Format: Hardback, 407 pages, height x width: 235x155 mm, 15 Tables, color; 17 Illustrations, color;
100 Illustrations, black and white; XVI, 407 p. 117 illus., 17 illus. in color.
Series: Advances in Mechanics and Mathematics 49
Pub. Date: 13-Sep-2023
ISBN-13: 9783031356544
This monograph presents the geometric foundations of continuum mechanics. An emphasis is placed on increasing the generality and elegance of the theory by scrutinizing the relationship between the physical aspects and the mathematical notions used in its formulation. The theory of uniform fluxes in affine spaces is covered first, followed by the smooth theory on differentiable manifolds, and ends with the non-smooth global theory. Because continuum mechanics provides the theoretical foundations for disciplines like fluid dynamics and stress analysis, the authorfs extension of the theory will enable researchers to better describe the mechanics of modern materials and biological tissues. The global approach to continuum mechanics also enables the formulation and solutions of practical optimization problems.
Foundations of Geometric Continuum Mechanics will be an invaluable resource for researchers in the area, particularly mathematicians, physicists, and engineers interested in the foundational notions of continuum mechanics
1. Introduction.-
2. Prelude: Finite Dimensional Systems.- Part I Algebraic Theory: Uniform Fluxes.-
3. Simplices in Affine Spaces and Their Boundaries.-
4. Uniform Fluxes in Affine Spaces.-
5. From Uniform Fluxes to Exterior Algebra.- Part II: Smooth Theory.-
6. Smooth Analysis on Manifolds: A Short Review.-
7. Interlude: Smooth Distributions of Defects.-
8. Smooth Fluxes.-
9. Frames, Body Points, and Spacetime Structure.-
10. Stresses.-
11. Smooth Electromagnetism on Manifolds.-
12. The Elasticity Problem.-
13. Symmetry and Dynamics.- Part III Non-Smooth, Global Theories.-
14. Banachable Space of Sections of Vector Bundles over Compact Manifolds.-
15. Manifolds of Sections and Embeddings.-
16. The General Framework for Global Analytic Stress Theory.-
17. Dual Spaces Corresponding to Spaces of Differentiable Sections of a Vector Bundle: Localization of Sections and Functionals.-
18. de Rham Currents.-
19. Interlude: Singular Distributions of Defects in Bodies.-
20. Vector-Valued Currents.-
21. The Representation of Forces by Stresses and Hyperstresses.-
22. Simple Forces and Stresses.-
23. Whitney's Geometric Integration Theory and Non-Smooth Bodies.-
24. Optimal Fields and Load Capacity of Bodies.- Index.
Format: Paperback / softback, 193 pages, height x width: 235x155 mm, weight: 320 g, 30 Tables, color;
30 Illustrations, color; 9 Illustrations, black and white; X, 193 p. 39 illus., 30 illus. in color.
Series: Surveys and Tutorials in the Applied Mathematical Sciences 12
Pub. Date: 23-Jul-2023
ISBN-13: 9783031345142
This primer on averaging theorems provides a practical toolbox for applied mathematicians, physicists, and engineers seeking to apply the well-known mathematical theory to real-world problems. With a focus on practical applications, the book introduces new approaches to dissipative and Hamiltonian resonances and approximations on timescales longer than 1/e.
Accessible and clearly written, the book includes numerous examples ranging from elementary to complex, making it an excellent basic reference for anyone interested in the subject. The prerequisites have been kept to a minimum, requiring only a working knowledge of calculus and ordinary and partial differential equations (ODEs and PDEs).
In addition to serving as a valuable reference for practitioners, the book could also be used as a reading guide for a mathematics seminar on averaging methods. Whether you're an engineer, scientist, or mathematician, this book offers a wealth of practical tools and theoretical insights to help you tackle a range of mathematical problems.
