Applying machine learning to the problem of choosing a heuristic to select the variable ordering for cylindrical algebraic decomposition

Cylindrical algebraic decomposition(CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. When using CAD, there is often a choice for the ordering placed on the variables. This can be important, with some problems infeasible with one variable ordering but easy with another. Machine learning is the process of fitting a computer model to a complex function based on properties learned from measured data. In this paper we use machine learning (specifically a support vector machine) to select between heuristics for choosing a variable ordering, outperforming each of the separate heuristics.Comment: 16 page

Recent advances in real geometric reasoning

In the 1930s Tarski showed that real quantifier elimination was possible, and in 1975 Collins gave a remotely practicable method, albeit with doubly-exponential complexity, which was later shown to be inherent. We discuss some of the recent major advances in Collins method: such as an alternative approach based on passing via the complexes, and advances which come closer to "solving the question asked" rather than "solving all problems to do with these polynomials"

Late Cretaceous - Early Palaeogene sandy-to-gravelly debris flows and their sediments in the Silesian Basin of the Alpine Tethys (Western Outer Carpathians, Istebna Formation)

The study focuses on Upper Cretaceous - Palaeocene deposits from the Beskid Śląski mountain range in southern Poland constituting the Istebna Beds. The Istebna Beds, also referred to as the Istebna Formation, are part of the Silesian tectonic unit, which forms the Outer Carpathian fold-and-thrust belt (part of the Alpine-Carpathian system). The results of qualitative and quantitative lithological-sedimentological studies were the basis for the interpretation of lithofacies types, sedimentary processes and palaeoenvironment as well as for the reconstruction of the architecture of the depositional system. The analysis conducted on the basis of field description of the deposits shows the prevalence (nearly 70%) of siliciclastic strata representing a sandstone-conglomerate association (S-C), which is the main subject of this work. The S-C lithofacies: sandstones, gravelly sandstones, sandy conglomerates and conglomerates constitute the deposits formed mostly by mass gravity-flows such as sandy-to-gravelly debris flows. The distribution of the coarse-clastic material indicates a sediment supply from southerly directions and implies the presence of an active source area in the rear part of the Silesian Basin. A succession of the sandstone-to-conglomerate deposits with the secondary participation of other lithofacies, with a thickness of approximately two thousand metres, indicates temporary increased diastrophic activity in the Silesian Ridge (source area) and the intense denudation of this area. The uplift of the alimentation area and its destruction coinciding with enforced relative regression and the uncovering of the proximal depositional zone of the basin led to resedimentation of the older intrabasinal material and repeated mass deposition together with delivery of extraclasts of pre-existing rocks and minerals. The lithofacies development of the sandstone-to-conglomerate debrites and the related sedimentary palaeotransport directions suggest an accumulation domain in the form of a linear apron depositional system developed in a deep-water setting. Experimental modelling of subaqueous sandy flows has contributed to a better understanding of the complex genesis of deep-water sediment gravity flows developing in depositional systems rich in sand material

Cohesive debrites of the Istebna Beds (Upper Senonian - Paleocene) West of the Skawa River

W pracy opisano debryty kohezyjne warstw istebniańskich zachodniej części jednostki śląskiej Karpat zewnętrznych. Przedstawiona litofacja reprezentuje utwory silikoklastyczne, zbudowane z mieszaniny mułowo-piaskowego spoiwa i rozproszonego materiału okruchowego frakcji psefitowej. Powstawanie debrytów kohezyjnych związane było z grawitacyjnymi spływami osadu, generowanymi na skłonie basenu sedymentacyjnego przez ruchy masowe. Prezentowane utwory deponowane były w formie fartuchów przez podwodne, kohezyjne spływy rumoszowe. Największy udział miąższościowy debrytów kohezyjnych (12%) stwierdzono w warstwach istebniańskich Beskidu Śląskiego, znaczący udział odnotowano w Beskidzie Małym (7%), a najmniejszy w Beskidzie Morawskim (2%). Z największą częstotliwością opisana litofacja występuje w obrębie piaskowców istebniańskich dolnych, a regułą jest także pojawianie się w stropie łupków istebniańskich dolnych debrytów kohezyjnych z litoklastami o największych rozmiarach. Około 25% opisanych utworów zawierało w składzie fazy rozproszonej materiał egzotykowy (egzotykowe debryty kohezyjne), w których dominowały skały krystaliczne, przede wszystkim gnejsy, łupki krystaliczne i granitoidy. Wśród znacznie rzadziej obserwowanych egzotyków skał osadowych stwierdzono wyraźne zwiększanie się z zachodu na wschód udziału różnego rodzaju wapieniCohesive debrites of the Istebna Beds occur within the western part of the Silesian Unit of the Outer Flysch Carpathians. The lithofacies described represents siliciclastic deposits consisting of mud-sandy matrix and scattered, psephite-size clasts. The cohesive debrites originated from sediment gravity flows that were generated on the slope of sedimentary basin by mass-movements and were deposed as aprons by submarine, cohesive debris flows. The highest content of the cohesive debrites (12% of the succession thickness) has been recorded within the Istebna Beds of the Silesian Beskid Mts, the medium in the Beskid Mały Mts (7%) and the lowest within the Moravian Beskid Mts (2%). Most frequently the lithofacies presented occurs within the Lower Istebna Sandstones, while the cohesive debrites with the biggest lithoclasts occur within the top part of the Lower Istebna Shales. The exotic material with dominating crystalline rocks, mostly gneisses, schists and granitoids, makes up 25% of the described deposits (exotic cohesive debrites). Various types of limestones prevail among much rarer clasts of sedimentary rocks and the amount of the limestones distinctly increases going to the eas

