1 Gradient and Hamiltonian systems


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  • exposé
1 Gradient and Hamiltonian systems 1.1 Gradient systems These are quite special systems of ODEs, Hamiltonian ones arising in conserva- tive classical mechanics, and gradient systems, in some ways related to them, arise in a number of applications. They are certainly nongeneric, but in view of their origin, they are common. A system of the form X ′ = −∇V (X) (1) where V : Rn → R is, say, C∞, is called, for obvious reasons, a gradient system.
  • various constants of motion of the approximation as candidates for lyapunov functions
  • x∗
  • initial condition x.
  • phase portrait
  • lyapunov function
  • taylor approximation
  • coordinates
  • point
  • system



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Nombre de lectures 23
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Philosophy 150: Introduction to Logic
Dave Barker-Plummer (dbp@stanford.edu) Cordura 213 (723-9030)
Welcome toIntroduction to Logic. In this course you will be introduced to the concepts and techniques used in mathematical logic. We will start right from the beginning, assuming no prior exposure to this or similar material, and progress through discussions of the proof and model theories of propositional and first-order logic.
The ability to reason is fundamental to human beings. Whatever the discipline or discourse it is important to be able to distinguish correct reasoning from incorrect reasoning. The consequences of incorrect reasoning can be minor, like getting lost on the way to a birthday party, or more significant, for example launching nuclear missiles at a flock of ducks, or permanently losing contact with a space craft.
The fundamental question that we will address in this course is "when does one statement necessarily follow from another" --- or in the terminology of the course, "when is one statement alogical consequenceof another". This is an issue of some importance, since an answer to the question would allow us to examine an argument presented in an article, for example, and to decide whether it really demonstrates the truth of the conclusion of the argument. Our own reasoning might also improve, since we would also be able to analyze our own arguments to see whether they really do demonstrate their conclusions.
We will proceed by giving a theory of truth, and of logical consequence, based on a formal language called FOL (the language of First-Order Logic). We adopt a formal language for making statements, since natural languages (like English, for example) are far to vague and ambiguous for us to analyze sufficiently. Armed with the formal language, we will be able to model the notions of truth, proof and consequence, among others.
While mathematical logic is technical in nature, the key concepts in the course will be developed by considering natural English statements, and we will focus the relationships between such statements and their FOL counterparts. The goal of the course is to show how natural English statements and arguments can be formalized and analyzed for correctness and truthfullness for example.
Course Outline
The course will divided into three sections:
1. PropositionalLogic (Part I of the text) 1. Atomicsentences 2. Booleanconnectives 3. Conditionals This section develops a simple theory of truth and proof that we will later extend to full first-order logic. The focus here is on simple English statements involving words likeand,or,notand constructions likeif...then...and...unless....We will also examine arguments whose validity depends on the meanings of these words and introduce rules for determining which arguments are valid and which invalid. 2. First-OrderLogic (Part II of the text) 1. Quantification 2. Quantifieralternation and scope,