs with spectral parameters follow from the associative yang baxter equation and its degenerations then we proceed to matrix extensions of the models and find out that some additional constraints are required for their construction we describe a matrix version of the reduced elliptic top and verify that the latter constraints are fulfilled in this case the construction of matrix extensions is naturally generalized to the monodromy preserving equation in this way we get matrix extensions of the painleve vi equation and its multidimensional analogues written in the form of non autonomous elliptic tops finally it is mentioned that the matrix valued variables can be replaced by elements of noncommutative associative algebra at the end of the paper we also describe special elliptic gaudin models which can be considered as matrix extensions of the reduced elliptic top

local monomialization of analytic maps abstract in this paper local monomialization theorems are proven for morphisms of complex and real analytic spaces this gives the generalization of the local monomialization theorem for morphisms of algebraic varieties over a field of characteristic zero proven in 16 and 18 to analytic spaces

a class of integrable flows on the space of symmetric matrices for a given skew symmetric real n n matrix n the bracket x y n xny ynx defines a lie algebra structure on the space sym n n of symmetric n n real matrices and hence a corresponding lie poisson structure the purpose of this paper is to investigate the geometry integrability and linearizability of the hamiltonian system dot x x 2 n or equivalently in lax form the equation dot x x xn nx on this space along with a detailed study of the poisson geometry itself if n has distinct eigenvalues it is proved that this system is integrable on a generic symplectic leaf of the lie poisson structure of sym n n this is established by finding another compatible poisson structure

canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant this article contains an elementary constructive proof of resolution of singularities in characteristic zero our proof applies in particular to schemes of finite type and to analytic spaces so we recover the great theorems of hironaka we introduce a discrete local invariant rm inv x a whose maximum locus determines a smooth centre of blowing up leading to desingularization to define rm inv x we need only to work with a category of local ringed spaces x x cal o x satisfying certain natural conditions if a in x then rm inv x a depends only on widehat cal o x a more generally rm inv x is defined inductively after any sequence of blowings up whose centres have only normal crossings with respect to the exceptional divisors and lie in the constant loci of rm inv x cdot the paper is self contained and includes detailed examples one of our goals is that the reader understand the desingularization theorem rather than simply know it is true

techniques for the study of singularities with applications to resolution of 2 dimensional schemes we give an overview of invariants of algebraic singularities over perfect fields we then show how they lead to a synthetic proof of embedded resolution of singularities of 2 dimensional schemes

integrable matrix equations related to pairs of compatible associative algebras we study associative multiplications in semi simple associative algebras over bb c compatible with the usual one an interesting class of such multiplications is related to the affine dynkin diagrams of skew5 tilde a 2 k 1 tilde d k tilde e 6 tilde e 7 and tilde e 8 type in this paper we investigate in detail the multiplications of the skew5 tilde a 2 k 1 type and integrable matrix odes and pdes generated by them

exact solutions of quantum mappings from the lattice kdv as multi dimensional operator difference equations the lattice kdv equation is defined on an associative algebra and a new initial value problem on the lattice kdv is considered this leads to two new families of iterative mappings discrete equations of motion which can be solved for all discrete times imposing a certain commutation relation on the algebra makes it possible to prove using a yang baxter structure that these mappings are quantum integrable maps in the classical limit the maps become a new class of integrable mappings in the sense of liouville arnold veselov lav

a simple algorithm for principalization of monomial ideals in this paper we give a simple constructive proof of principalization of monomial ideals and the global analog this also gives an algorithm for principalization

poincare series of resolutions of surface singularities let x spec r be a resolution of singularities of a normal surface singularity spec r with integral exceptional divisors e 1 e r we consider the poincare series g h n t n where h n r x o x n 1 e 1 n r e r we show that if r m has characteristic zero and pic 0 x is a semi abelian variety then the poincare series g is rational however we give examples to show that this series can be irrational if either of these conditions fails

logistic regression diagnostics a maximum likelihood fit of a logistic regression model and other similar models is extremely sensitive to outlying responses and extreme points in the design space we develop diagnostic measures to aid the analyst in detecting such observations and in quantifying their effect on various aspects of the maximum likelihood fit the elements of the fitting process which constitute the usual output parameter estimates standard errors residuals etc will be used for this purpose with a properly designed computing package for fitting the usual maximum likelihood model the diagnostics are essentially free for the asking in particular good data analysis for logistic regression models need not be expensive or time consuming

resolution of singularities of real analytic vector fields in dimension three let be an analytic vector field defined in a real analytic manifold of dimension three we prove that all the singularities of can be made elementary by a finite number of blowing ups in the ambient space

samples of noncommutative products in certain differential equations a set of associative noncommutative products is considered in different differential equations of the ordinary and partial types a method of separation of variables is considered for a large set of those systems the products involved include for example some products and some products based on nijenhuis tensors which are embedded in the differential equations of the laplace poisson lax and schr dinger styles a comment on the products of reshetikhin jambor sykora type is also given in relation to products of vey type

pairs of compatible associative algebras classical yang baxter equation and quiver representations given an associative multiplication in matrix algebra compatible with the usual one or in other words a linear deformation of the matrix algebra we construct a solution to the classical yang baxter equation we also develop a theory of such deformations and construct numerous examples it turns out that these deformations are in one to one correspondence with representations of certain algebraic structures which we call m structures we also describe an important class of m structures related to the affine dynkin diagrams of a d e type these m structures and their representations are described in terms of quiver representations

resolution of singularities for 3 folds in positive characteristic a concise complete proof of resolution of singularities of 3 folds in positive characteristic 5 is given abhyankar first proved this theorem in 1966

resolution of singularities of threefolds in positive characteristic i reduction to local uniformization on artin schreier and purely inseparable coverings abstract in dimension 3 the desingularization problem is reduced to local uniformization on artin schreier and purely inseparable coverings of regular space

double poisson vertex algebras and non commutative hamiltonian equations we develop the formalism of double poisson vertex algebras local and non local aimed at the study of non commutative hamiltonian pdes this is a generalization of the theory of double poisson algebras developed by van den bergh which is used in the study of hamiltonian odes we apply our theory of double poisson vertex algebras to non commutative kp and gelfand dickey hierarchies we also construct the related non commutative de rham and variational complexes