Proof without Words: Area and the Projection Theorem of a Right Triangle

Sidney H. Kung

doi:10.2307/2690464

Proof without Words: Area and the Projection Theorem of a Right Triangle

Mathematics Magazine ◽

10.2307/2690464 ◽

1993 ◽

Vol 66 (1) ◽

pp. 13

Author(s):

Sidney H. Kung

Keyword(s):

Projection Theorem

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Projection Theorem for Discrete-Time Quantum Walks

Electronic Proceedings in Theoretical Computer Science ◽

10.4204/eptcs.315.5 ◽

2020 ◽

Vol 315 ◽

pp. 48-58

Author(s):

Václav Potoček

Keyword(s):

Discrete Time ◽

Quantum Walks ◽

Projection Theorem ◽

Time Quantum

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The Quasi-Isomorphism Theorem

The Plaid Model ◽

10.23943/princeton/9780691181387.003.0019 ◽

2019 ◽

pp. 185-194

Author(s):

Richard Evan Schwartz

Keyword(s):

Affine Transformation ◽

Equivalence Theorem ◽

Isomorphism Theorem ◽

Orbit Equivalence ◽

Geometric Properties ◽

Projection Theorem ◽

Canonical Bijection ◽

Arithmetic Graph

This chapter proves the Quasi-Isomorphism Theorem modulo two technical lemmas, which will be dealt with in the next two chapters. Section 18.2 introduces the affine transformation TA from the Quasi-Isomorphism Theorem. Section 18.3 defines the graph grid GA = TA(Z2) and states the Grid Geometry Lemma, a result about the basic geometric properties of GA. Section 18.4 introduces the set Z* that appears in the Renormalization Theorem and states the main result about it, the Intertwining Lemma. Section 18.5 explains how the Orbit Equivalence Theorem sets up a canonical bijection between the nontrivial orbits of the plaid PET and the orbits of the graph PET. Section 18.6 reinterprets the orbit correspondence in terms of the plaid polygons and the arithmetic graph polygons. Everything is then put together to complete the proof of the Quasi-Isomorphism Theorem. Section 18.7 deduces the Projection Theorem (Theorem 0.2) from the Quasi-Isomorphism Theorem.

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