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This monograph summarizes and extends a number of results on the topology of trigonal curves in geometrically ruled surfaces. An emphasis is given to various applications of the theory to a few related areas, most notably singular plane curves of small degree, elliptic surfaces, and Lefschetz fibrations (both complex and real), and Hurwitz equivalence of braid monodromy factorizations. The approach relies on a close relation between trigonal curves/elliptic surfaces, a certain class of ribbon graphs, and subgroups of the modular group, which provides a combinatorial framework for the study of geometric objects. A brief summary of the necessary auxiliary results and techniques used and a background of the principal problems dealt with are included in the text. The book is intended to researchers and graduate students in the field of topology of complex and real algebraic varieties.

Curves, Plane. --- Topological degree. --- Curves, Plane --- Topological degree --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Degree, Topological --- Degree (Topology) --- Degree theory --- Algebraic topology --- Higher plane curves --- Plane curves --- Braid Monodromy. --- Dessin d’Enfant. --- Elliptic Surface. --- Fundamental Group. --- Lefschetz Fibration. --- Modular Group. --- Monodromy Factorization. --- Plane Sextic. --- Real Variety. --- Trigonal Curve.

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This monograph aims to give a self-contained introduction into the whole field of topological analysis: Requiring essentially only basic knowledge of elementary calculus and linear algebra, it provides all required background from topology, analysis, linear and nonlinear functional analysis, and multivalued maps, containing even basic topics like separation axioms, inverse and implicit function theorems, the Hahn-Banach theorem, Banach manifolds, or the most important concepts of continuity of multivalued maps. Thus, it can be used as additional material in basic courses on such topics. The main intention, however, is to provide also additional information on some fine points which are usually not discussed in such introductory courses. The selection of the topics is mainly motivated by the requirements for degree theory which is presented in various variants, starting from the elementary Brouwer degree (in Euclidean spaces and on manifolds) with several of its famous classical consequences, up to a general degree theory for function triples which applies for a large class of problems in a natural manner. Although it has been known to specialists that, in principle, such a general degree theory must exist, this is the first monograph in which the corresponding theory is developed in detail.

Topological degree. --- Topological spaces. --- Fredholm operators. --- Algebraic topology. --- Topology --- Operators, Fredholm --- Linear operators --- Spaces, Topological --- Degree, Topological --- Degree (Topology) --- Degree theory --- Algebraic topology --- Analysis. --- Banach Manifold. --- Fredholm. --- Hahn-Banach Theorem. --- Implicit Function Theorem. --- Inverse Function Theorem. --- Linear Functional Analysis. --- Multivalued Map. --- Nonlinear Functional Analysis. --- Nonlinear Inclusion. --- Separation Axiom. --- Topology. --- Triple Degree.

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This book discusses current challenges related to teaching geography, mainly at the secondary school and higher education level. Focusing on a range of current topics, different methods, techniques, materials, applications, and approaches to geography education with a regional Central European perspective, the book makes an original contribution to the field. Most of the chapters aims at the practical development of the themes such as geography curriculum (Part I), global education, inquiry-based education, project-based learning, case studies, powerful teaching (Part II), using of information and communication technologies (Part III) in geography teaching. The final part (Part IV) covers some geopolitical, and socio-geographical aspects of the aforementioned Central European former communist countries from the point of view how to teach them with various methods. Therefore, the book can appeal to many geography or science students, researchers and educators studying geography education around the world.

Nonlinear boundary value problems. --- Functional differential equations. --- Topological degree. --- Degree, Topological --- Degree (Topology) --- Degree theory --- Differential equations, Functional --- Curriculums (Courses of study). --- Science Education. --- Curriculum Studies. --- Teaching and Teacher Education. --- Education. --- Education --- Science education. --- Teaching. --- Curricula. --- Algebraic topology --- Differential equations --- Functional equations --- Boundary value problems --- Differential equations, Nonlinear --- Science --- Curriculum planning. --- Study and teaching. --- Curriculum development --- Instructional systems --- Planning --- Science education --- Scientific education --- Curricula --- Design --- Education—Curricula. --- Didactics --- Instruction --- Pedagogy --- School teaching --- Schoolteaching --- Pedagogical content knowledge --- Training --- Core curriculum --- Courses of study --- Curricula (Courses of study) --- Curriculums (Courses of study) --- Schools --- Study, Courses of --- Educació -- Currículums --- Ciència -- Ensenyament