1. Introduction. -
2. First Order Periodic Averaging. -
3. Periodic
Solutions. -
4. Second Order Periodic Averaging. -
5. First Order General
Averaging. -
6. Approximations on Timescales Longer than 1/ . -
7. Averaging
over Spatial Variables. -
8. Hamiltonian Resonances. -
9. Quasi-Periodic
Solutions and Tori. -
10. Averaging for Partial Differential Equations.
Format: Paperback / softback, 178 pages, height x width: 240x168 mm, 34 Tables, color; 34 Illustrations, color;
39 Illustrations, black and white; XII, 178 p. 73 illus., 34 illus. in color
Series: Oberwolfach Seminars 51
Pub. Date: 11-Sep-2023
ISBN-13: 9783031355493
These lecture notes are dedicated to the mathematical modelling, analysis and computation of interfaces and free boundary problems appearing in geometry and in various applications, ranging from crystal growth, tumour growth, biological membranes to porous media, two-phase flows, fluid-structure interactions, and shape optimization.
We first give an introduction to classical methods from differential geometry and systematically derive the governing equations from physical principles. Then we will analyse parametric approaches to interface evolution problems and derive numerical methods which will be thoroughly analysed. In addition, implicit descriptions of interfaces such as phase field and level set methods will be analysed. Finally, we will discuss numerical methods for complex interface evolutions and will focus on two phase flow problems as an important example of such evolutions.
Introduction.- Some notions from differential geometry.- Modeling.- Parametric approaches for geometric evolution equations and interfaces.- Implicit approaches for interfaces.- Numerical methods for complex interface evolutions.- Exercises.
Format: Paperback / softback, 263 pages, height x width: 235x155 mm, 4 Illustrations, color;
9 Illustrations, black and white; XV, 263 p. 13 illus., 4 illus. in color.,
Series: UNITEXT 154
Pub. Date: 11-Sep-2023
ISBN-13: 9783031359750
This textbook is devoted to second order linear partial differential equations. The focus is on variational formulations in Hilbert spaces. It contains elliptic equations, including the biharmonic problem, some useful notes on functional analysis, a brief presentation of Sobolev spaces and their properties, some basic results on Fredholm alternative and spectral theory, saddle point problems, parabolic and linear Navier-Stokes equations, and hyperbolic and Maxwell equations. Almost 80 exercises are added, and the complete solution of all of them is included. The work is mainly addressed to students in Mathematics, but also students in Engineering with a good mathematical background should be able to follow the theory presented here. This second edition has been enriched by some new sections and new exercises; in particular, three important equations are now included: the biharmonic equation, the linear Navier-Stokes equations and the Maxwell equations.
1. Introduction.-
2. Second order linear elliptic equations.-
3. A bit of functional analysis.-
4. Weak derivatives and Sobolev spaces.-
5. Weak formulation of elliptic PDEs.-
6. Technical results.-
7. Additional results.-
8. Saddle points problems.-
9. Parabolic PDEs.-
10. Hyperbolic PDEs.- Appendix A: Partition of unity.- Appendix B: Lipschitz continuous and smooth domains.- Appendix C: Integration by parts for smooth functions and vector ?elds.- Appendix D: Reynolds transport theorem.- Appendix E: Gronwall lemma.- Appendix F: Necessary and su?cient conditions for the well-posedness of the variational problem.
Format: Hardback, 382 pages, height x width: 235x155 mm, 3 Illustrations, color;
73 Illustrations, black and white; X, 382 p. 76 illus., 3 illus. in color
Series: Applied Mathematical Sciences 216
Pub. Date: 08-Oct-2023
ISBN-13: 9783031313141
Enlarged, updated, and extensively revised, this second edition illuminates specific problems of nonlinear elasticity, emphasizing the role of nonlinear material response. Opening chapters discuss strings, rods, and shells, and applications of bifurcation theory and the calculus of variations to problems for these bodies. Subsequent chapters cover tensors, three-dimensional continuum mechanics, three-dimensional elasticity , general theories of rods and shells, and dynamical problems. Each chapter includes interesting, challenging, and tractable exercises.
