inseparable local uniformization abstract it is known since the works of zariski in the early 40s that desingularization of varieties along valuations called local uniformization of valuations can be considered as the local part of the desingularization problem it is still an open problem if local uniformization exists in positive characteristic and dimension larger than three in this paper we prove that zariski local uniformization of algebraic varieties is always possible after a purely inseparable extension of the field of rational functions and therefore any valuation can be uniformized by a purely inseparable alteration

assouad type dimensions and homogeneity of fractals we investigate several aspects of the assouad dimension and the lower dimension which together form a natural dimension pair in particular we compute these dimensions for certain classes of self affine sets and quasiself similar sets and study their relationships with other notions of dimension such as the hausdorff dimension for example we also investigate some basic properties of these dimensions including their behaviour regarding unions and products and their set theoretic complexity

entropy and data compression schemes some new ways of defining the entropy of a process by observing a single typical output sequence as well as a new kind of shannon mcmillan breiman theorem are presented this provides a new and conceptually very simple ways of estimating the entropy of an ergodic stationary source as well as new insight into the workings of such well known data compression schemes as the lempel ziv algorithm

the model theory of separably tame valued fields a henselian valued field k is called separably tame if its separable algebraic closure ksep is a tame extension that is the ramification field of the normal extension ksep k is separable algebraically closed every separable algebraically maximal kaplansky field is a separably tame field but not conversely in this paper we prove ax kochen ershov principles for separably tame fields this leads to model completeness and completeness results relative to the value group and residue field as the maximal immediate extensions of separably tame fields are in general not unique the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy ax kochen ershov principles our approach also yields alternate proofs of known results for separably closed valued fields

efficient block predictor corrector methods with a small number of corrections recently various classes of predictor corrector methods have been proposed as being suitable for solving nonstiff ordinary differential equations in a parallel environment this paper shows that methods based on a low order predictor and a runge kutta corrector are not efficient and that if predictor corrector methods are to be used efficiently for solving nonstiff problems in parallel then high order predictors are required examples of methods with high order predictors are given and their efficiency properties are studied in terms of stability and local error theory

hausdorff dimension of planar self affine sets and measures let x bigcup varphi i x be a strongly separated self affine set in mathbb r 2 or one satisfying the strong open set condition under mild non conformality and irreducibility assumptions on the matrix parts of the varphi i we prove that dim x is equal to the affinity dimension and similarly for self affine measures and the lyapunov dimension the proof is via analysis of the dimension of the orthogonal projections of the measures and relies on additive combinatorics methods

functionally fitted block methods for ordinary differential equations abstract functionally fitted methods generalize collocation techniques to integrate exactly a chosen set of linearly independent functions in this paper we propose a new type of functionally fitted block methods for ordinary differential equations the basic theory for the proposed methods is established first we derive two sufficient conditions to ensure the existence of the functionally fitted block methods discuss their equivalence to collocation methods in a special case and independence on integration time for a set of separable basis functions we then obtain some basic characteristics of the methods by taylor series expansions and show that the order of accuracy of the r point functionally fitted block method is at least r for general ordinary differential equations experimental results are conducted to demonstrate the validity of the functionally fitted block methods

quantitative recurrence properties in conformal iterated function systems abstract we consider a quantitative version of poincare s recurrence theorem in a conformal iterated function system let i i be a conformal iterated function system on 0 1 d with a countable index set denote by j the attractor of let f 0 1 d r be a positive function s n f x be the sum f x f w 1 1 x f w 1 w n 1 1 x analogous to an ergodic sum and consider the set of points for which the inequality x w 1 w n 0 1 d x w 1 w n 1 x e s n f x is realized for infinitely many n n with w 1 w n n the set above contains the points x whose orbit returns very close to x infinitely many times with the quality of approximation depending on the time n and the point x this can be viewed as a quantitative version of poincare s recurrence theorem it is shown that its hausdorff dimension is the solution to some pressure function the setting considered in this paper includes the quantitative recurrence properties in the dynamical systems of b adic expansions continued fraction expansion as well as some dynamical systems defined on fractal sets applying the main result to diophantine approximation gives a partial answer to a question by k mahler

