multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

matrix displacement decompositions and applications to toeplitz linear systems using the approach of bozzo di fiore and zellini new matrix displacement decomposition formulas are introduced it is shown how an arbitrary square matrix a can be expressed as sums of products of hessenberg algebra matrices and high level block matrices whose submatrices are hessenberg algebra matrices and have variable sizes in most cases these block factors are block diagonal matrices then these formulas are used in sequential and parallel solution of toeplitz systems

an infinite pebble game and applications we generalize the pebble game to infinite directed acyclic graphs and use this generalization to give new and shorter proofs of the following well known results 1 that unbounded memory increases the power of logics of programs and 2 that there exists a context free grammar with infinite index

the a core and a cover of a group abstract this paper provides a comprehensive investigation of the cellular approximation functor cell a g in the category of groups approximating a group g by a group a we also study related notions such as a injection a generation and a constructibility of a group g and we find several interesting connections with the schur multiplier h 2 g z our constructions are direct and are given in a slow and detailed manner

nontrivial definability by flow chart programs the ability of flow chart programs to define relations and functions nondefinable by open first order formulas is studied an example of a unary not locally finite structure t with equality such that every flow chart is equivalent in t to a loop free flow chart is shown but the same does not hold for some flow charts equipped with counters or recursion from this it is deduced that deterministic dynamic logic of regular programs is strictly weaker than nondeterminitic dynamic logic it is also proved that flow charts with recursion but without counters are able to define nontrivial functions in any structure provided a nontrivial computable function exists over this structure

unbounded program memory adds to the expressive power of first order programming logic it is proved that no logic of programs with unbounded memory is reducible to a bounded memory programming logic this is achieved by carefully analyzing firstorder definability in the algebra of finite binary trees

computation by while programs on topological partial algebras the language of while programs is a fundamental model for imperative programming on any data type it leads to a generalisation of the theory of computable functions on the natural numbers to the theory of computable functions on any many sorted algebra the language is used to express many algorithms in scientific computing where while programs are applied to continuous data in the theory of data continuous data types are modelled by topological many sorted algebras we study both exact and approximate computations by while programs and while programs with arrays over topological many sorted algebras with partial operations first we establish that partial operations are necessary in order to compute a wide range of continuous functions we prove basic continuity properties of our abstract computability any partial function computable over a partial topological algebra by a while array program is continuous any set semicomputable or computable over a partial topological algebra by a while array program is open or clopen respectively secondly we contrast exact and approximate computations the class of functions that can be computed exactly can be quite limited we show that on connected total algebras the while and while array computable functions are precisely those that are explicitly definable by terms we show that for certain general classes of topological algebras the total functions that can be approximated by while programs are precisely those that can be effectively approximated by terms this property we call generalised weierstrass approximation an application of this result is that a function on the set r of reals is computable in the sense of computable analysis if and only if it is while program approximable on a simple algebra based on r

definability by programs in first order structures introduction ti 1 preliminarv detinitilms and results algebraic characterizations of the unwind property a structure with the unwind property for every iterative but not every recursive 1 program the truth table property structures with the truth table property but not the unwind property for every iterative program 4ppendix 1 the pebble game on infinite dags ippendix 2 remaining proofs references

on the foundations of parameter estimation for generalized partial linear models with b splines and continuous optimization generalized linear models are widely used in statistical techniques as an extension generalized partial linear models utilize semiparametric methods and augment the usual parametric terms with a single nonparametric component of a continuous covariate in this paper after a short introduction we present our model in the generalized additive context with a focus on the penalized maximum likelihood and the penalized iteratively reweighted least squares p irls problem based on b splines which is attractive for nonparametric components then we approach solving the p irls problem using continuous optimization techniques they have come to constitute an important complementary approach alternative to the penalty methods with flexibility for choosing the penalty parameter adaptively in particular we model and treat the constrained p irls problem by using the elegant framework of conic quadratic programming the method is illustrated using a small numerical example

homotopical localizations of spaces for a map f of spaces dror farjoun and the author have constructed an f localization functor where a space y is called f local when map f y is an equivalence this very general construction gives all known idempotent homotopy functors of spaces the main theorem of this paper shows that f localization functors always preserve fiber sequences of connected h spaces up to small error terms for instance the localization with respect to the n th morava k theory preserves such fiber sequences up to error terms with at most three nontrivial homotopy groups this implies for example that a k 1 homology equivalence of h spaces must induce an isomorphism of v 1 periodic homotopy groups results are also obtained on the a nullification or a periodization functors which are just the f localization functors for the maps f from spaces a to points two spaces are said to have the same nullity when they give the same nullification functors and it is shown that arbitrary sets of nullity classes have both least upper bounds and greatest lower bounds the a nullifications of nilpotent postnikov spaces are completely determined

computational complexity of real functions in this chapter we introduce the formal definitions of computational complexity of real functions we first review the definitions of computable real numbers and computable real functions and their basic properties then the oracle turing machine is introduced as the formal model for computing real functions this model allows us to define the complexity measures for computing real functions in a natural way the class of polynomial time computable real functions is then defined and several characterizations of this class will be given other complexity classes of real numbers and real functions such as np real functions and log space computable real functions will be defined in later chapters

