Geometric Constructions - Mathematical and Statistical ...
Philosophy of Constructions Constructions using compass and straightedge have a long history in Euclidean geometry. Their use reflects the basic axioms of this system. However, the stipulation that these be the only tools used in a construction is artificial and only has meaning if one views the process of construction as an application of logic.
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Higher Geometry - Mathematical & Statistical Sciences
www-math.ucdenver.eduThree Point Geometry Axioms for the Three Point Geometry: 1. There exist exactly 3 points in this geometry. 2. Two distinct points are on exactly one line. 3. Not all the points of the geometry …
The Chinese Remainder Theorem - UC Denver
www-math.ucdenver.eduby 3, and remainder 3 when divided by 7. We are looking for a number which satisfies the congruences, x ≡ 2 mod 3, x ≡ 3 mod 7, x ≡ 0 mod 2 and x ≡ 0 mod 5.
Proof Methods - Mathematical and Statistical Sciences
www-math.ucdenver.eduWriting mathematical proofs is therefore an art form (the art of convincing) and a social process since it is directed at people (the readers). A mathematical proof of a statement strongly depends on who the proof is written for. Proofs for a research audience are quite different from those found in textbooks. And even textbook proofs look ...
Linear Codes
www-math.ucdenver.educonvert from the generator matrix to the parity-check matrix and vice-versa. This is easily done. Theorem 2: If G = [I k A] is the generator matrix (in standard form) for the [n,k]-code C, then H = [-AT I n-k] is the parity check matrix for C. Proof.
Equivalence Relations - Mathematical and Statistical Sciences
www-math.ucdenver.edu1) a is an upper bound for B, and 2) a R x for every upper bound x for B. Example: Reals with the usual ordering. B = {x | 5 < x < 7 } 8, 7.5, 11, 7.00001 are all upper bounds of B. 7 is also an upper bound, and 7 ≤ any of the upper bounds, so 7 …
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