Denition 3.16 Inverse of a Power Series. Let
f(x)=anxn from 0 to be a formal power series.
Then g(x) = bnxn from 0 to , such that f(x)g(x)=1
we write 1/f(x)=f-1(x)
Denition 3.1 a recurrence relation
is a sequence (an)n0 together with
a relations an= f(an-
Walk - in a graph in G is a sequence of vertices v1,v2,
Trees - A tree is a connected graph
with no cycles.
Denition 3.1 A graph G =
(V(G),E(G) is a set of V(G) of
vertices together with a set E(G) of
edges, where E(G) is a subset of:
Corollary 3.3 In any
if A and B are not disjoint sets, then
we obtain |AuB| = |A| + |B| - |AnB|
Inclusion-Exclusion Principle:
Binomial Theorem
Binomial Theorem: For all
(complex) numbers a,b, and for
all natura; numbers n, we have :
Multinomial Number
Multinomial Theorem - f
Theorem 4.5 A code C is d-errordetecting if and only if (C)d+1
Error Detection and Error
Correction
Theorem 4.11 Let H be a binary matrix in which no column consists
entirely of zeros, and in which no two columns are the same. Then
the code C with H as th
repetition is allowed: n x n x n .
r times, so nr
Given n distinct objects, how many
ways are there to choose r of these
objects when the order they are
chose in is important and:
Theorem 1.12 For every n and r,
repetition is not allowed: (n)r = n x
Addit
The labels are colours
Denition 3.23 A k-colouring of a graph
G=(V,E) is a labelling f:(V->cfw_1,2,k
Lemma 3.25 For every n-vertex graph
G, (G) (G) and (G) n/(G)
The vertices of one colour form a
colour class
The chromatic number (G) is the
least k such t
solve for x1 and x2 explicitly as
functions of t, and then eliminate t.
Analytic Method:
Theorem 2.5.3 Jordan Normal Curve A simple closed curve divides the
plane into two regions, one of which is bounded, and the other is unbounded.
solve the dierential
Theorem 5.3.3 T = Rn i rank [B|AB|An-1B] = n
Basic objects of study in Control
Theory are undetermined
dierential equations.
Denition - The system above is said to be controllable at time T if for
every pair of vectors x0, x1 in Rn, there exists a control
system of ordinary dierential
equations
1. Every solution of x=Ax tends to zero as t-> i for
all in (A), Re()<0. Moreover, in this case, the
solutions converge exponentially to 0: there exists a
>0 and M>0 such that for all t0, |x(t)|Me-t|x(0)|
considerin
Corollary 3.4.1 The equilibrium point 0 of the system x=Ax is
exponentially stable if for all in (), Re()<0. The equilibrium point 0 of
the system x=Ax is unstable if there exists a in (A) such that Re()>0.
Denition: Let A be in Rnxn, and let be in (). Th
Theorem 4.1.1 consider the dierential equation
x=f(x,t), x(t0)=C
Theorem 4.6.1 Let f be globally lipschitz in x (with constant L) uniformly
in t. Let x1,x2 be solutions to the equation x = f(x,t), for t in [t0,t1], with
where f:R2->R is such that there ex
Derivative of Maps from Rn to
Rm
Denition 5.26
Lemma 5.27 Let U Rm be open, c in U and f:U->Rm be
dierentiable at c. Then the derivative of f at c is unique
Proposition 5.22 Let fn:(a,b)->R be a sequence of dierentiable functions on (a,b), such that there
Denition 4.57 Let (X,dX),(Y,dY) be metric spaces. f:X->Y is said to
be uniformly continuous if for every >0, there exists a >0 such
that for all x,y in X satisfying dX(x,y)<, there holds that dY(f(x),f(y)<
Corollary 4.20 Let (X,dX),(Y,dY) be metric spaces
The series an 1-> is called
Deniton 3.47: Let (an)n in N be a
convergent if (sn)n in N converges
Denition 3.1: Let (an)n in N be a
sequence in a normed space (X,|.|).
The sequence (sn)n in N of partial
in (X,|.|). Then we write
an 1-> = lim(sn) n->
sequen
D1 (positive deniteness) - For all
x,y in X, d(x,y)0. For all x in X,
d(x,x)=0. if x,y in X are such that
d(x,y)=0, then x=y.
N1 (positive deniteness) - For all x
in X, |x|0. If x in X is such that |x|=0,
then x=0 (the zero vector in X)
Denition 1.9 a Nor
Denition 2.43 Let (X,d) be a metric space. A subset K of X is said to be
compact if every sequence in K has a convergent subsequence with limit
in K, that is, if (xn)n in N is a sequence such that xn in K for n in N, then
Lemma 2.50 Every bounded sequence