The use of divergent series in history

In this thesis the author presents a history of non-convergent series which, in the past, played an important role in mathematics. Euler\u27s formula, Stirling\u27s series and Poincare\u27s theory are examined to show the development of asymptotic series, a subdivision of divergent series

The calculus according to S. F. Lacroix (1765-1843)

Silvestre François Lacroix (Paris. 1765 - ibid., 1843) was not a prominent mathematical researcher, but he was certainly a most influential mathematical book author. His most famous book is a monumental Traité du calcul différentiel et du calcul intégral (three large volumes, 1797-1800; a second edition appeared in 1810-1819) - an encyclopaedic appraisal of 18th-century calculus. He also published many textbooks, one of which is closely associated to this large Traité: the Traité élémentaire du calcul différentiel et du calcul intégral (first edition in 1802; four more editions in Lacroix's lifetime; four posthumous editions). Although most historians acknowledge the great influence of Lacroix's large Traité in early 19th-century mathematics it has not been thoroughly studied. This thesis is a contribution for correcting this omission. The focus is on its first edition, but the second edition and the Traité élémentaire, are also addressed. The thesis starts with a short biography of Lacroix, followed by an overview of the first edition of the large Traité. Next corne five chapters where particular aspects are analyzed in detail: the foundations of the calculus, analytic and differential geometry, approximate integration and conceptions of the integral, types of solutions of differential equations (singular/complete/general integrals, geometrical interpretations, and generality of arbitrary functions), and three aspects related to finite differences and series (the use of subscript indices, types of solutions of finite difference equations, and mixed difference equations); for all these aspects Lacroix's treatment is compared to the 18th-century background, and to his likely sources. Then we examine how the large Traité was adapted to a textbook - the Traité élémentaire, we take a look at the second edition of the large Traité, and conclude the body of the thesis with some final remarks

Coordinate space methods for the evaluation of Feynman diagrams in lattice field theory

We describe an efficient position space technique to calculate lattice Feynman integrals in infinite volume. The method applies to diagrams with massless propagators. For illustration a set of two-loop integrals is worked out explicitly. An important ingredient is an observation of Vohwinkel that the free lattice propagator can be evaluated recursively and is expressible as a linear function of its values near the origin.Comment: 27 pages, postscript file, now compressed & uuencode

A fragment on Euler\u27s constant in Ramanujan\u27s lost notebook

A formula for Euler’s constant found in Ramanujan’s lost notebook and also in a problem he submitted to the Journal of the Indian Mathematical Society is proved and discussed

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

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

The calculus according to S. F. Lacroix (1765-1843)

Silvestre François Lacroix (Paris. 1765 - ibid., 1843) was not a prominent mathematical researcher, but he was certainly a most influential mathematical book author. His most famous book is a monumental Traité du calcul différentiel et du calcul intégral (three large volumes, 1797-1800; a second edition appeared in 1810-1819) - an encyclopaedic appraisal of 18th-century calculus. He also published many textbooks, one of which is closely associated to this large Traité: the Traité élémentaire du calcul différentiel et du calcul intégral (first edition in 1802; four more editions in Lacroix's lifetime; four posthumous editions). Although most historians acknowledge the great influence of Lacroix's large Traité in early 19th-century mathematics it has not been thoroughly studied. This thesis is a contribution for correcting this omission. The focus is on its first edition, but the second edition and the Traité élémentaire, are also addressed. The thesis starts with a short biography of Lacroix, followed by an overview of the first edition of the large Traité. Next corne five chapters where particular aspects are analyzed in detail: the foundations of the calculus, analytic and differential geometry, approximate integration and conceptions of the integral, types of solutions of differential equations (singular/complete/general integrals, geometrical interpretations, and generality of arbitrary functions), and three aspects related to finite differences and series (the use of subscript indices, types of solutions of finite difference equations, and mixed difference equations); for all these aspects Lacroix's treatment is compared to the 18th-century background, and to his likely sources. Then we examine how the large Traité was adapted to a textbook - the Traité élémentaire, we take a look at the second edition of the large Traité, and conclude the body of the thesis with some final remarks.EThOS - Electronic Theses Online ServiceGBUnited Kingdo