on the l q spectrum of planar self affine measures the author was supported by the epsrc grant ep j013560 1 this work was started whilst the author was an epsrc funded phd student at the university of st andrews and he expresses his gratitude for the support he found there

algebraic independence of elements in immediate extensions of valued fields abstract refining a constructive combinatorial method due to maclane and schilling we give several criteria for a valued field that guarantee that all of its maximal immediate extensions have infinite transcendence degree if the value group of the field has countable cofinality then these criteria give the same information for the completions of the field the criteria have applications to the classification of valuations on rational function fields we also apply the criteria to the question of which extensions of a maximal valued field algebraic or of finite transcendence degree are again maximal in the case of valued fields of infinite p degree we obtain the worst possible examples of nonuniqueness of maximal immediate extensions fields which admit an algebraic maximal immediate extension as well as one of infinite transcendence degree

structure of equilibrium states on self affine sets and strict monotonicity of affinity dimension a fundamental problem in the dimension theory of self affine sets is the construction of high dimensional measures which yield sharp lower bounds for the hausdorff dimension of the set a natural strategy for the construction of such high dimensional measures is to investigate measures of maximal lyapunov dimension these measures can be alternatively interpreted as equilibrium states of the singular value function introduced by falconer whilst the existence of these equilibrium states has been well known for some years their structure has remained elusive particularly in dimensions higher than two in this article we give a complete description of the equilibrium states of the singular value function in the three dimensional case showing in particular that all such equilibrium states must be fully supported in higher dimensions we also give a new sufficient condition for the uniqueness of these equilibrium states as a corollary giving a solution to a folklore open question in dimension three we prove that for a typical self affine set in r 3 removing one of the affine maps which defines the set results in a strict reduction of the hausdorff dimension

ledrappier young formula and exact dimensionality of self affine measures in this paper we solve the long standing open problem on exact dimensionality of self affine measures on the plane we show that every self affine measure on the plane is exact dimensional regardless of the choice of the defining iterated function system in higher dimensions under certain assumptions we prove that self affine and quasi self affine measures are exact dimensional in both cases the measures satisfy the ledrappier young formula

a variable step implicit block multistep method for solving first order odes a new four point implicit block multistep method is developed for solving systems of first order ordinary differential equations with variable step size the method computes the numerical solution at four equally spaced points simultaneously the stability of the proposed method is investigated the gauss seidel approach is used for the implementation of the proposed method in the pe ce m mode the method is presented in a simple form of adams type and all coefficients are stored in the code in order to avoid the calculation of divided difference and integration coefficients numerical examples are given to illustrate the efficiency of the proposed method

on the distribution of long term time averages on symbolic space the pressure was studied in a rather abstract theory as an important notion of the thermodynamic formalism the present paper gives a more concrete account in the case of symbolic spaces including subshifts of finite type we relate the pressure of an interaction function to its long term time averages through the hausdorff and packing dimensions of the subsets on which has prescribed long term time average values functions with values in d are considered for those depending only on finitely many symbols we get complete results unifying and completing many partial results

dynamical models for circle covering brownian motion and poisson updating we consider two dynamical variants of dvoretzky s classical problem of random interval coverings of the unit circle the latter having been completely solved by l shepp in the first model the centers of the intervals perform independent brownian motions and in the second model the positions of the intervals are updated according to independent poisson processes where an interval of length l is updated at rate l where 0 is a parameter for the model with brownian motions a special case of our results is that if the length of the nth interval is c n then there are times at which a fixed point is not covered if and only if c 2 and there are times at which the circle is not fully covered if and only if c 3 for the poisson updating model we obtain analogous results with c and c 1 instead we also compute the hausdorff dimension of the set of exceptional times for some of these questions

