Critical connectedness of thin arithmetical discrete planes

An arithmetical discrete plane is said to have critical connecting thickness if its thickness is equal to the infimum of the set of values that preserve its $2$-connectedness. This infimum thickness can be computed thanks to the fully subtractive algorithm. This multidimensional continued fraction algorithm consists, in its linear form, in subtracting the smallest entry to the other ones. We provide a characterization of the discrete planes with critical thickness that have zero intercept and that are $2$-connected. Our tools rely on the notion of dual substitution which is a geometric version of the usual notion of substitution acting on words. We associate with the fully subtractive algorithm a set of substitutions whose incidence matrix is provided by the matrices of the algorithm, and prove that their geometric counterparts generate arithmetic discrete planes.Comment: 18 pages, v2 includes several corrections and is a long version of the DGCI extended abstrac

Combinatorics of Pisot Substitutions

Siirretty Doriast

Digital Analytical Geometry: How do I define a digital analytical object?

International audienceThis paper is meant as a short survey on analytically de-ned digital geometric objects. We will start by giving some elements on digitizations and its relations to continuous geometry. We will then explain how, from simple assumptions about properties a digital object should have, one can build mathematical sound digital objects. We will end with open problems and challenges for the future

Brentanian continua and their boundaries

This dissertation focuses on how a specific conceptual thread of the history of mathematics unfolded throughout the centuries from its original account in Ancient Greece to its demise in the Modern era due to new mathematical developments and, finally, to its revival in the work of Brentano. In particular, we shall discuss how the notion of continuity and the connected notion of continua and boundaries developed through the ages until Brentano's revival of the original Aristotelian account against the by then established mathematical orthodoxy. Thus, this monograph hopes to fill in a gap in the present state of Brentanian scholarship as well as to present a thorough account of this specific historical thread

Hegel’s Theory of Quantity

Prior to Hegel\u27s time, no philosopher had ever tried to state what quantity or number was. Hegel undertakes a monumental definition in some of his longest chapters in the Science of Logic. In this paper, the author describes the nature of Hegel\u27s project and how Hegel derives quantity from the more primitive concept of quality. The author also reviews Hegel\u27s critique of calculus, which hereto has received very little attention in English. Finally the author shows how, according to Hegel, the concept of measure derives from the unity of quality and quantity. This is the second in a planned nine-part series that explicates how Hegel\u27s logic functions, using a specially designed series of graphic illustrations that memorialize each and every official progression in the Science of Logic

