By Nagpal, Radhika; Meyer, Albert R

**Read or Download 6.042J / 18.062J Mathematics for Computer Science (SMA 5512), Fall 2002 PDF**

**Similar mathematics books**

**Periodic solutions of nonlinear wave equations with general nonlinearities**

Authored via prime students, this complete, self-contained textual content provides a view of the cutting-edge in multi-dimensional hyperbolic partial differential equations, with a specific emphasis on difficulties during which smooth instruments of research have proved necessary. Ordered in sections of steadily expanding levels of hassle, the textual content first covers linear Cauchy difficulties and linear preliminary boundary worth difficulties, ahead of relocating directly to nonlinear difficulties, together with surprise waves.

**Chinese mathematics competitions and olympiads: 1981-1993**

This booklet includes the issues and suggestions of 2 contests: the chinese language nationwide highschool pageant from 198182 to 199293, and the chinese language Mathematical Olympiad from 198586 to 199293. China has a good checklist within the overseas Mathematical Olympiad, and the ebook comprises the issues which have been used to spot the group applicants and choose the chinese language groups.

- Data About Us: Statistics, Grade 6 (Connected Mathematics 2, Teacher's Guide)
- General Facts about Random Dynamical Systems
- New Trends In Mathematical Physics: In Honour Of The Salvatore Rionero 70th Birthday : Proceedings Of The International Meeting, Naples, Italy 24 - 25 January 2003
- Conference on the Numerical Solution of Differential Equations: Dundee 1973 (Lecture Notes in Mathematics)

**Additional resources for 6.042J / 18.062J Mathematics for Computer Science (SMA 5512), Fall 2002**

**Example text**

2. In every graph, there are an even number of vertices of odd degree. Proof. Partitioning the vertices into those of even degree and those of odd degree, we know � � � d(v) = d(v) + d(v) v∈V d(v) is odd d(v) is even The value of the lefthand side of this equation is even, and the second summand on the righthand side is even since it is entirely a sum of even values. So the ﬁrst summand on the righthand side must also be even. But since it is entirely a sum of odd values, it must must contain an even number of terms.

A partition of a set A is a collection of subsets {A1 , . . , Ak } such that any two of them are disjoint (for any i �= j, Ai ∩ Aj = ∅) and such that their union is A. Let R be an equivalence relation on the set A. For an element a ∈ A, let [a] denote the set {b ∈ A given a ∼R b}. We call this set the equivalence class of a under R. We call a a representative of [a]. 3. The sets [a] for a ∈ A constitute a partition of A. That is, for every a, b ∈ A, either [a] = [b] or [a] ∩ [b] = ∅. Proof. Consider some arbitrary a, b ∈ A.

So (a − c) = (a − b) + (b − c) = (k1 + k2 )m. Therefore m | (a − c). The equivalence class of a is the set [a] = {b ∈ Z | a ≡ b (mod m)}, or {km + a | k ∈ Z}. It turns out that we can extend a lot of standard arithmetic to work modulo m. In fact, we can deﬁne notions of sum and product for the equivalence classes mod m. For example, we deﬁne [a] + [b], given two equivalence classes mod m, to be the equivalence class [a + b]. This is not as obvious as it seems: notice that the result is given in terms of a and b, two selected representatives from the equivalence classes, but that it is supposed to apply to the equivalence classes themselves.