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Maarten Abbink (TU Delft) Beamer Sample April 1, 2014 1 / 16
Sample presentation using Beamer
Delft University of Technology
Maarten Abbink
April 1, 2014

Outline
1 First Section
Section 1 - Subsection 1
2 Second Section
Section 2 - Last Subsection

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Section 1 - Subsection 1 - Page 1
Example
0 1 2 3 4 5 6 7 8
n
0
1
2
3
4
5
6u(n)
u(n) = [3, 1, 4]n

Deﬁnition
Let n be a discrete variable, i.e. n ∈ Z. A 1-dimensional periodic
number is a function that depends periodically on n.
u(n) = [u0, u1, . . . , ud−1]n =



u0 if n ≡ 0 (mod d)
u1 if n ≡ 1 (mod d)
...
ud−1 if n ≡ d − 1 (mod d)
d is called the period.

Example
f (n) = − 1
2, 1
3 n
n2 + 3n − [1, 2]n
=
−1
3n2 + 3n − 2 if n ≡ 0 (mod 2)
−1
2n2 + 3n − 1 if n ≡ 1 (mod 2)
0 1 2 3 4 5 6 7
n
−3
−2
−1
0
1
2
3
4
5f(n)=−1
2
,1
3n
n2
+3n−[1,2]n
−1
2
n2
+ 3n − 1
−1
3
n2
+ 3n − 2

Deﬁnition
A polynomial in a variable x is a linear combination of powers of x:
f (x) =
g
i=0
ci xi

Definition
A polynomial in a variable x is a linear combination of powers of x:
f (x) =
g
i=0
ci xi
Definition
A quasi-polynomial in a variable x is a polynomial expression with
periodic numbers as coefficients:
f (n) =
g
i=0
ui (n)ni
with ui (n) periodic numbers.

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1 First Section
2 Second Section

Example
0 1 2 3 4 5 6 7
x
0
1
2
3
4
5
6
7
y
p =3
x + y ≤ p
p f (p)
3 5

Example
0 1 2 3 4 5 6 7
x
0
1
2
3
4
5
6
7
y
p =4
x + y ≤ p
p f (p)
3 5
4 8

Example
0 1 2 3 4 5 6 7
x
0
1
2
3
4
5
6
7
y
p =5
x + y ≤ p
p f (p)
3 5
4 8
5 10

Example
0 1 2 3 4 5 6 7
x
0
1
2
3
4
5
6
7
y
p =6
x + y ≤ p
p f (p)
3 5
4 8
5 10
6 13

Example
0 1 2 3 4 5 6 7
x
0
1
2
3
4
5
6
7
y
p =6
x + y ≤ p
p f (p)
3 5
4 8
5 10
6 13
5
2
p + −2,
−5
2 p

• The number of integer points in a parametric polytope Pp of
dimension n is expressed as a piecewise a quasi-polynomial of
degree n in p (Clauss and Loechner).

• More general polyhedral counting problems:
Systems of linear inequalities combined with ∨, ∧, ¬, ∀, or ∃
(Presburger formulas).

• More general polyhedral counting problems:
Systems of linear inequalities combined with ∨, ∧, ¬, ∀, or ∃
(Presburger formulas).
• Many problems in static program analysis can be expressed as
polyhedral counting problems.

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A picture made with the package TiKz
Example

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Section 2 - subsection 1 - page 1
Alertblock
This page gives an example with numbered bullets (enumerate)
in an ”Example” window:
Example
Discrete domain ⇒ evaluate in each point
Not possible for
1 parametric domains

Section 2 - subsection 1 - page 1
Alertblock
This page gives an example with numbered bullets (enumerate)
in an ”Example” window:
Example
Discrete domain ⇒ evaluate in each point
Not possible for
1 parametric domains
2 large domains (NP-complete)

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