74 Matching Annotations
  1. Aug 2023
    1. "But there's a very famous theorem in topology called the Jordan curve theorem. You have a plane and on it a simple curve that doesn't intersect and closes—in other words, a loop. There's an inside and an outside to the loop." As Riehl draws this, it seems obvious enough, but here's the problem: No matter how much your intuition tells you that there must be an inside and an outside, it's very hard to prove mathematically that this holds true for any loop that can be drawn.

      How does one concretely define "inside" and "outside"? This definition is part of the missing space between the intuition and the mathematical proof.

  2. May 2023

      Integral of the limit of uniform converging function is the same as the integral of the lmit. Pay attention to conditions: 1. Closed and bounded interval \([a, b]\). 2. Sequence of CONTINUOUS functions. 3. Of course, converges uniformly.

    1. 4.7-3 Uniform Boundedness Theorem.

      A sequence of linear operator such that, pointwise the value of the linear operators are bounded is enough for the norm of the operator to be bounded for the limit of the sequence of linear operators.

    2. 4.4-1 Riesz's Theorem (Functionals on C[a, b]).

      Every Bounded linear functionals on C[a, b] is Riemann Integrals.

    3. 4.8-4 Theorem (Strong and weak convergence).

      Strong convergence, equivalences and converse.

    4. 4.3-3 Theorem (Bounded linear functionals).

      Bounded Linear Functional Theorems in Banach Space.

    5. 4.3-2 Hahn-Banach Theorem (Normed spaces).

      Hahn Banach Theorem in normed vector spaces

    6. 4.2-1 Hahn-Banach Theorem (Extension of linear functionals).



  3. Apr 2023
    1. Theorem 5.7.

      Complexity of the Floyd warshall Algorithm.

    2. Theorem 11.9 (Triangularity Property).

      the incidence matrix of a spanning tree graph can be, lower triangular.

    3. Theorem 11.2 (Spanning Tree Property).
    4. Theorem 11.1 (Cycle Free Property).
    5. Theorem 9.5 (Weak Duality Theorem)

      (9.11) is literally the dual LP formulation of the min cost flow problem onthe network.

    6. Theorem 9.4 (Complementary Slackness Optimality Conditions)

      Some flow is optimal, if and only if, there is some potential, where, the reduced costs for the potential satisfies the complementary slackness conditions for the duality of the linear program of the min cos flow problem.

    1. 2.10-4 Theorem (Dual space).

      The dual spaces theorem

    2. 2.10-2 Theorem (Completeness)
    3. 2.10-1 Theorem (Space B(X, Y».
    4. 2.9-1 Theorem (Dimension of X*).

      The space and the dual space is having the same dimension.

    5. 2.8-3 Theorem (Continuity and boundedness)

      It's an example of theorem 2.7-9. A linear operator has boundedness and continuity being an equivalent conditions.

    6. 2.7-9 Theorem (Continuity and boundedness)

      Let T be a linear mapping between 2 normed space, then:

      1. T is continous if and only if it's bounded.
      2. T is continous at a single point then it's continous.
    7. 2.7-8 Theorem (Finite dimension).

      Linear operator, and bounded linear operators are equivalent when the vector space is finite dimensional.

    8. 2.6-9 Theorem (Range and null space).
      1. Range is a vector space.
      2. If the dim of the range is less than infinitely, then the dim of the range is \(\le\) dim of the domain.
    9. 2.5-6 Theorem (Continuous mapping)

      Continuous mapping preserves compactness in finite dimensional spaces.

    10. 2.5-5 Theorem (Finite dimension)

      Compact Closed unit ball in a normed spaces would mean that we have finite dimension.

    11. 2.5-3 Theorem (Compactness).

      compactness is euivalent to closed and boundedness in finite dimensional spaces.

    1. 18.1 Theorem.

      extreme value theorem. A continuous function attains some type of minimum and maximum over an compact set in its domain.

    2. 11.4 Theorem.

      Monotone Subsequence Theorem for real Sequences.



    1. 1.9 Theorem (attainment of a minimum)

      The existence of a minimizer for functions.



