Sample presentation using Beamer Delft University of Technology

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

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, 2014 2 / 16

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, 2014 3 / 16

Section 1 - Subsection 1 - Page 1 Example u(n) 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 n u(n) = [3, 1, 4] n Maarten Abbink (TU Delft) Beamer Sample October 10, 2014 4 / 16

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, 2014 5 / 16

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) 5 1 3 n2 + 3n 2 ] n n2 + 3n [1, 2]n f(n) = [ 1, 1 2 3 4 3 2 1 0 0 1 2 3 4 5 6 7 n 1 2 1 2 n2 + 3n 1 3 Maarten Abbink (TU Delft) Beamer Sample October 10, 2014 6 / 16

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, 2014 7 / 16

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, 2014 7 / 16

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, 2014 8 / 16

Section 1 - Subsection 2 - Page 1 Example 7 6 5 y 4 3 2 1 0 0 1 2 3 4 5 6 7 x p =3 x + y p p f (p) 3 5 Maarten Abbink (TU Delft) Beamer Sample October 10, 2014 9 / 16

Section 1 - Subsection 2 - Page 1 Example 7 6 5 y 4 3 2 1 0 0 1 2 3 4 5 6 7 x p =4 x + y p p f (p) 3 5 4 8 Maarten Abbink (TU Delft) Beamer Sample October 10, 2014 9 / 16

Section 1 - Subsection 2 - Page 1 Example 7 6 5 y 4 3 2 1 0 0 1 2 3 4 5 6 7 x p =5 x + y p p f (p) 3 5 4 8 5 10 Maarten Abbink (TU Delft) Beamer Sample October 10, 2014 9 / 16

Section 1 - Subsection 2 - Page 1 Example 7 6 5 y 4 3 2 1 0 0 1 2 3 4 5 6 7 x p =6 x + y p p f (p) 3 5 4 8 5 10 6 13 Maarten Abbink (TU Delft) Beamer Sample October 10, 2014 9 / 16

Section 1 - Subsection 2 - Page 1 Example 7 6 5 y 4 3 2 1 0 0 1 2 3 4 5 6 7 x p =6 x + y p p f (p) 3 5 4 8 5 10 6 13 [ 5 2 p + 2, 5 ] 2 p Maarten Abbink (TU Delft) Beamer Sample October 10, 2014 9 / 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). Maarten Abbink (TU Delft) Beamer Sample October 10, 2014 10 / 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, 2014 10 / 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). Many problems in static program analysis can be expressed as polyhedral counting problems. Maarten Abbink (TU Delft) Beamer Sample October 10, 2014 10 / 16

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, 2014 11 / 16

Section 1 - Subsection 3 - Page 1 A picture made with the package TiKz Example Maarten Abbink (TU Delft) Beamer Sample October 10, 2014 12 / 16

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, 2014 13 / 16

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, 2014 14 / 16

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, 2014 14 / 16

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, 2014 15 / 16

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, 2014 16 / 16