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Grammars
https://www.dpmms.cam.ac.uk/~tef10/cam_only/langs-and-automata/3-0.html13 Aug 2009: S aSa. S bSb. Sε. generates the language of palindromes over the alphabet Σ = {a, b}. ... Examples here are 'swim', 'walk', 'cat' and 'dog'. Notice that there is a grammar that generates the language of palindromes over Σ = {a, b} even though this -
EXAMPLE SHEET 1 (LECTURES 1–6) GALOIS THEORY MICHAELMAS 2009 ...
https://www.dpmms.cam.ac.uk/study/II/Galois/2009-2010/ex1.pdf18 Oct 2009: 8. Find the minimal polynomials over Q of the complex numbers 53, i+. ... Deduce that the only elements of K(X) which are algebraic over K areconstants. -
Exercises
https://www.dpmms.cam.ac.uk/~tef10/cam_only/langs-and-automata/3-0-1.html13 Aug 2009: 3.0.1 Exercises. Provide a context-free grammar for regular expressions over the alphabet {a, b}. -
PART II REPRESENTATION THEORYSHEET 4 Unless otherwise stated, all ...
https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2009-2010/repex4.pdf8 Oct 2009: PART II REPRESENTATION THEORYSHEET 4. Unless otherwise stated, all vector spaces are finite-dimensional over C. ... : a,b,x Fp. of 3 3 upper unitriangular matrices over the finite field Fp of p elements (p prime).Show that G has p -
The Thought-experiment
https://www.dpmms.cam.ac.uk/~tef10/cam_only/langs-and-automata/1-2.html13 Aug 2009: Let us try some examples. Let L be the set of strings over the alphabet {a, b} which have the same number of as as bs. ... Let L be the set of strings over the alphabet {a, b} which have an even number of as and an even number of bs. -
More about bombs
https://www.dpmms.cam.ac.uk/~tef10/cam_only/langs-and-automata/2-1-1.html13 Aug 2009: The thought-experiment swiftly persuades us that the set of palindromes over an alphabet Σ is not regular (unless Σ contains only one character of course!). ... Let L be the language of palindromes over {a, b}. It isn't regular, but there is no obvious -
Michaelmas Term 2009 T.A. Fisher Linear Algebra: Example Sheet ...
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2009-2010/lin_alg-09-2.pdf30 Oct 2009: 5. Let A and B be n n matrices over a field F. ... 13. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices. -
Languages Recognised by Machines
https://www.dpmms.cam.ac.uk/~tef10/cam_only/langs-and-automata/1-1.html13 Aug 2009: The language {aa, ab, ba, bb} is the language of two-letter words over Σ. ... for characters in alphabets;. We tend to use lower-case letters from near the end of the alphabet - like 'u', 'v' and 'w'- for variables to range over strings;. -
Michaelmas Term 2009 T.A. Fisher Linear Algebra: Preliminaries This…
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2009-2010/lin_alg-09-0.pdf9 Oct 2009: In whichof these cases is U a vector space over R?(a) x1 > 0.(b) either x1 = 0 or x2 = 0.(c) x1 x2 = 0.(d) x1 x2 = 1.(e) x1 ... 2. Determine which of the following sets of sequences of real numbers (xn) form vector spaces over R.(a) xn is bounded.(b) xn -
IA Probability Examples Sheet 3, Lent 2009 W. T. ...
https://www.dpmms.cam.ac.uk/study/IA/Probability/2008-2009/probex.2008.3.pdf19 Feb 2009: a) When the culture has been going for just over n minutes, what is the probabilitythat no white cells have yet appeared? ... b) What is the probability that the entire culture eventually dies out?
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