the hausdorff and dynamical dimensions of self affine sponges a dimension gap result we construct a self affine sponge in mathbb r 3 whose dynamical dimension i e the supremum of the hausdorff dimensions of its invariant measures is strictly less than its hausdorff dimension this resolves a long standing open problem in the dimension theory of dynamical systems namely whether every expanding repeller has an ergodic invariant measure of full hausdorff dimension more generally we compute the hausdorff and dynamical dimensions of a large class of self affine sponges a problem that previous techniques could only solve in two dimensions the hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge implying that sponges with a dimension gap represent a nonempty open subset of the parameter space

a localized jarnik besicovitch theorem abstract fundamental questions in diophantine approximation are related to the hausdorff dimension of sets of the form x r x where 1 and x is the diophantine approximation exponent of an irrational number x we go beyond the classical results by computing the hausdorff dimension of the sets x r x f x where f is a continuous function our theorem applies to the study of the approximation exponents by various approximation families it also applies to functions f which are continuous outside a set of prescribed hausdorff dimension

a dynamical dimension transference principle for dynamical diophantine approximation diophantine approximation in dynamical systems concerns the diophantine properties of the orbits in classic diophantine approximation the powerful mass transference principle established by beresnevich and velani provides a general principle to the dimension for a limsup set in this paper we aim at finding a general principle for the dimension of the limsup set arising in a general expanding dynamical system more precisely let x t be a topological dynamical system where x is a compact metric space and t x rightarrow x is an expanding continuous transformation given y o in x we consider the following limsup set mathcal w t f driven by the dynamical system x t begin aligned big x in x x in b z e s n f log t z text for some z in t n y o text with infinitely many n in mathbb n big end aligned where log t is a function reflecting the local conformality of the transformation t f is a non negative continuous function over x and s n f log t z denotes the ergodic sum f log t z cdots f log t t n 1 z by proposing a dynamical ubiquity property assumed on the system x t we obtain that the dimensions of x and mathcal w t f are both related to the bowen manning mccluskey formulae namely the solution to the pressure functions begin aligned texttt p t log t 0 text and texttt p t log t f 0 text respectively end aligned we call this phenomenon a dynamical dimension transference principle because of its partial analogy with the mass transference principle this general principle unifies and extends some known results which were considered only separatedly before these include the b adic expansions expanding rational maps over julia sets inhomogeneous diophantine approximation on the triadic cantor set and finite conformal iterated function systems

valuations on rational function fields that are invariant under permutation of the variables abstract we study and characterize the class of valuations on rational functions fields that are invariant under permutation of the variables and can be extended to valuations with the same property whenever a finite number of new variables is adjoined the gaus valuation is in this class which constitutes a natural generalization of the concept of gaus valuation further we apply our characterization to show that the most common ad hoc generalization of the gaus valuation is also in this class

the algebra and model theory of tame valued fields a henselian valued field k is called a tame field if its algebraic closure k is a tame extension that is the ramification field of the normal extension k k is algebraically closed every algebraically maximal kaplansky field is a tame field but not conversely we develop the algebraic theory of tame fields and then prove ax kochen ershov principles for tame fields this leads to model completeness and completeness results relative to value group and residue field as the maximal immediate extensions of tame fields will in general not be unique the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy ax kochen ershov principles the results of this paper have been applied to gain insight in the zariski space of places of an algebraic function field and in the model theory of large fields

diophantine approximation and badly approximable sets let x d be a metric space and d a compact subspace of x which supports a non atomic finite measure m we consider natural classes of badly approximable subsets of loosely speaking these consist of points in which stay clear of some given set of points in x the classical set bad of badly approximable numbers in the theory of diophantine approximation falls within our framework as do the sets bad i j of simultaneously badly approximable numbers under various natural conditions we prove that the badly approximable subsets of have full hausdorff dimension applications of our general framework include those from number theory classical complex p adic and formal power series and dynamical systems iterated function schemes rational maps and kleinian groups

