Pascal\u27s Mystic Hexagon in Tropical Geometry

Pascal\u27s mystic hexagon is a theorem from projective geometry. Given six points in the projective plane, we can construct three points by extending opposite sides of the hexagon. These three points are collinear if and only if the six original points lie on a nondegenerate conic. We attempt to prove this theorem in the tropical plane

Affine and Projective Planes

In this thesis, we investigate affine and projective geometries. An affine geometry is an incidence geometry where for every line and every point not incident to it, there is a unique line parallel to the given line. Affine geometry is a generalization of the Euclidean geometry studied in high school. A projective geometry is an incidence geometry where every pair of lines meet. We study basic properties of affine and projective planes and a number of methods of constructing them. We end by prov- ing the Bruck-Ryser Theorem on the non-existence of projective planes of certain orders

Units in irregular elementary abelian group rings

If P is a regular prime and A an elementary abelian p-group, every unit in the integral group ring of A is a product of units coming from cyclic subgroups of A

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

Reaction patterns of secondary school students to the symbolism of mathematics

Thesis (Ed.M.)--Boston Universit