Format: Paperback / softback, 269 pages, height x width: 235x155 mm, 15 Illustrations, color;
24 Illustrations, black and white; XV, 269 p. 39 illus., 15 illus. in color
Series: Mathematics Study Resources 3
Pub. Date: 21-Apr-2023
ISBN-13: 9783662666425
Why is the squaring of the circle, why is the division of angles with compass and ruler impossible? Why are there general solution formulas for polynomial equations of degree 2, 3 and 4, but not for degree 5 or higher? This textbook deals with such classical questions in an elementary way in the context of Galois theory. It thus provides a classical introduction and at the same time deals with applications. The point of view of a constructive mathematician is consistently adopted: To prove the existence of a mathematical object, an algorithmic construction of that object is always given. Some statements are therefore formulated somewhat more cautiously than is classically customary; some proofs are more elaborately conducted, but are clearer and more comprehensible. Abstract theories and definitions are derived from concrete problems and solutions and can thus be better understood and appreciated. The material in this volume can be covered in a one-semester lecture on algebra right at the beginning of mathematics studies and is equally suitable for first-year students at the Bachelor's level and for teachers. The central statements are already summarised and concisely presented within the text, so the reader is encouraged to pause and reflect and can repeat content in a targeted manner. In addition, there is a short summary at the end of each chapter, with which the essential arguments can be comprehended step by step, as well as numerous exercises with an increasing degree of difficulty.
1. introduction.-
2. the fundamental theorem of algebra.-
3. impossibility of squaring the circle.-
4. impossibility of cube doubling and angle division.-
5. on the constructability of regular n-corners.-
6. on the solvability of polynomial equations.- A constructive mathematics.- B linear algebra.- C analysis.
Format: Hardback, 217 pages, height x width: 235x155 mm, 38 Tables, color;
36 Illustrations, color; 9 Illustrations, black and white; XII, 217 p. 45 illus., 36 illus. in color.
Series: Problem Books in Mathematics
Pub. Date: 30-Mar-2023
ISBN-13: 9783031245862
This book is aimed to undergraduate STEM majors and to researchers using ordinary differential equations. It covers a wide range of STEM-oriented differential equation problems that can be solved using computational power series methods. Many examples are illustrated with figures and each chapter ends with discovery/research questions most of which are accessible to undergraduate students, and almost all of which may be extended to graduate level research. Methodologies implemented may also be useful for researchers to solve their differential equations analytically or numerically. The textbook can be used as supplementary for undergraduate coursework, graduate research, and for independent study.
1. Introduction: The Linear ODE: x' = bx + c.
2. Egg 1: The Quadratic ODE: x' = ax2 + bx + c.
3. Egg 2: The First Order Exponent ODE: x' = xr.
4. Egg 3: The First Order Sine ODE: x' = sin x.
5. Egg 4: The Second Order Exponent ODE: x'' = xr.
6. Egg 5: The Second Order Sine ODE - The Single Pendulum.
7. Egg 6: Newton's Method and the Steepest Descent Method.
8. Egg 7: Determining Power Series for Functions through ODEs.
9. Egg 8: The Periodic Planar ODE: x' = y + ax2 + bxy + cy2 ; y' = x + dx2 + exy + fy2.-
10. Egg 9: The Complex Planar Quadratic ODE: z' = az2 + bz + c.-
11. Egg 10: Newton's N-Body Problem.
12. Egg 11: ODEs and Conservation Laws.
13. Egg 12: Delay Differential Equations.-
14. An Overview of Our Dozen ODEs.-
15. Appendix
1. A Review of Maclaurin Polynomials and Power Series.
2. The Dog Rabbit Chasing Problem.
3. A PDE Example: Burgers' Equation.- References.
Format: Hardback, 277 pages, height x width: 240x168 mm, 42 Illustrations,
color; 28 Illustrations, black and white; XII, 277 p. 70 illus., 42 illus. in color.,
Series: Synthesis Lectures on Mathematics & Statistics
Pub. Date: 27-Mar-2023
ISBN-13: 9783031246807
This book is intended for a first-semester course in calculus, which begins by posing a question: how do we model an epidemic mathematically? The authors use this question as a natural motivation for the study of calculus and as a context through which central calculus notions can be understood intuitively. The book’s approach to calculus is contextual and based on the principle that calculus is motivated and elucidated by its relevance to the modeling of various natural phenomena. The authors also approach calculus from a computational perspective, explaining that many natural phenomena require analysis through computer methods. As such, the book also explores some basic programming notions and skills.
1. A Context for Calculus.-
2. The Derivative.-
3. Differential Equations.-
4. Accumulation functions and the integral.-
5. Techniques of Integration.
Format: Paperback / softback, 121 pages, height x width: 235x155 mm, weight: 215 g, 1 Illustrations, black and white; VIII, 121 p. 1 illus.
Series: SpringerBriefs in Statistics
Pub. Date: 04-Feb-2023
ISBN-13: 9789811994852
This book provides a self-contained introduction of mixed-effects models and small area estimation techniques. In particular, it focuses on both introducing classical theory and reviewing the latest methods. First, basic issues of mixed-effects models, such as parameter estimation, random effects prediction, variable selection, and asymptotic theory, are introduced. Standard mixed-effects models used in small area estimation, known as the Fay-Herriot model and the nested error regression model, are then introduced. Both frequentist and Bayesian approaches are given to compute predictors of small area parameters of interest. For measuring uncertainty of the predictors, several methods to calculate mean squared errors and confidence intervals are discussed. Various advanced approaches using mixed-effects models are introduced, from frequentist to Bayesian approaches. This book is helpful for researchers and graduate students in fields requiring data analysis skills as well as in mathematical statistics.
Introduction.- General Mixed-Effects Models and BLUP.- Measuring Uncertainty of Predictors.- Basic mixed-effects Models for Small Area Estimation.- Hypothesis Tests and Variable Selection.- Advanced Theory of Basic Small Area Models.- Small Area Models for Non-normal Response Variables.- Extensions of Basic Small Area Models.
Format: Hardback, 205 pages, height x width: 235x155 mm, VII, 205 p.
Series: PNLDE Subseries in Control 102
Pub. Date: 09-Apr-2023
ISBN-13: 9783031245824
This monograph explores a dual variational formulation of solutions to nonlinear diffusion equations with general nonlinearities as null minimizers of appropriate energy functionals. The author demonstrates how this method can be utilized as a convenient tool for proving the existence of these solutions when others may fail, such as in cases of evolution equations with nonautonomous operators, with low regular data, or with singular diffusion coefficients. By reducing it to a minimization problem, the original problem is transformed into an optimal control problem with a linear state equation. This procedure simplifies the proof of the existence of minimizers and, in particular, the determination of the first-order conditions of optimality. The dual variational formulation is illustrated in the text with specific diffusion equations that have general nonlinearities provided by potentials having various stronger or weaker properties. These equations can represent mathematical models to various real-world physical processes. Inverse problems and optimal control problems are also considered, as this technique is useful in their treatment as well.
Introduction.- Nonlinear Diffusion Equations with Slow and Fast Diffusion.- Weakly Coercive Nonlinear Diffusion Equations.- Nonlinear Diffusion Equations with a Noncoercive Potential.- Nonlinear Parabolic Equations in Divergence Form with Wentzell Boundary Conditions.- A Nonlinear Control Problem in Image Denoising.- An Optimal Control Problem for a Phase Transition Model.- Appendix.- Bibliography.- Index.