scaling scenery of m n invariant measures abstract we study the scaling scenery and limit geometry of invariant measures for the non conformal toral endomorphism x y m x mod 1 n y mod 1 that are bernoulli measures for the natural markov partition we show that the statistics of the scaling can be described by an ergodic cp chain in the sense of furstenberg invoking the machinery of cp chains yields a projection theorem for bernoulli measures which generalises in part earlier results by hochman shmerkin and ferguson jordan shmerkin we also give an ergodic theoretic criterion for the dimension part of falconer s distance set conjecture for general sets with positive length using cp chains and hence verify it for various classes of fractals such as self affine carpets of bedford mcmullen lalley gatzouras and baranski class and all planar self similar sets

a generalized multifractal spectrum of the general sierpinski carpets in this paper we study a class of subsets of the general sierpinski carpets for which the limiting frequency of a horizontal fibre falls into a prescribed closed interval we obtain the explicit expression for the hausdorff dimension of these subsets in terms of the parameters of the construction and give necessary and sufficient conditions for the corresponding hausdorff measure to be positive finite

elimination of ramification i the generalized stability theorem we prove a general version of the stability theorem if k is a valued field such that the ramification theoretical defect is trivial for all of its finite extensions and if f k is a finitely generated transcendental extension of valued fields for which equality holds in the abhyankar inequality then the defect is also trivial for all finite extensions of f this theorem is applied to eliminate ramification in such valued function fields it has applications to local uniformization and to the model theory of valued fields in positive characteristic

diophantine approximation of the orbit of 1 in the dynamical system of beta expansions we consider the distribution of the orbits of the number 1 under the transformations as varies mainly the size of the set of for which a given point can be well approximated by the orbit of 1 is measured by its hausdorff dimension the dimension of the following set is determined where is a given point in and is a sequence of integers tending to infinity as for the proof of this result the notion of the recurrence time of a word in symbolic space is introduced to characterise the lengths and the distribution of cylinders the set of with a common prefix in the expansion of 1 in the parameter space

the ergodic theory of shrinking targets to any dynamical system equipped with a metric we associate a class of well approximable sets in the case of an expanding rational map of the riemann sphere acting on its julia set we estimate and in some cases compute the hausdorff dimension of the associated well approximable sets the methods used show a clear link between distortion properties and the type of results obtained in this paper via ergodic theory and ubiquity

limit theorems for partially hyperbolic systems we consider a large class of partially hyperbolic systems containing among others affine maps frame flows on negatively curved manifolds and mostly contracting diffeomorphisms if the rate of mixing is sufficiently high the system satisfies many classical limit theorems of probability theory

a multifractal mass transference principle for gibbs measures with applications to dynamical diophantine approximation let mu be a gibbs measure of the doubling map t of the circle for a mu generic point x and a given sequence r n subset of r consider the intervals t n x r n mod 1 t n x r n mod 1 in analogy to the classical dvoretzky covering of the circle we study the covering properties of this sequence of intervals this study is closely related to the local entropy function of the gibbs measure and to hitting times for moving targets a mass transference principle is obtained for gibbs measures that are multifractal such a principle was proved by beresnevich and velani ann math 164 2006 971 992 for monofractal measures in the symbolic language we completely describe the combinatorial structure of a typical relatively short sequence in particular we can describe the occurrence of atypical relatively long words our results have a direct and deep number theoretical interpretation via inhomogeneous dyadic diophantine approximation by numbers belonging to a given dyadic diophantine class less

the graph and range singularity spectra of b adic independent cascade functions abstract with the iso holder sets of a function we naturally associate subsets of the graph and the range of the function we compute the hausdorff dimension of these subsets for a class of statistically self similar multifractal functions namely the b adic independent cascade functions

hausdorff dimension of the limit sets of some planar geometric constructions abstract we determine the hausdorff and box dimension of the limit sets for some class of planar non moran like geometric constructions generalizing the bedford mcmullen general sierpinski carpets the class includes affine constructions generated by an arbitrary partition of the unit square by a finite number of horizontal and vertical lines as well as some non affine examples e g the flexed sierpinski gasket

