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20cs700 [2020-09-15] – [Synopsis] Martin Ziegler20cs700 [2020-09-17] – [Synopsis] Martin Ziegler
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 **II. Computability theory over the Reals**:  \\ **II. Computability theory over the Reals**:  \\
 a)   Computing Real Numbers: three equivalent notions, counter/examples, oracle-computable reals/limit Lemma ([[https://drive.google.com/file/d/1goACIp9v1Lva1m7tUx8E7_bS2xawIfUT|video]]) \\ a)   Computing Real Numbers: three equivalent notions, counter/examples, oracle-computable reals/limit Lemma ([[https://drive.google.com/file/d/1goACIp9v1Lva1m7tUx8E7_bS2xawIfUT|video]]) \\
-b)  Computing Real Sequences: semi-decidability / strong undecidability of equality, no computable sequence contains all computable reals  [[https://drive.google.com/file/d/14TqLkBtmdQy5A-bLUDb0juOAmQKxcW-j|video]])\\ +b)  Computing Real Sequences: semi-decidability / strong undecidability of equality, no computable sequence contains all computable reals  ([[https://drive.google.com/file/d/14TqLkBtmdQy5A-bLUDb0juOAmQKxcW-j|video]])\\ 
-c)  Computing Real Functions: closure under composition and restriction and sequences, necessarily continuous, computable Weierstrass Theorem \\+c)  Computing Real Functions: closure under composition and restriction and sequences, necessarily continuous, computable Weierstrass Theorem ([[https://drive.google.com/file/d/1bD4sUmWErYefwmnXKibD08s04XPTDW6W|video]])\\
 d)  Advanced Un/Computility of Real Functions: uncomputable Argmin/Root Finding, uncomputable Derivative, uncomputable ODE, uncomputable, Wave Equation, compactness in Real Computation \\ d)  Advanced Un/Computility of Real Functions: uncomputable Argmin/Root Finding, uncomputable Derivative, uncomputable ODE, uncomputable, Wave Equation, compactness in Real Computation \\
 e)  Multi-Functions & Enrichment: generalized restriction, fundamental theorem of algebra, fuzzy sign, Archimedian property, linear algebra, analytic functions \\ e)  Multi-Functions & Enrichment: generalized restriction, fundamental theorem of algebra, fuzzy sign, Archimedian property, linear algebra, analytic functions \\