Univariate Real Root Isolation in Multiple Extension Fields

We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in Bα ∈ L[y], where L = (Q(α1,..., αℓ) is an algebraic extension of the rational numbers. Our bounds are single exponential in ℓ and match the ones presented in [34] for the case ℓ = 1. We consider two approaches. The first, indirect approach, using multivariate resultants, computes a univariate polynomial with integer coefficients, among the real roots of which are the real roots of Bα. The Boolean complexity of this approach is ÕB(N 4ℓ+4), where N is the maximum of the degrees and the coefficient bitsize of the involved polynomials. The second, direct approach, tries to solve the polynomial directly, without reducing the problem to a univariate one. We present an algorithm that generalizes Sturm algorithm from the univariate case, and modified versions of well known solvers that are either numerical or based on Descartes ’ rule of sign. We achieve a Boolean complexity of ÕB(min{N 4ℓ+7, N 2ℓ2 +6}) and ÕB(N 2ℓ+4), respectively. We implemented the algorithms in C as part of the core library of mathematica and we illustrate their efficiency over various data sets

Advances on the continued fractions method using better estimations of positive root bounds

We present an implementation of the Continued Fractions (CF) real root isolation method using a recently developed upper bound on the positive values of the roots of polynomials. Empirical results presented in this paper verify that this implementation makes the CF method always faster than the Vincent-Collins-Akritas bisection method1, or any of its variants. © Springer-Verlag Berlin Heidelberg 2007

Early Cretaceous deposits of the proto-Silesian Basin in Polish-Czech Flysch Carpathians

The proto-Silesian Basin was well developed within the Alpine Tethys during the Late Jurassic times and existed as undivided entity until the significant Late Cretaceous reorganization. The deposits originated within this basin were incorporated into different structural units: Silesian, Subsilesian and Skole. The calciturbiditic Cieszyn Limestone Formation is the oldest Cretaceous flischoidal sequence of the proto-Silesian Basin. This calciturbiditic sedimentation passed gradually into younger siliciclastic deposition

Verovice Formation deposits during Early Cretaceous sedimentological regimes in the western part of the proto-Silesian Basin (Moravia, the Czech Republic)

Sedimentological development of the Verovice Formation (Czech Moravia) is presented, according to Early Cretaceous geotectonic-eustatic changes in the proto-Silesian Basin. Those processes were a consequence of a gradual rebuilding the architecture of the Alpine-Carpathian realm. The geological events well correspond with the global sequence stratigraphy

Upper Jurassic--Lower Cretaceous source rocks in the western part of the Flysch Carpathians

Przedmiotem rozważań przedstawionego artykułu jest analiza możliwości odnalezienia górnojurajsko-dolnokredowych skał macierzystych w zachodniej części Karpat fliszowych, na terenie Polski i Republiki Czeskiej. Górnojurajskie margle mikułowskie stanowią skały macierzyste światowej klasy. Te bogate w substancję organiczną skały z TOC 0.2-10% zasilały ropy naftowe w basenie wiedeńskim i pod nasunięciem karpackim w Czechach i Austrii. Podobne skały macierzyste prawdopodobnie występują w głębokich strefach pod nasunięciem w Polsce. Utwory formacji wędryńskiej i grodziskiej nie zawierają większych ilości węgla organicznego. Większą zawartość TOC stwierdzono natomiast w formacji wierzowskiej i najwyższej części formacji grodziskiej. Utwory te reprezentują globalne anoksyczne wydarzenie oceaniczne OAE 1b. Utwory formacji wierzowskiej mogą stanowić potencjalne skały macierzyste dla systemów obejmujących różnego wieku skały zbiornikowe Karpat zewnętrznych.The analysis of possibility to find Upper Jurassic--Lower Cretaceous source rocks in the western part of Flysch Carpathians in Poland and Czech Republic was the subject of present paper. The Upper Jurassic organic-rich Mikulov marls represent world-class source rocks. These 1400 m thick organic-rich rocks with TOC value 0.2-10% sourced oils in the Vienna Basin and Carpathian subthrust in Czech Republic and Austria. The similar source rocks perhaps exist in the deeper subthrust areas in Poland. The Vendryne Formation rocks do not contain significant amount of organic carbon. The increased TOC was encountered within the Verovice Formation and uppermost part of Hradiste Formatio. These rocks represent global anoxic event OAE 1b. Verovice Formation contains potential source rocks for the Outer Carpathians systems with reservoirs various in age