twostep by twostep pirk type pc methods with continuous output formulas this paper deals with parallel predictor corrector pc iteration methods based on collocation runge kutta rk corrector methods with continuous output formulas for solving nonstiff initial value problems ivps for systems of first order differential equations at nth step the continuous output formulas are used not only for predicting the stage values in the pc iteration methods but also for calculating the step values at n 2 th step in this case the integration processes can be proceeded twostep by twostep the resulting twostep by twostep tbt parallel iterated rk type pirk type methods with continuous output formulas twostep by twostep pirkc methods or tbtpirkc methods give us a faster integration process fixed stepsize applications of these tbtpirkc methods to a few widely used test problems reveal that the new pc methods are much more efficient when compared with the well known parallel iterated rk methods pirk methods parallel iterated rk type pc methods with continuous output formulas pirkc methods and sequential explicit rk codes dopri5 and dop853 available from the literature

counting the number of distinct distances of elements in valued field extensions abstract the defect of valued field extensions is a major obstacle in open problems in resolution of singularities and in the model theory of valued fields whenever positive characteristic is involved we continue the detailed study of defect extensions through the tool of distances which measure how well an element in an immediate extension can be approximated by elements from the base field we show that in several situations the number of essentially distinct distances in fixed extensions or even just over a fixed base field is finite and we compute upper bounds we apply this to the special case of valued functions fields over perfect base fields in particular this provides important information used in forthcoming research on the ramification theory of two dimensional valued function fields

every place admits local uniformization in a finite extension of the function field we prove that every place p of an algebraic function field f k of arbitrary characteristic admits local uniformization in a finite extension f of f we show that f f can be chosen to be galois after a finite purely inseparable extension of the ground field k instead of being galois the extension can also be chosen such that the induced extension fp fp of the residue fields is purely inseparable and the value group of f only gets divided by the residue characteristic if f lies in the completion of an abhyankar place then no extension of f is needed our proofs are based solely on valuation theoretical theorems which are of particular importance in positive characteristic they are also applicable when working over a subring r k and yield similar results if r is regular and of dimension smaller than 3

entropy preservation under markov coding using the notion of topological entropy for non compact sets we prove that for a c1 map with a finite markov partition the corresponding coding map preserves topological entropy of subsets we also provide an example of a piecewise linear conformal repeller with a markov coding decreasing topological entropy these results are generalized to the notions of u dimensions

a mass transference principle and the duffin schaeffer conjecture for hausdorff measures a hausdorff measure version of the duffin schaeffer conjecture in metric number theory is introduced and discussed the general conjecture is established modulo the original conjecture the key result is a mass transference principle which allows us to transfer lebesgue measure theoretic statements for lim sup subsets of r k to hausdorff measure theoretic statements in view of this the lebesgue theory of lim sup sets is shown to underpin the general hausdorff theory this is rather surprising since the latter theory is viewed to be a subtle refinement of the former

dimension theory of iterated function systems let si l i 1 be an iterated function system ifs on d with attractor k let denote the one sided full shift over the alphabet 1 l we define the projection entropy function h on the space of invariant measures on e associated with the coding map k and develop some basic ergodic properties about it this concept turns out to be crucial in the study of dimensional properties of invariant measures on k we show that for any conformal ifs respectively the direct product of finitely many conformal ifss without any separation condition the projection of an ergodic measure under is always exactly dimensional and its hausdorff dimension can be represented as the ratio of its projection entropy to its lyapunov exponent respectively the linear combination of projection entropies associated with several coding maps furthermore for any conformal ifs and certain affine ifss we prove a variational principle between the hausdorff dimension of the attractors and that of projections of ergodic measures 2008 wiley periodicals inc

dimensions and waiting times for gibbs measures for shifts of finite type we relate the waiting time between two different orbits one chosen according to an ergodic measure the other according to a gibbs measure to billingsley dimensions of generic sets this is achieved by computing billingsley dimensions of saturated sets in terms of a relative entropy which satisfies a pointwise ergodic result as a by product a similar result is obtained for match lengths that are dual quantities of waiting times