GEOMETRY OF CRITICAL LOCI

2001 ◽  
Vol 63 (3) ◽  
pp. 533-552 ◽  
Author(s):  
LÊ DŨNG TRÁNG ◽  
HÉLÈNE MAUGENDRE ◽  
CLAUDE WEBER
Keyword(s):  
Initial Data ◽  
Finite Subset ◽  
Intersection Number ◽  
Rational Numbers ◽  
Topological Invariant ◽  
Normal Surface ◽  
Critical Locus ◽  
Discriminant Locus ◽  
Definition Of ◽  
Analytic Curves

Let(formula here)be the germ of a finite (that is, proper with finite fibres) complex analytic morphism from a complex analytic normal surface onto an open neighbourhood U of the origin 0 in the complex plane C2. Let u and v be coordinates of C2 defined on U. We shall call the triple (π, u, v) the initial data.Let Δ stand for the discriminant locus of the germ π, that is, the image by π of the critical locus Γ of π.Let (Δα)α∈A be the branches of the discriminant locus Δ at O which are not the coordinate axes.For each α ∈ A, we define a rational number dα by(formula here)where I(–, –) denotes the intersection number at 0 of complex analytic curves in C2. The set of rational numbers dα, for α ∈ A, is a finite subset D of the set of rational numbers Q. We shall call D the set of discriminantal ratios of the initial data (π, u, v). The interesting situation is when one of the two coordinates (u, v) is tangent to some branch of Δ, otherwise D = {1}. The definition of D depends not only on the choice of the two coordinates, but also on their ordering.In this paper we prove that the set D is a topological invariant of the initial data (π, u, v) (in a sense that we shall define below) and we give several ways to compute it. These results are first steps in the understanding of the geometry of the discriminant locus. We shall also see the relation with the geometry of the critical locus.

scholarly journals Upaya Meningkatkan Kreativitas Anak melalui Permainan Finger Painting Usia 5-6 Tahun di TK Harapan Bunda Pekanbaru

2018 ◽  
Vol 2 (01) ◽  
pp. 31-39
Author(s):  
Jumilah Jumilah ◽  
Sean Marta Efastri ◽  
Siti Fadillah
Keyword(s):  
Initial Data ◽  
Key Words ◽  
Average Increase ◽  
Research Results ◽  
Class Actions ◽  
Analysis Technique ◽  
Determination Methods ◽  
Definition Of ◽  
Children's Development ◽  
Children’S Development

Abstrak Kreativitas adalah kemampuan untuk menciptakan sesuatu yang baru yang berupa gagasan maupun karya nyata. Adapun tujuan dari penelitian yaitu untuk mendapatkan data perkembangan kreativitas anak  melalui permainan finger painting. Metode penelitian yang digunakan adalah Penelitian Tindakan Kelas ( PTK) . Subjek yang diambil untuk penelitian  terdiri dari 15 anak di TK Harapan Bunda. Teknik pengumpulan data dalam penelitian ini adalah melalui lembar observasi, dan dokumentasi. Teknik analisis data yaitu dengan menggunakan rumus persentase= f/n x100 % . Hasil penelitian data yang diperoleh  dalam meningkatkan perkembangan kreativitas anak melalui permainan finger painting, sebelum tindakan adalah 59 %  ( Mulai Berkembang / MB ) dan meningkat pada siklus I dengan kategori BSH “ Berkembang Sesuai Harapan”  dengan rata – rata peningkatan 71 % dan pada siklus II mengalami peningkatan dengan kategori BSB “ Berkembang Sangat Baik” yaitu rata-rata 84 %. Jadi peningkatan kreativitas anak dari data awal 59 %  ke data siklus II 84 %, terjadi peningkatan sebanyak 25 %.    Kata  Kunci :  Kreativitas Anak,   permainan  finger painting Abstract Creativity is the ability to create a new something which is in the form of ideas or real works. There is a purpose of definition that is to get data of children's development creativity through Finger painting games. Determination methods used are definition of class actions (ptk). Observation sheet, and documentation. Analysis technique is using the percentage formula f / n x 100%. The research results obtained in improving the development of children's creativity through finger painting, before the action was 59% (developing / mb) and increasing in the first cycle with the bsh category "developing as expected" with an average increase of 71% and in the second cycle experiencing an increase with the bsb category "develops very well" which is an average of 84%. So increasing children's creativity from the initial data of 59% to data cycle ii 84%, an increase of 25%. Key words : children's creativity, finger painting game

Signal Recovery by Linear Estimation

2019 ◽  
pp. 260-385
Author(s):  
John Stillwell
Keyword(s):  
Human Activities ◽  
Mathematical Concept ◽  
Rational Numbers ◽  
Precise Definition ◽  
Computable Analysis ◽  
Signal Recovery ◽  
Linear Estimation ◽  
Definition Of ◽  
Computable Sequence ◽  
Informal Description

This chapter develops the basic results of computability theory, many of which are about noncomputable sequences and sets, with the goal of revealing the limits of computable analysis. Two of the key examples are a bounded computable sequence of rational numbers whose limit is not computable, and a computable tree with no computable infinite path. Computability is an unusual mathematical concept, because it is most easily used in an informal way. One often talks about it in terms of human activities, such as making lists, rather than by applying a precise definition. Nevertheless, there is a precise definition of computability, so this informal description of computations can be formalized.

