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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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Next Subsection
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
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Section 1 - Subsection 1 - Page 2
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.
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Section 1 - Subsection 1 - Page 3
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
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Section 1 - Subsection 1 - Page 4
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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Section 1 - Subsection 1 - Page 4
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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Next Subsection
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 2 - Page 1
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
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Section 1 - Subsection 2 - Page 1
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
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Section 1 - Subsection 2 - Page 1
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
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Section 1 - Subsection 2 - Page 1
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
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Section 1 - Subsection 2 - Page 1
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
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Section 1 - Subsection 2 - Page 2
• 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 2 - Page 2
• 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 2 - Page 2
• 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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Next Subsection
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 3 - Page 1A picture made with the package TiKz
Example
1
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Next Subsection
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 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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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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Next Subsection
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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Last Page
Summary
End of the beamer demowith a tidy TU Delft lay-out.
Thank you!
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