Presentation Simplex

7/23/2019 Presentation Simplex

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Maarten Abbink (TU Delft) Beamer Sample April 1, 2014 1 / 16

Sample presentation using BeamerDelft University of Technology

Maarten Abbink

April 1, 2014


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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



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Section 1 - Subsection 1 - Page 1

Example

0 1 2 3 4 5 6 7 8n

0123

456u(n)

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

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DefinitionLet 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, . . . , ud1]n =

u0 ifn 0 (mod d)

u1 ifn 1 (mod d)...

ud1 ifn d 1 (mod d)

d is called the period.

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Example

f(n) =

12 , 13nn2 + 3n [1, 2]n

=

13 n

2 + 3n 2 ifn 0 (mod 2)

12 n2 + 3n 1 ifn 1 (mod 2)

0 1 2 3 4 5 6 7n

3

2

1

0

1

2

3

4

5

f(n)=

1 2

,1 3

n

n2

+

3n

[1,

2]n

1

2n2 + 3n 1

1

3n2 + 3n 2

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Definition

A polynomial in a variable xis a linear combination of powers ofx:

f(x) =

gi=0

cixi

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Definition

A polynomial in a variable xis a linear combination of powers ofx:

f(x) =

gi=0

cixi

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

f(n) =

g

i=0 u

i(n)n

i

with ui(n) periodic numbers.

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Next Subsection



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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+yp

p f(p)

3 5

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Example

0 1 2 3 4 5 6 7x

0

1

2

3

4

5

6

7

y

p =4

x+yp

p f(p)

3 54 8

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Example

0 1 2 3 4 5 6 7x

0

1

2

3

4

5

6

7

y

p =5

x+yp

p f(p)

3 54 85 10

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Example

0 1 2 3 4 5 6 7x

0

1

2

3

4

5

6

7

y

p =6

x+yp

p f(p)

3 54 85 106 13

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Example

0 1 2 3 4 5 6 7x

0

1

2

3

4

5

6

7

y

p =6

x+yp

p f(p)

3 54 85 106 13

5

2p+

2,

5

2

p

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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).

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More generalpolyhedral counting problems:

Systems of linear inequalities combined with ,,,,or (Presburger formulas).

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More generalpolyhedral counting problems:

Systems of linear inequalities combined with ,,,,or (Presburger formulas).

Many problems instatic program analysiscan be expressed aspolyhedral counting problems.


S
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Next Subsection




S i 1 S b i 3 P 1
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Section 1 - Subsection 3 - Page 1A picture made with the package TiKz

Example


N t S b ti
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Next Subsection




S ti 2 b ti 1 1
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Section 2 - subsection 1 - page 1

AlertblockThis page gives an example with numbered bullets (enumerate)in an Example window:

Example

Discrete domain evaluate in each pointNot possible for

1 parametric domains


S ti 2 b ti 1 1
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Section 2 - subsection 1 - page 1

AlertblockThis 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)


Ne t S b e ti
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Next Subsection




Last Page
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Last Page

Summary

End of the beamer demowith a tidyTU Delft lay-out.

Thank you!

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Presentation Simplex