Preface*
Chapter
1. Background*
Chapter
2. The Equations of Motion for Extensible Strings*
Chapter
3. Elementary Problems for Elastic Strings*
Chapter
4. Planar Steady-State Problems for Elastic Rods*
Chapter
5. Introduction to Bifurcation Theory and it's Applications to Elasticity*
Chapter
6. Global Bifurcation Problems for Strings and Rods*
Chapter
7. Variational Methods*
Chapter
8. Theory of Rods Deforming in Space*
Chapter
9. Spatial Problems for Rods*
Chapter
10. Axisymmetric Equilibria of Shells*
Chapter
11. Tensors*
Chapter
12. 3-Dimensional Continuum*
Chapter
13. 3-Dimensional Theory of Nonlinear Elasticity*
Chapter
14. Problems in Nonlinear Elasticity*
Chapter
15. Large-Strain Plasticity*
Chapter
16. General Theories of Rods*
Chapter
17. General Theories of Shells*
Chapter
18. Dynamical Problems*
Chapter
19. Appendix: Topics in Linear Analysis*
Chapter
20. Appendix: Local Nonlinear Analysis*
Chapter
21. Appendix: Degree Theory and it's Applications* References* Index
Format: Hardback, 696 pages, height x width: 235x155 mm, 6 Illustrations, color;
8 Illustrations, black and white; XVIII, 696 p. 14 illus., 6 illus. in color.
Series: Springer Proceedings in Mathematics & Statistics 426
Pub. Date: 05-Oct-2023
ISBN-13: 9783031320088
The goal of the 2019 conference on Stochastic Processes and Algebraic Structures held in SPAS2019, Vasteras, Sweden, from September 30th to October 2nd 2019 was to showcase the frontiers of research in several important topics of mathematics, mathematical statistics, and its applications. The conference has been organized along the following tracks:
1. Stochastic processes and modern statistical methods in theory and practice,
2. Engineering Mathematics,
3. Algebraic Structures and applications.
This book highlights the latest advances in algebraic structures and applications focused on mathematical notions, methods, structures, concepts, problems, algorithms, and computational methods for the natural sciences, engineering, and modern technology. In particular, the book features mathematical methods and models from non-commutative and non-associative algebras and rings associated to generalizations of differential calculus, quantum deformations of algebras, Lie algebras, Lie superalgebras, color Lie algebras, Hom-algebras and their n-ary generalizations, semi-groups and group algebras, non-commutative and non-associative algebras and computational algebra interplay with q-special functions and q-analysis, topology, dynamical systems, representation theory, operator theory and functional analysis, applications of algebraic structures in coding theory, information analysis, geometry and probability theory.
The book gathers selected, high-quality contributed chapters from several large research communities working on modern algebraic structures and their applications. The chapters cover both theory and applications, and are illustrated with a wealth of ideas, theorems, notions, proofs, examples, open problems, and results on the interplay of algebraic structures with other parts of Mathematics. The applications help readers grasp the material, and encourage them to develop new mathematical methods and concepts in their future research. Presenting new methods and results, reviews of cutting-edge research, open problems, and directions for future research, will serve as a source of inspiration for a broad range of researchers and students.
Chapter
1. Adimi, H., Makhlouf, A.: Index of Hom-Lie algebras.
Chapter
2. Amri, H., Makhlouf, A.: On ternary (Hom-)Nambu-Poisson algebras.
Chapter
3. Alekseev, A., Arutyunov, A., Silvestrov, S.: On ( , )-derivations of group
algebra as category characters.
Chapter
4. Armakan, A., Silvestrov, S.:
Color hom-Lie algebras, color hom-Leibniz algebras and color omni-hom-Lie
algebras.
Chapter
5. Armakan, A., Silvestrov, S.: Killing Forms on color
Hom-Lie algebras.
Chapter
6. Bakayoko, I., Silvestrov, S.:
Hom-prealternative superalgebras.
Chapter
7. Ilwoo, C., Jorgensen, P.E.T:
Spectral Analysis of Equations over Quaternions.
Chapter
8. Djinja, D.,
Silvestrov, S., Behakanira Tumwesigye, A.: Multiplication and linear integral
operators on Lp spaces representing polynomial covariant type commutation
relations.
Chapter
9. Djinja, D., Silvestrov, S., Behakanira Tumwesigye, A.:
Representations of polynomial covariant type commutation relations by
piecewise function multiplication and composition operators.
Chapter
10.