  4. Mar 2023
    1. 2.4-5 Theorem (Equivalent norms).

      In a finite dimensional space, every norm is Equivalent

    2. 2.4-3 Theorem (Closedness)

      Every finite dimensional Banach space is closed.

    3. 2.4-2 Theorem (Completeness).

      Every finite dimension subspace of the normed space is complete, so are their subspace.

    4. 2.3-2 Theorem (Completion)

      Alternative explanations from Walfram Math world: here.

      In brief, you can use an isometry map from a banach space to another subspace in a different Banach space such that it's dense.

    5. 2.3-1 Theorem (Subspace of a Banach space).

      Similar to 1.4-7



  5. Feb 2023
    1. Theorem 6.6.

      complexity of the max flow labeling algorithm.

    2. Theorem 6.7.

      Min cut capacity and the cardinality of maximum number of arc disjoint path from source to destination.

    3. Theorem 6.3 (Max-Flow Min-Cut Theorem).
    4. Theorem 6.4 (Augmenting Path Theorem).

      Aug Path theorem

    5. Theorem 6.5 (Integrality Theorem).

      Network flow integers theorem.

    6. Theorem 6.8
    7. Theorem 5.1 (Shortest Path Optimality Conditions).

      There is a linear programming interpretation via the network flow standard form and the reduction of shortest path to the network flow problem at the end of chapter 5.2.

    8. Theorem 3.7 (Augmenting Cycle Theorem).

      Given 2 feasible flow, \(x, x^\circ\), it's possible to construct \(x\) with at most m directed cycles on the residual graph \(G(x^\circ)\).

    9. Theorem 3.8 (Negative Cycle Optimality Theorem).
    1. 5.1-4 Theorem (Contraction on a ball)

      Paculiar, I have no idea about the significant of this theorem and why it's stated here. How does this variation of the theorem even make inuitive sense?

    2. 5.1-2 Banach Fixed Point Theorem
    3. 1.4-7 Theorem (Complete subspace)


    4. 1.4-6 Theorem (Closure, closed set).



  6. Local file Local file
    1. Theorem 3.2.5

      Characterization of a limit point of a set via a limit that occurs inside of the set.



    1. Bell’s theorem is aboutcorrelations (joint probabilities) of stochastic real variables and therefore doesnot apply to quantum theory, which neither describes stochastic motion nor usesreal-valued observables

      strong statement, what do people think about this? is it accepted by anyone or dismissed?

    1. Theorem 29


      metric equivalences preserves sequential convergence and openedness of sets.

    2. Theorem 24

      A convergence sequence is a Cauchy sequence under a certain metric.

    3. Theorem 27

      sequential continuity of a mapping in the metric space.

    4. Theorem 13

      For an accumulation point in a subset \(M\) of space \(X\), any epsilon balls with central isolation has ifninitely many points and there is a sequence in such epsilon ball such that it converges to the singularity but never really being on the singularity.

      Singularity: \(x_0\) as described in the proof.

    5. Theorem 25

      The equivalence of the sequential closedness of a set and the epsilon ball closedness of a set.



  7. Nov 2022
    1. it became clear that Fermat's Last Theorem could be proven as a corollary of a limited form of the modularity theorem (unproven at the time and then known as the "Taniyama–Shimura–Weil conjecture"). The modularity theorem involved elliptic curves, which was also Wiles's own specialist area.[15][16]

      Elliptical curves are also use in Ed25519 which are purportedly more robust to side channel attacks. Could there been some useful insight from Wiles and the modularity theorem?

  8. Jan 2022
    1. Central Limit Theorem

      the Central Limit Theorem tells us the sampling distribution of X̄ is closely approximated to a normal distribution.

  9. May 2021
  10. Jun 2019
  11. mitpressonpubpub.mitpress.mit.edu mitpressonpubpub.mitpress.mit.edu
    1. Or do you recall jotting down formulae for molecules and compounds in the margins of your chemistry textbook?

      I can't help but think of one of the biggest and longest standing puzzles in mathematics in Fermat's Last Theorem. He famously wrote in the margin of a book that he had a proof. but that it was too large to fit in the margin.


  12. Jun 2017