A Fuzzy Ontology-Based Platform for Flexible Querying

2015 ◽  
Vol 6 (3) ◽  
pp. 12-26 ◽  
Author(s):  
Amira Aloui ◽  
Amel Grissa Touzi
Keyword(s):  
Fuzzy Logic ◽  
Initial Data ◽  
Data Retrieval ◽  
Conceptual Clustering ◽  
Large Databases ◽  
Flexible Queries ◽  
Definition Of ◽  
Preliminary Classification ◽  
Flexible Querying

Flexible queries have recently received increasing attention to better characterize the data retrieval. In this paper, a new flexible querying approach using ontological knowledge is proposed. This approach presents an FCA based methodology for building ontologies from scratch then interrogating them intelligently through the fusion of conceptual clustering, fuzzy logic, and FCA. The main contribution is the definition of the ontology rom classes resulting from a preliminary classification of the data and not the initial data. The data cleansing provides a simple ontology and an optimal research of relevant data taking into account the preferences cited by the user in his initial interrogation. To realize this approach, a new platform called “FO-FQ Tab plug-in” is implemented. This plug-in is integrated within the ontology editor Protégé to allow building fuzzy ontologies from large databases and querying them intelligently

scholarly journals DEFINITION OF SEDIMENTOLOGICAL MODEL IN LIMITED INITIAL DATA SET CONDITIONS

2018 ◽  
pp. 21-26
Author(s):  
A. V. Podnebesnykh
Keyword(s):  
Initial Data ◽  
Low Cost ◽  
Central Africa ◽  
Main Direction ◽  
Geological Exploration ◽  
Data Set ◽  
Definition Of

The article considers the main procedures which essentially increase the validity of sedimentological models in limited initial data set conditions. The Bentiu formation (Central Africa) is studied as an example of the research. Using simple and low-cost methods it is possible to identify not only the main direction of sedimentary material displacement but also the main characteristics of alluvial systems that will let to plan exploitation of reservoir formation of such type correctly even on geological exploration stage.

scholarly journals An information technique for segmentation of military assets in conditions of uncertainty of initial data

2021 ◽  
pp. 6-18
Author(s):  
С.І. Березіна ◽  
О.І. Солонець ◽  
Кювон Лі ◽  
М.В. Борцова
Keyword(s):  
Initial Data ◽  
Poor Quality ◽  
Fuzzy Logic Systems ◽  
Suggested Technique ◽  
Segmentation Methods ◽  
Probabilistic Values ◽  
The Common ◽  
Conventional Methods ◽  
Definition Of ◽  
The Military

To solve the applied task of detecting military assets in aerospace images the presented paper investigates the processes of constructing segmented maps of the images. The goal is to develop an information technique for detecting military assets in conditions of uncertainty of initial data. To achieve the goal, the following tasks were formulated: 1) to analyze usability of the existing segmentation methods for automatic detection of military assets in the images; 2) if the existing methods are inapplicable, to develop a new algorithm to solve the problem. In the paper the following methods are used: the methods of digital image processing, the methods of Boolean algebra and fuzzy sets, the methods of statistical analysis. The following results are received. Analysis of the known segmentation methods showed that due to camouflage coloring of the military assets, similarity of their color characteristics to those of underlying surfaces and due to the presence of large number of textured fragments in the images those methods provide segmented maps of poor quality. Among the common problems arising when conventional methods are used there are wrong segmentation, when the received contours do not coincide with the borders of the objects of interest; oversegmentation, when there are a lot of minor segments which produce "litter" objects; undersegmentation, when potentially possible segments are missed etc. As the conventional methods are inapplicable, in the paper it is suggested using the fuzzy logic systems. For each pixel the probability of the fact that the pixel belongs to the object or to the background is calculated. For making decision whether a pixel belongs to the object the production rules based on the chosen most significant factors (probabilistic values of spectral sub-bands, belonging of the neighboring pixels to the object, jumps of brightness in spectral sub-bands on the object's borders) are constructed. Conclusion. The suggested technique ensures high-quality definition of objects' borders, thus considerably increasing the reliability of military assets recognition.

Solutions to Selected Exercises

2019 ◽  
pp. 447-608
Author(s):  
John Stillwell
Keyword(s):  
Human Activities ◽  
Mathematical Concept ◽  
Rational Numbers ◽  
Computability Theory ◽  
Precise Definition ◽  
Computable Analysis ◽  
Infinite Path ◽  
Definition Of ◽  
Computable Sequence ◽  
Informal Description

This chapter develops the basic results of computability theory, many of which are about noncomputable sequences and sets, with the goal of revealing the limits of computable analysis. Two of the key examples are a bounded computable sequence of rational numbers whose limit is not computable, and a computable tree with no computable infinite path. Computability is an unusual mathematical concept, because it is most easily used in an informal way. One often talks about it in terms of human activities, such as making lists, rather than by applying a precise definition. Nevertheless, there is a precise definition of computability, so this informal description of computations can be formalized.