some relationships between logics of programs and complexity theory abstract we show that some open problems concerning comparative schematology and logics of programs are equivalent to open problems in complexity theory in particular we show that p space p time holds if and only if flow diagrams with arrays are of the same computational power as recursive procedures both conditions are equivalent to the statement that programming logics based on the mentioned classes of program schemes have equal expressive power a similar characterization is given for some other equalities between complexity classes

a necessary and sufficient condition in order that a herbrand interpretation be expressive relative to recursive programs it is proved that a recursive program without counters is able to enumerate all elements in any herbrand interpretation it follows that all recursive program domains in a herbrand interpretation can be defined by first order formulas iff there are first order formulas expressing integer arithmetic in that interpretation

nullification and cellularization of classifying spaces of finite groups in this note we discuss the effect of the bz p nullification p bz p and the bz p cellularization cw bz p over classifying spaces of finite groups and we relate them with the corresponding functors with respect to moore spaces that have been intensively studied in the last years we describe p bz p bg by means of a covering fibration and we classify all finite groups g for which bg is bz p cellular we also carefully study the analogous functors in the category of groups and their relationship with the fundamental groups of p bz p bg and cw bz p bg

some questions about expressiveness and relative completeness in hoare s logic abstract a well known result of cook asserts the completeness of hoare s logic for while programs relative to any expressive structure in this paper we present a wide and natural class of structures whose members are either expressive or make hoare s logic strongly incomplete relative to them in the sense that a trivially true partial correctness assertion is not hoare derivable from the first order theory of the structure the definition of this class is related to the so called unwind property for while programs and its behaviour follows from quite general sufficient conditions for strong relative incompleteness we state also two questions about the connections among inexpressiveness relative incompleteness and strong relative incompleteness and point out the seeming difficulty of answering them

central extensions and generalized plus constructions we describe the effect of homological plus constructions on the homotopy groups of eilenberg maclane spaces in terms of universal central extensions

idempotent transformations of finite groups we describe the action of idempotent transformations on finite groups we show that finiteness is preserved by such transformations and enumerate all possible values such transformations can assign to a fixed finite simple group this is done in terms of the first two homology groups we prove for example that except special linear groups such an orbit can have at most 7 elements we also study the action of monomials of idempotent transformations on finite groups and show for example that orbits of this action are always finite

on approximate and algebraic computability over the real numbers abstract we consider algebraic and approximate computations of partial real functions r d r algebraic computability is defined by means of parameter free finite algorithmic procedures the notion of approximate computability is a straightforward generalization of the ko friedman approach based on oracle turing machines to functions with not necessarily recursively open domains the main results of the paper give characterizations of approximate computability by means of the passing sets of finite algorithmic procedures i e characterizations from the algebraic point of view some consequences and also modifications of the concepts are discussed finally two variants of arithmetical hierarchies over the reals are considered and used to classify and mutually compare the domains graphs and ranges of algebraically resp approximately computable real functions

the unwind property in certain algebras in the present paper we investigate abstract algebras having the so called unwind property i e algebras where every total computable function can be defined by open formulas in an adequate first order language a method of proving the unwind property is given for a special kind of algebra applying this method we prove that there exists a structure satisfying the unwind property for all flow charts but not for recursive procedures we prove also that algebraically closed and real closed fields with infinite degrees of transcendence over their prime subfields satisfy the unwind property for all schemas

effective metric spaces and representations of the reals based on standard notions of classical recursion theory a natural model of approximate computability for partial functions between effective metric spaces is presented it generalizes the ko friedman approach to computability of real functions by means of oracle turing machines follows the main ideas of weihrauch s type 2 theory of effectivity but it avoids the explicit use of representations the topological arithmetical hierarchy is introduced and shown to be strict if the underlying space contains an effectively discrete sequence the domains of computable functions are exactly the 2 sets of this hierarchy if the space admits a finitary stratification finally this framework is used to investigate and characterize the standard representations of the real numbers they are just those functions from the name space onto the reals which have both computable extensions and inversions that are computable as relations

computing algebraic formulas using a constant number of registers it is shown that over an arbitrary ring the functions computed by polynomial size algebraic formulas are also computed by polynomial length algebraic straight line programs that use only three registers this was previously known for boolean formulas d a barrington j comput system sci 38 1989 pp 150 164 which are equivalent to algebraic formulas over the ring gf 2 for formulas over arbitrary rings the result is an improvement over previous methods that require the number of registers to be logarithmic in the size of the formulas in order to obtain polynomial length straight line programs moreover the straight line programs that arise in these constructions have the property that they consist of statements whose actions on the registers are linear and bijective a consequence of this is that the problem of determining the iterated product of n3 times 3 matrices is complete under p projections for algebraic nc 1 also when the ring is gf 2 the programs that arise in the c