https://en.wikipedia.org/wiki/Infinite_monkey_theorem

Gemignani, Michael C. Elementary Topology. 2nd ed. Addison-Wesley Series in Mathematics. Reading, MA: Addison-Wesley Publishing Company, 1971.

Annotations URL<br /> Alternate annotations URL

for - quote - no cloning theorem - quantum mechanics - extended to consciousness and qualia - Frederico Faggin - hard problem of consciousness - no cloning theorem and private inner world of qualia - Frederico Faggin quote - no cloning theorem - quantum mechanics - extended to consciousness and qualia - Frederico Faggin - (see below) - What I feel what I feel is private. - What you feel is private. - You cannot transfer it to me - In order to tell you what I feel, I must translate that private feeling into classical information bit saying what I say. - The symbols must be this. - They must be sharable. - They must be copyable to share. You need to copy. Yeah. - My inner experience cannot be copied. And there is something in physics that cannot be copy. - In Quantum state, there is the "no cloning theorem", which says do not copy. - Not only that, but the maximum information that you can get if you make a measurement of the quantum state is one bit per quantum bit. - Olivas theorem says that and we have or Labor's theorem ourselves. What I can say about what I feel is much, much less

Mordell's theorem indicates that the group of rational points on an elliptic curve is finitely generated.

The Nagell-Lutz theorem provides a constructive algorithm on elliptic curves for finding all of the ration points of finite order.

• for: Thomas Theorem, The definition of the situation, William Isaac Thomas, Dorothy Swain Thomas, definition - Thomas Theorem, definition - definition of the situation, conflicting belief systems - Thomas theorem, learned something new - Thomas theorem

• definition: Thomas Theorem

• definition: definition of the situation
  • "The Thomas theorem is a theory of sociology which was formulated in 1928 by William Isaac Thomas and Dorothy Swaine Thomas:

If men define situations as real, they are real in their consequences.[1]

In other words, the interpretation of a situation causes the action. This interpretation is not objective. Actions are affected by subjective perceptions of situations. Whether there even is an objectively correct interpretation is not important for the purposes of helping guide individuals' behavior.|

• comment
• learned something new

• key insight: polarization

  • Behaviors subsequently are enacted out of a set of beliefs.
  • If there are a multitude of conflicting belief systems emerged from different cultures, then real conflicts can emerge out of the disharmony of conflicting beliefs
  • This is a very important insight into the polarization we see in the world today

• adjacency between:

  • polarization
  • Thomas Theorem

• adjacency statement

  • polarization can be explained by the Thomas Theorem

• reference

  • https://en.wikipedia.org/wiki/Thomas_theorem#:~:text=The%20Thomas%20theorem%20is%20a,This%20interpretation%20is%20not%20objective.

reply to u/Sensitive-Binding at https://www.reddit.com/r/antinet/comments/17ck5la/what_is_it_with_index_cards/

Some of it may involve the difference in available space versus other forms of writing on larger pages of paper. Similarly, many find that there is less pressure to write something short on Twitter or similar social media platforms because there is less space in the user interface that your mind feels the need to fill up. One can become paralyzed by looking at the larger open space on a platform like WordPress with the need to feel like they should write more.

With index cards you fill one up easily enough, and if there's more, you just grab another card and repeat.

cross reference with Fermat's Last Theorem being easier to suggest in a margin than actually writing it out in full.

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

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.

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.

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

Strong convergence, equivalences and converse.

Bounded Linear Functional Theorems in Banach Space.

Hahn Banach Theorem in normed vector spaces

open set, closed complement and vice versa. In finite Euclidean space.

Complexity of the Floyd warshall Algorithm.

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

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

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.

The dual spaces theorem

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

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

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.

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

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.

Continuous mapping preserves compactness in finite dimensional spaces.

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

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

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

Monotone Subsequence Theorem for real Sequences.

A Necessary conditions for a function to he differentiable at a point.

The existence of a minimizer for functions.

In a finite dimensional space, every norm is Equivalent

Every finite dimensional Banach space is closed.

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

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.

Similar to 1.4-7

no negative cycles optimal solutions min cost flow theorem

The complexity of the FIFO label correcting algorithms.

We implicitly assume that the flow is positive due to how the algorithm works.

Subgradient sum rules

Normal cone of the non smooth level plot of a function.

complexity of the max flow labeling algorithm.

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

Aug Path theorem

Network flow integers theorem.

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.

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

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?

theorem

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

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

Theorem:

metric equivalences preserves sequential convergence and openedness of sets.

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

sequential continuity of a mapping in the metric space.

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.

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

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?

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

🔥 Kareem Carr 🔥 on Twitter. (n.d.). Twitter. Retrieved 1 May 2021, from https://twitter.com/kareem_carr/status/1383925269132582912

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.

https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem

https://edupediapublications.org/journals/index.php/IJR/article/download/.../3589