14.2 Spaces of incidence vectors 175 ¡ ¢ the whole linear space; since the dimension

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• Abstract algebra
• Polynomial
• Linear independence
• Vector space
• Theorems and definitions in linear algebra
• Eigenvalue algorithm
• Algebra
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[removed]The Hales–Jewett Theorem[removed]Many players: the matrix product problem One of the highest known lower

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10.6 Monotone circuits 123 For two boolean functions f and g in n variables, we write f 6 g if f (x) 6 g(x) for all x ∈ {0, 1}n .

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• Algebraic logic
• Boolean algebra

Exercises[removed](!) (Ray-Chaudhuri–Wilson[removed]Prove the following uniform version of Theorem 14.13: if A1 , . . . , Am is a¡ k-uniform

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• Polynomial
• Vector space
• Euclidean space
• Algebra
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R Foundations and Trends in Theoretical Computer Science Vol. 9, No–123 c 2013 S. Jukna and I. Sergeev

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• Circuit complexity
• Linear algebra
• Computational complexity theory
• Arithmetic circuit complexity
• Binary operations
• Boolean algebra
• Matrix
• Rank
• Exclusive or
• Theoretical computer science
• Algebra
• Mathematics

[removed]Chains and Antichains[removed]Union-free families A family of sets F is called r-union-free if A0 6⊆ A1 ∪ A2 ∪ · · · ∪ Ar holds for

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• Antichain
• Structure
• Mathematics
• Order theory
• Integer sequences

2.3 Density of 0-1 matrices 27 Our goal is to show that |X| 6 k. Note that, for each x ∈ X we can choose Yx ⊆ V2 so that 1 6 |Yx | 6 r,

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• Szemerédi regularity lemma
• Dyadic cubes
• Constructible universe
• Lemmas
• Functional analysis

12.3 The isolation lemma[removed]

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• Isolation lemma
• Operator theory
• Quantum mechanics
• Theorems and definitions in linear algebra
• Group action
• Mathematics
• Lemmas
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Entropy of Operators or Nechiporuk for Depth-2 Circuits Stasys Jukna Institute of Mathematics, Vilnius, Lithuania University of Frankfurt, Germany

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• Entropy
• Statistical theory
• Circuit
• FP
• Probability and statistics
• Computer science
• Applied mathematics
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• Randomness
• Theory of computation