Topological invariant and definition method for detecting isomorphism in planar kinematic chains

2020 ◽  
pp. 095440622096078
Author(s):  
Rongjiang Cui ◽  
Zhizheng Ye ◽  
Liang Sun ◽  
Chuanyu Wu
Keyword(s):  
Planetary Gear ◽  
Topological Invariant ◽  
Detection Methods ◽  
Kinematic Chains ◽  
Essential Step ◽  
Planetary Gear Trains ◽  
Gear Trains ◽  
Configuration Synthesis ◽  
Definition Of ◽  
Isomorphism Identification

Isomorphism identification is an essential step in mechanism configuration synthesis. Although various detection methods have been proposed, some of them can only effectively identify kinematic chains (KCs) within 10 links or complex programs that are needed to identify multilink KCs. In this study, a new isomorphism identification method is proposed based on the distance concept of graphs and the graph theory definition of isomorphism. In addition to two complex 21- and 28-link planar simple-joint KCs (PSKCs), the proposed algorithm is tested on the complete atlas of 8-link 1-DOF, 9-link 2-DOF, 10-link 1-DOF, 12-link 1-DOF, and 13-link 2-DOF PSKCs. The algorithm is also tested on 6-link 1-DOF and 7-link 1-DOF planetary gear trains (PGTs) to detect isomorphism. All results are in agreement with those of the existing literature. The method is fully automated via a computer program and has been verified to be reliable and efficient.

scholarly journals Geometry, Coordinatization and Cardinality of the Rational Numbers from Physical Perspective

2021 ◽  
Vol 2090 (1) ◽  
pp. 012037
Author(s):  
Kaushik Ghosh
Keyword(s):  
Line Segment ◽  
Statistical Physics ◽  
Rational Numbers ◽  
Partition Functions ◽  
Actual Process ◽  
Real Numbers ◽  
Hausdorff Topology ◽  
Straight Line ◽  
Definition Of ◽  
Positive Integers

Abstract In this article, we will first discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric continuum. According to this definition, points are not regarded as the basic constituents of a line segment and a line segment is considered to be a fundamental geometric object. This definition is in particular suitable to coordinatize different points on the straight line preserving the order properties of real numbers. Geometrically fundamental nature of line segments are required in physical theories like the string theory. We will construct a new topology suitable for this alternate definition of the straight line as a geometric continuum. We will discuss the cardinality of rational numbers in the later half of the article. We will first discuss what we do in an actual process of counting and define functions well-defined on the set of all positive integers. We will follow an alternate approach that depends on the Hausdorff topology of real numbers to demonstrate that the set of positive rationals can have a greater cardinality than the set of positive integers. This approach is more consistent with an actual act of counting. We will illustrate this aspect further using well-behaved functionals of convergent functions defined on the finite dimensional Cartezian products of the set of positive integers and non-negative integers. These are similar to the partition functions in statistical physics. This article indicates that the axiom of choice can be a better technique to prove theorems that use second-countability. This is important for the metrization theorems and physics of spacetime.

scholarly journals Introduction to Liouville Numbers

2017 ◽  
Vol 25 (1) ◽  
pp. 39-48
Author(s):  
Adam Grabowski ◽  
Artur Korniłowicz
Keyword(s):  
Positive Integer ◽  
Algebraic Number ◽  
Finite Sequence ◽  
Rational Numbers ◽  
Liouville Number ◽  
Liouville Numbers ◽  
Constant Sequence ◽  
Transcendental Numbers ◽  
Finite Sequences ◽  
Definition Of

Summary The article defines Liouville numbers, originally introduced by Joseph Liouville in 1844 [17] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and It is easy to show that all Liouville numbers are irrational. Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is defined in Section 6 quite generally as the sum for a finite sequence {ak}k∈ℕ and b ∈ ℕ. Based on this definition, we also introduced the so-called Liouville number as substituting in the definition of L(ak, b) the constant sequence of 1’s and b = 10. Another important examples of transcendental numbers are e and π [7], [13], [6]. At the end, we show that the construction of an arbitrary Lioville constant satisfies the properties of a Liouville number [12], [1]. We show additionally, that the set of all Liouville numbers is infinite, opening the next item from Abad and Abad’s list of “Top 100 Theorems”. We show also some preliminary constructions linking real sequences and finite sequences, where summing formulas are involved. In the Mizar [14] proof, we follow closely https://en.wikipedia.org/wiki/Liouville_number. The aim is to show that all Liouville numbers are transcendental.

Adding fractions using the definition of addition of rational numbers and the Euclidean algorithm

1973 ◽  
Vol 20 (1) ◽  
pp. 27-28
Author(s):  
Laurence Sherzer
Keyword(s):  
Rational Numbers ◽  
Euclidean Algorithm ◽  
Common Denominator ◽  
Efficient Approach ◽  
First Finding ◽  
Long Time ◽  
Definition Of

Adding fractions by first finding the lowest common denominator has for a long time been assumed to be the best approach to this operation. Is this really the most efficient approach? Is it pedagogically the most practical?

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