Sample presentation using Beamer Delft University of Technology

Transcription

1 Sample presentation using Beamer Delft University of Technology Maarten Abbink October 10, 2014 Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

2 Outline 1 First Section Section 1 - Subsection 1 Section 1 - Subsection 2 Section 1 - Subsection 3 2 Second Section Section 2 - Subsection 1 Section 2 - Last Subsection Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

3 Next Subsection 1 First Section Section 1 - Subsection 1 Section 1 - Subsection 2 Section 1 - Subsection 3 2 Second Section Section 2 - Subsection 1 Section 2 - Last Subsection Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

4 Section 1 - Subsection 1 - Page 1 Example u(n) n u(n) = [3, 1, 4] n Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

5 Section 1 - Subsection 1 - Page 2 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 0 if n 0 (mod d) u 1 if n 1 (mod d) u(n) = [u 0, u 1,..., u d 1 ] n =. u d 1 if n d 1 (mod d) d is called the period. Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

6 Section 1 - Subsection 1 - Page 3 Example f (n) = [ 1 2, 1 ] { 3 n n2 + 3n [1, 2] n 1 = 3 n2 + 3n 2 if n 0 (mod 2) 1 2 n2 + 3n 1 if n 1 (mod 2) n2 + 3n 2 ] n n2 + 3n [1, 2]n f(n) = [ 1, n n2 + 3n 1 3 Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

7 Section 1 - Subsection 1 - Page 4 Definition A polynomial in a variable x is a linear combination of powers of x: f (x) = g c i x i i=0 Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

8 Section 1 - Subsection 1 - Page 4 Definition A polynomial in a variable x is a linear combination of powers of x: f (x) = g c i x i i=0 Definition A quasi-polynomial in a variable x is a polynomial expression with periodic numbers as coefficients: f (n) = g u i (n)n i i=0 with u i (n) periodic numbers. Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

9 Next Subsection 1 First Section Section 1 - Subsection 1 Section 1 - Subsection 2 Section 1 - Subsection 3 2 Second Section Section 2 - Subsection 1 Section 2 - Last Subsection Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

10 Section 1 - Subsection 2 - Page 1 Example y x p =3 x + y p p f (p) 3 5 Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

11 Section 1 - Subsection 2 - Page 1 Example y x p =4 x + y p p f (p) Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

12 Section 1 - Subsection 2 - Page 1 Example y x p =5 x + y p p f (p) Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

13 Section 1 - Subsection 2 - Page 1 Example y x p =6 x + y p p f (p) Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

14 Section 1 - Subsection 2 - Page 1 Example y x p =6 x + y p p f (p) [ 5 2 p + 2, 5 ] 2 p Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

15 Section 1 - Subsection 2 - Page 2 The number of integer points in a parametric polytope P p of dimension n is expressed as a piecewise a quasi-polynomial of degree n in p (Clauss and Loechner). Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

16 Section 1 - Subsection 2 - Page 2 The number of integer points in a parametric polytope P p 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). Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

17 Section 1 - Subsection 2 - Page 2 The number of integer points in a parametric polytope P p 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). Many problems in static program analysis can be expressed as polyhedral counting problems. Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

18 Next Subsection 1 First Section Section 1 - Subsection 1 Section 1 - Subsection 2 Section 1 - Subsection 3 2 Second Section Section 2 - Subsection 1 Section 2 - Last Subsection Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

19 Section 1 - Subsection 3 - Page 1 A picture made with the package TiKz Example Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

20 Next Subsection 1 First Section Section 1 - Subsection 1 Section 1 - Subsection 2 Section 1 - Subsection 3 2 Second Section Section 2 - Subsection 1 Section 2 - Last Subsection Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

21 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 Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

22 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) Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

23 Next Subsection 1 First Section Section 1 - Subsection 1 Section 1 - Subsection 2 Section 1 - Subsection 3 2 Second Section Section 2 - Subsection 1 Section 2 - Last Subsection Maarten Abbink (TU Delft) Beamer Sample October 10, / 16

24 Last Page Summary End of the beamer demo with a tidy TU Delft lay-out. Thank you! Maarten Abbink (TU Delft) Beamer Sample October 10, / 16