Strength of Materials

Strength of materials is the science of engineering methods for calculating the strength, rigidity and durability of machine and structure elements. Elements of mechanical engineering and building structures during operation are subjected to the force action of different nature. These forces are either applied directly to the element or transmitted through joint elements. For normal operation of engineering structure or machine, each element must be of such sizes and shapes that it can withstand the load on it, without fracture (strength), not changing in size (rigidity), retaining its original shape (durability). Strength of materials is theoretical and experimental science. Experiment – theory – experiment – such is the dialectic of the development of the science of solids resistance to deformation and fracture. However, the science of strength of materials does not cover all the issues of deformable bodies mechanics. Other related disciplines are also involved: structural mechanics of core systems, elasticity theory and plasticity theory. Strength of materials is general engineering science, in which, on the basis of experimental data concernimg properties of materials, on one hand, and rules of theoretical mechanics, physics and higher mathematics, on the other, the general methods of calculating rational sizes and shapes of engineering structures elements, taking into account the size and character of loads acting on them are studied. Strength of materials tasks are solved by simple mathematical methods, with a number of assumptions and hypotheses, as well as with the use of experimental data. Strength of materials has independent importance, as the subject, knowledge of which are required for all engineering specialties. It is the basis for studying all sections of structural mechanics, the basis for studying the course of machine parts, etc. Strength of materials is the scientific basis of engineering calculations, without which at rescent time it is impossible to design and create all the variety of modern mechanical engineering and civil engineering structures. The peculiarity of this course book is its focus on performing the term paper in strength of materials, which includes 14 tasks covering the entire course. The manual summarizes the main material for the topic of each task, outlines the statement of the task, and examples of solutions. The appendices provide the example of term paper structure (title page, contents, example of solving the task) and reference materials needed for its performance. All this will contribute to deeper course learning and independent performance of the term paper.INTRODUCTION...5 How to choose the task ...6 1. BASIC CONCEPTS OF STRENGTH OF MATERIALS ...7 2. CENTRAL TENSION AND COMPRESSION OF DIRECT RODS (BARS) ...13 Task 1 Strength calculation and displacement determination under tension and compression...19 Example of solving the task 1 Strength calculation and displacement determination under tension and compression ...22 Task 2 Calculation of statically indeterminate rod (bar) system under tensile-compression ...26 Example of solving the task 2 Calculation of statically indeterminate rod (bar) system under tensilecompression ...29 3. GEOMETRIC CHARACTERISTICS OF PLANE SECTIONS ...33 Task 3 Determination of axial moments of inertia of plane sections ...37 Example of solving the task 3 Determination of axial moments of inertia of plane sections ...40 4. SHEAR. TORSION ...43 Task 4 Shaft calculation for torsion...47 Example of solving the task 4 Shaft calculation for torsion (strength and rigidity) ...50 5. COMPLEX STRESSED STATE ...55 Task 5 Analysis of plane stressed state ...58 Example of solving the task 5 Analysis of plane stressed state ...60 6. STRAIGHT TRANSVERSE BENDING ...65 Task 6 Drawing the diagrams of shear (cutting) force and bending moment for cantilever beam ...76 Example of solving the task 6 Drawing the diagrams of shear (cutting) force and bending moment for cantilever beam ...79 Task 7 Diagraming of shear (cutting) force and bending moment for simply supported beam ...82 Task 8 Strength calculation under the bending of beams ...85 Task 9 Calculation for strength and determining displacements during the bending of beams ...85 Example of solving the task 7 and 8 Diagraming of shear (cutting) force and bending moment for simply supported beam. Strength calculation under the bending of beams ...88 7. DETERMINATION OF DISPLACEMENTS UNDER BENDING ...94 Example of solving the task 9 by the method of initial parameters ...108 Example of solving the task 9 by Mohr method ...110 8. STATICALLY INDETERMINATE SYSTEMS ...114 Task 10 Calculation of statically indeterminate frame ...120 Example of solving the task 10 using the force method ...123 Example of solving the task 10 by the metod of minimum potential energy of deformation ...128 9. EVALUATION OF STRESSES AND DISPLACEMENTS AT OBLIQUE BENDING ...130 Task 11 Choosing the beam section at oblique bending deformation ...134 Example of solving the task 11 Choosing the beam section at oblique bending deformation ...137 10. JOINT ACTION OF BENDING WITH TORSION ...144 Task 12 Calculation of the shaft for bending with torsion...146 Example of solving the task 12 Calculation of the shaft for bending with torsion ...149 11. STABILITY OF CENTRALLY-COMPRESSED RODS ...154 Task 13 Calculation of stability of compressed rod ...160 Example of solving the task 13 Calculation of stability of compressed rod ...162 12. DYNAMIC LOADS. DETERMINING IMPACT STRESSED AND DISPLACEMENTS ...165 Task 14 Determining maximum dynamic stresses and displacements under the impact ...169 Example of solving the task 14.1 ...172 Example of solving the task 14.2 ...175 List of references and recommended literature ...178 Annexes ...179 MAIN DEFINITIONS OF STRENGTH OF MATERIALS ...187 MAIN FORMULAS OF STRENGTH OF MATERIALS ...191 PERSONALITIES ...195 MAIN SYMBOLS OF STRENGTH OF MATERIALS ...230 UKRAINIAN-ENGLISH VOCABULARY OF BASIC TERMS ...23

Drawing division : emerging and developing multiplicative structure in low-attaining students representational strategies

This thesis examines the particular difficulties with multiplicative thinking experienced by students with very low attainment in school mathematics, and the representational strategies they use for multiplication and division-based tasks.\ud Selected students in two mainstream secondary schools, all performing significantly below age-related expectations in mathematics, placed in ‘bottom sets’, and described by their teachers as having particularly weak numeracy, received a series of tuition sessions (individual or paired). These involved ongoing qualitative diagnosis of their arithmetical strengths and weaknesses, and personalised, flexible learning support, delivered by the author. Students engaged mainly in division-based scenario tasks designed to encourage their engagement in multiplicative thinking, and explored various visuospatial representational strategies tailored to their specific areas of conceptual and procedural difficulty.\ud Multimodal audiovisual data collected from tuition sessions was analysed qualitatively across multiple analytic dimensions using a microgenetic approach. This led to the development of an adaptable framework for the analysis of nonstandard visuospatial representations of arithmetical structures and relationships. Analysis of changes in individual students’ strategies provided insight into some possible learning trajectories for multiplicative thinking. Parallel comparison of students’ varied representational strategies resulted in evidence for the psychological power of certain fundamental representation types, such as unit arrays and containers.\ud The main findings of this thesis concern: the fundamentally componential nature of the concept and practice of division, the potential difficulties this causes in understanding, and the importance of modelling and manipulating unitary multiplicative structures; and the relationship between representational strategies, economy and efficiency in carrying out multiplication and division-based tasks.\ud Conclusions are drawn on the relationship between the development of representational strategies and multiplicative thinking. Recommendations are given regarding learning and teaching practice for students with severe and milder difficulties in mathematics, and particularly the nature of 1:1 support provision for those considered to have Special Educational Needs