The Science of Learning Mathematical Proofs

類似書籍推薦給您

College students struggle with the switch from thinking of mathematics as a calculation based subject to a problem solving based subject. This book describes how the introduction to proofs course can be taught in a way that gently introduces students to this new way of thinking. This introduction utilizes recent research in neuroscience regarding how the brain learns best. Rather than jumping right into proofs, students are first taught how to change their mindset about learning, how to persevere through difficult problems, how to work successfully in a group, and how to reflect on their learning. With these tools in place, students then learn logic and problem solving as a further foundation. Next various proof techniques such as direct proofs, proof by contraposition, proof by contradiction, and mathematical induction are introduced. These proof techniques are introduced using the context of number theory. The last chapter uses Calculus as a way for students to apply the proof techniques they have learned. Request Inspection Copy Contents: Preface to Students Preface to Professors Pedagogical Notes for Professors Brain Growth Team Building Setting Goals Logic Problem Solving Study Techniques Pre-proofs Direct Proofs (Even, Odd, & Divides) Direct Proofs (Rational, Prime, & Composite) Direct Proofs (Square Numbers & Absolute Value) Direct Proofs (GCD & Relatively Prime) Proof by Division into Cases Proof by Division into Cases (Quotient Remainder Theorem) Forward-Backward Proofs Proof by Contraposition Proof by Contradiction Proof by Induction Proof by Induction Part II Calculus Proofs Mixed Review Appendices: 100# Task Activity Sheet Answers for Hiking Activity Escape Room Proof for Exercise 17.11 Selected Proofs from all Chapters Proof Methods Proof Template Homework Log Bibliography Index

The Mathematics of Open Quantum Systems

類似書籍推薦給您

This book presents new developments in the open quantum systems theory with emphasis on applications to the (frequent) measurement theory. In the first part of the book, the uniqueness theorems for the solutions to the restricted Weyl commutation relations braiding unitary groups and semi-groups of contractions are discussed. The major theme involves an intrinsic characterization of the simplest symmetric operator solutions to the Heisenberg uncertainty relations, the problem posed by Jørgensen and Muhly, followed by the proof of the uniqueness theorems for the simplest solutions to the restricted Weyl commutation relations. The detailed study of unitary invariants of the corresponding dissipative and symmetric operators opens up a look at the classical Stone-von Neumann uniqueness theorem from a new angle and provides an extended version of the uniqueness result relating various realizations of a differentiation operator on the corresponding metric graphs. The second part of the book is devoted to mathematical problems of the quantum measurements under continuous monitoring. Among the topics discussed are the complementarity of the Quantum Zeno effect and Exponential Decay scenario in frequent quantum measurements, and a rigorous treatment, within continuous monitoring paradigm, of the celebrated "double-slit experiment" where the renowned exclusive and interference measurement alternatives approach in quantum theory is presented in a way that is accessible for mathematicians. One of the striking applications of the generalized (1-stable) central limit theorem is the mathematical evidence of exponential decay of unstable states of the quantum pendulum under continuous monitoring. Sample Chapter(s) Preface Chapter 1: INTRODUCTION Contents: Acknowledgments Preface Representations of Operator Commutation Relations: Introduction Preliminaries and Basic Definitions The Commutation Relations and Character-Automorphic Functions The Differentiation Operator on Metric Graphs The Magnetic Hamiltonian The Livšic Function s(Ḋ,DΘ)(z) The Weyl-Titchmarsh Function M(Ḋ,DΘ)(z) The Model Dissipative Operators The Characteristic Function S(Ḋ,Ḓ,DΘ)(z) The Transmission Coefficient and the Characteristic Function Uniqueness Results Dissipative Solutions to the CCR Main Results Unitary Dynamics on the Full Graph Continuous Monitoring, Quantum Measurements: Continuous Monitoring of the Quantum Systems The Quantum Zeno versus Anti-zeno Effect Alternative The Quantum Zeno Effect versus Exponential Decay Alternative Preliminaries: Probabilities versus Amplitudes Massless Particles on a Ring Continuous Monitoring with Interference Continuous Monitoring with No Interference The Self-adjoint Dilation General Open Quantum Systems on a Ring Operator Coupling Limit Theorems Appendices: The Characteristic Function for Rank-One Perturbations Prime Symmetric Operators A Functional Model of a Triple The Spectral Analysis of the Model Dissipative Operator Transformation Laws The Invariance Principle The Operator Coupling and the Multiplication Theorem Stable Laws References Index Readership: Graduates and Researchers interested in mathematical physics, operator theory, quantum physics.