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Sample presentation using BeamerDelft University of Technology

Maarten Abbink

April 1, 2014

Outline

1 First SectionSection 1 - Subsection 1Section 1 - Subsection 2Section 1 - Subsection 3

2 Second SectionSection 2 - Subsection 1Section 2 - Last Subsection


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Section 1 - Subsection 1 - Page 1

Example

0 1 2 3 4 5 6 7 8n

0123456u(n)

u(n) = [3, 1, 4]n



Definition

Let n be a discrete variable, i.e. n ∈ Z. A 1-dimensional periodicnumber 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) = −[12 ,

13

]nn2 + 3n − [1, 2]n

=

{−1

3n2 + 3n − 2 if n ≡ 0 (mod 2)

−12n

2 + 3n − 1 if n ≡ 1 (mod 2)

0 1 2 3 4 5 6 7n

−3

−2

−1

0

1

2

3

4

5f(n)=

−[ 1 2,1 3

] nn2+3n

−[1,2] n

−12n2 + 3n− 1

−13n2 + 3n− 2



Definition

A polynomial in a variable x is a linear combination of powers of x :

f (x) =

g∑

i=0

cixi

Definition

A quasi-polynomial in a variable x is a polynomial expression withperiodic numbers as coefficients:

f (n) =

g∑

i=0

ui (n)ni

with ui (n) periodic numbers.


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Example

0 1 2 3 4 5 6 7x

0

1

2

3

4

5

6

7

y

p =3

x + y ≤ p

p f (p)

3 5

4 85 106 13

5

2p +

[−2,−5

2

]

p



Example

0 1 2 3 4 5 6 7x

0

1

2

3

4

5

6

7

y

p =4

x + y ≤ p

p f (p)

3 54 8

5 106 13

5

2p +

[−2,−5

2

]

p



Example

0 1 2 3 4 5 6 7x

0

1

2

3

4

5

6

7

y

p =5

x + y ≤ p

p f (p)

3 54 85 10

6 13

5

2p +

[−2,−5

2

]

p



Example

0 1 2 3 4 5 6 7x

0

1

2

3

4

5

6

7

y

p =6

x + y ≤ p

p f (p)

3 54 85 106 13

5

2p +

[−2,−5

2

]

p



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

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

• Many problems in static program analysis can be expressed aspolyhedral counting problems.


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Section 1 - Subsection 3 - Page 1A picture made with the package TiKz

Example

1


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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 pointNot possible for

1 parametric domains

2 large domains (NP-complete)


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Last Page

Summary

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