Ernst, T.: On generalized q-hyperbolic functions in the spirit of Kapteyn,
with corresponding q-Lie group.
Chapter
11. Hounkonnou, M.N., Houndedji,
G.D., Silvestrov, S.: Double constructions of biHom-Frobenius algebras.-
Chapter
12. Kitouni, A., Silvestrov, S.: On Classification of
(n+1)-dimensional n-Hom-Lie Algebras with nilpotent twisting maps.
Chapter
13. Kitouni, A., Silvestrov, S.: On the Classification of (n+1)-dimensional
n-Hom-Lie Algebras for n=4,5,6 and nilpotent twisting map with 2-dimensional
kernel.
Chapter
14. Langlois-Remillard, A.: Deforming algebras with
anti-involution via twisted associativity.
Chapter
15. Muhumuza, A.K.,
Mango, J.M., Kakuba, G., Lundengard, K., Malyarenko, A., Silvestrov, S.: The
Wishart Distribution on Symmetric cones.
Chapter
16. Muhumuza, A.K.,
Silvestrov, S.: Symmetric Group Properties of Extreme Points of Vandermonde.-
Chapter
17. Tumwesigye, A.B., Silvestrov, S.: Commutants in Crossed Products
for Piecewise Constant Function Algebras Related to Multiresolution
Analysis.
Chapter
18. Gomez-Olvera, M.D., Lopez-Ramos, J.A., Torrecillas,
B.: Secure Group Communications using Twisted Group Rings.
Chapter
19.
Prakasj, O., Islam, H., Verma, R.K.: Constacyclic and skew constacyclic codes
over a finite commutative non-chain ring.
Chapter
20. Waweru, D.K., Maingi,
D.M: Two-Sided Noncommutative Groebner Basis on Quiver Algebras.
Chapter
21.
Silvestrov, S., Rajkovic, P.M., Marinkovic, S.D.: Wallis type formula and a
few versions of the number p in q-calculus.
Chapter
22. Silvestrov, S.,
Rajkovic, P.M.: On the q-Rodrigues formula - New Form for Fast Computing.-
Chapter
23. Silvestrov, S., Zargeh, C.: HNN-extension of involutive
multiplicative Hom-Lie algebras.
Chapter
24. Tanana, B., Cassy, B.,
Silvestrov, S.: About compact monothetic semirings and compact monothetic
C-semirings.
Format: Paperback / softback, 126 pages, height x width: 235x155 mm, weight: 219 g, VIII, 126 p.
Series: Lecture Notes in Mathematics 2336
Pub. Date: 01-Aug-2023
ISBN-13: 9783031368561
This book focuses on the study of the volume of vector fields on Riemannian manifolds. Providing a thorough overview of research on vector fields defining minimal submanifolds, and on the existence and characterization of volume minimizers, it includes proofs of the most significant results obtained since the subjectfs introduction in 1986. Aiming to inspire further research, it also highlights a selection of intriguing open problems, and exhibits some previously unpublished results. The presentation is direct and deviates substantially from the usual approaches found in the literature, requiring a significant revision of definitions, statements, and proofs.
A wide range of topics is covered, including: a discussion on the conditions for a vector field on a Riemannian manifold to determine a minimal submanifold within its tangent bundle with the Sasaki metric; numerous examples of minimal vector fields (including those of constant length on punctured spheres); a thorough analysis of Hopf vector fields on odd-dimensional spheres and their quotients; and a description of volume-minimizing vector fields of constant length on spherical space forms of dimension three.
Each chapter concludes with an up-to-date survey which offers supplementary information and provides valuable insights into the material, enhancing the reader's understanding of the subject. Requiring a solid understanding of the fundamental concepts of Riemannian geometry, the book will be useful for researchers and PhD students with an interest in geometric analysis.
1. Introduction. -
2. Minimal Sections of Tensor Bundles. -
3. Minimal
Vector Fields of Constant Length on the Odd-Dimensional Spheres. -
4. Vector
Fields of Constant Length of Minimum Volume on the Odd-Dimensional Spherical
Space Forms. -
5. Vector Fields of Constant Length on Punctured Spheres.