1 The How, When, and Why of Mathematics Spotlight: George Polya Tips on Doing Homework 2 Logically Speaking 3 Introducing the Contrapositive and Converse 4 Set Notation and Quantifiers Tips on Quantification 5 Proof Techniques Tips on Definitions …… 10 Relations Tips on Reading Mathematics 11 Partitions Tips on Putting It All Together 12 Order in the Reals 13 Consequences of the Completeness of R Tips: You Solved ItNow What? 14 Functions, Domain, and Range Spotlight: The Definition of Function 15 Functions, One-to-One, and Onto 16 Inverses 17 Images and Inverse Images Spotlight: Minimum or Infimum? 18 Mathematical Induction 19 Sequences 20 Convergence of Sequences of Real Numbers 21 Equivalent Sets 22 Finite Sets and an Infinite Set 23 Countable and Uncountable Sets 24 The Cantor–Schroder–Bernstein Theorem Spotlight: The Continuum Hypothesis ……

 What is image segmentation? Separation of image domain based on contents.  General image segmentation: ENERGY METHOD Mumford-Shah (MS) Model CURVE EVOLUTION Snake, Geodesic Active-Contour, Chan-Vese Model, Level Set Method BRAIN ANEURYSM SEGMENTATION ENERGY METHOD - REVISITED Approximations of Mumford-Shah Model

Problem set 1 Solutions Due: Monday February 7 at 9 PM Problem 1. The connectives A(and), V(or), and =(implies)come often not only in com uter programs, but also everyday speech. But devices that compute the nand operation

Notes for recitation 5 1 Well-ordering principle Every non-empty set of natural numbers has a minimum element Do you believe this statement? Seems obvious, right? Well, it is. But dont fail to realize how tight it is. Crucially, it talks about a non-empty set -otherwise, it would clearly be false. And it also talks about natural

In this example, the domain is the set fa,b, c, d, el and the range is the set Y= (1, 2, 3, 4, 5/. Related elements are joined by an arrow. This relation is a function because every element on the left is related to exactly one element on the right. In graph-theoretic

Dynamic order statistics OS-SELECT(i, S): returns the ith smallest element in the dynamic set S. OS-RANK(x, S): returns the rank of x ∈ S in the sorted order of S’s elements. IDEA: Use a red-black tree for the set S, but keep subtree sizes in the nodes

G( V, E ) where G ::= graph, V = V( G ) ::= finite nonempty set of vertices, and E = E( G ) ::= finite set of edges

1 Equivalence Relations 【Definition】A relation R is defined on a set S if for every pair of elements (a, b), a, b S, a R b is either true or false. If a R b is true, then we say that a is related to b

16 选择 FLOTRAN 各自由度相应的求解器 命令：FLDATA18, METH, Lable, Value 功能：选择每个自由度的解算方法 菜单：Main Menu>Preprocessor>FLOTRAN Set Up>CFD Solver Controls>VX Solver CFD Main Menu>Preprocessor>FLOTRAN Set Up>CFD Solver Controls>VY Solver CFD

Production requires inputs — factors of production — for example, labour and capital equipment. A production set is the set of outputs feasible given a particular combination of inputs. The production function