T(W) ⊆ W

T = [ [1, 1],
      [0, 1] ]

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Partial-Sum Probabilities ⇄ Successive Measurements in Bell Experiments

Figure: Left — running-partial-sum Markov transitions; Right — Bell measurement settings and entangled state. (Generated diagram.)

Overview

Both the classical partial-sum process and the quantum Bell experiment produce probabilities for sequences of outcomes.
The useful analogy is that both systems evolve probabilities over a space of possible “paths” (integer sums vs. measurement outcomes). However, the
crucial difference is that Bell correlations require non-classical probability structure: no classical Markov chain or hidden-local-variable model can reproduce them.

1. The Partial-Sum (Markov) Model — short description

Suppose we draw integers uniformly from {1,…,N} with replacement and keep a running sum \(S_k\). The evolution is
classical and Markovian:

\(P(S_k = s) = \sum_{i=1}^{N} P(S_{k-1} = s-i)\cdot \frac{1}{N}.\)
      

To ask for the probability of reaching exactly \(n\) at some step is to condition on allowed paths through the integer-state lattice; paths that overshoot are excluded.
This process is classical: transition probabilities are nonnegative, normalized, and depend only on the previous state (Markov property).

2. Bell (CHSH-style) Experiment — short description

Two distant parties, Alice and Bob, share an entangled state \(\lvert\psi\rangle\). Each party chooses a measurement setting (Alice: \(a\) or \(a’\); Bob: \(b\) or \(b’\))
and obtains outcomes \(A,B\in\{+1,-1\}\). Quantum mechanics predicts joint probabilities

\(P(A,B \mid a,b) = \langle\psi \rvert \; \big(M_A^{(a)} \otimes M_B^{(b)}\big) \; \lvert\psi\rangle,\)

where \(M_A^{(a)},M_B^{(b)}\) are measurement operators (projectors).
      

The key empirical fact: for certain choices of \((a,a’,b,b’)\), the correlations violate the CHSH inequality and reach up to \(2\sqrt{2}\) (Tsirelson bound).

3. Classical Hidden-Variable / Markov-Factorization Assumption

A classical model with a hidden variable \(\lambda\) (and local Markov-like transitions) assumes the joint probability factorizes as:

\(P(A,B \mid a,b) \;=\; \int d\lambda \; \rho(\lambda)\; P(A\mid a,\lambda)\; P(B\mid b,\lambda).\)
      

This expresses locality and a classical probabilistic structure. If such a representation exists, all CHSH-type correlations obey the classical bound of 2.

4. CHSH inequality (derivation sketch)

Define correlators
\(\;E(a,b) = \sum_{A,B=\pm1} AB \, P(A,B\mid a,b).\)
Under the classical factorization with deterministic \(\pm1\) responses (or by convexity for probabilistic responses),
one can show the CHSH combination satisfies:

\(S \;=\; E(a,b) + E(a,b') + E(a',b) - E(a',b') \;\le\; 2.\)
      

The short intuitive proof: for a fixed \(\lambda\) and deterministic outcomes \(A(a,\lambda),B(b,\lambda)\in\{\pm1\}\),
the quantity
\[
Q(\lambda) = A(a,\lambda)\big[ B(b,\lambda) + B(b’,\lambda) \big] + A(a’,\lambda)\big[ B(b,\lambda) – B(b’,\lambda) \big]
\]
can only be \(\pm2\). Averaging over \(\lambda\) yields \(|S|\le 2\).

5. Quantum prediction violates the classical bound

For the two-qubit singlet state and spin (or polarization) measurements at appropriate angles one finds

\(S_{\text{quantum}} = 2\sqrt{2} > 2.\)
      

Therefore no model of the factorized classical form (and hence no classical Markov chain producing local factorized joint probabilities) can reproduce these correlations.

6. Why this forbids any classical Markov-chain representation

A Markov chain (or any classical sequential stochastic process) defines joint distributions built from nonnegative transition probabilities and local conditionalization on prior states.
If you attempt to represent the Bell scenario with a classical Markov chain / path model, you would need to assign joint probabilities
\(P(A,B\mid a,b)\) that simultaneously satisfy all measurement-setting marginals and the factorization/locality condition.
But because quantum correlations violate CHSH, no such global assignment of classical nonnegative transition probabilities exists.

Concretely:

• Partial-sum Markov processes: probabilities are built from local, stepwise transition kernels (nonnegative, normalized).
• Any classical hidden-variable or Markov description that respects locality must obey Bell (CHSH) bounds.\li>
• Quantum correlations (experimentally verified) violate those bounds, so they cannot be written as expectations over local Markov transitions.

7. Side-by-side summary (compact)

Feature	Partial-Sum / Markov	Bell / Quantum
State space	Integer sums, classical lattice	Hilbert space (amplitudes)
Allowed transitions	Nonnegative transition kernel, Markov	Unitary + measurement (non-commuting)
Path weighting	Sum of nonnegative path probabilities	Amplitude interference (complex), not representable as simple path probabilities
Bell/CHSH	Always satisfies CHSH bound \(|S|\le2\)	Can achieve \(|S|=2\sqrt2>2\)

8. Intuition: interference & non-commutativity vs. classical conditioning

Classical path models add probabilities for disjoint paths. Quantum mechanics adds complex amplitudes that can interfere, producing correlations that cannot be decomposed into a convex mixture of local deterministic paths.
Mathematically this is tied to the non-commutativity of measurement operators and the fact that a global joint distribution for all possible measurement outcomes (for all settings) that is both local and reproduces quantum marginals does not exist.

9. Optional: short worked example (CHSH angles)

For the singlet state choose measurement directions such that
\(\theta_{a,b}=0^\circ,\; \theta_{a,b’}=90^\circ,\; \theta_{a’,b}=45^\circ,\; \theta_{a’,b’}=135^\circ\).
The quantum correlator for spin-1/2 is \(E(\alpha,\beta) = -\cos(\theta_{\alpha\beta})\).
Then

\(S = -\cos 0^\circ – \cos 90^\circ – \cos 45^\circ + \cos 135^\circ
= -1 – 0 – \tfrac{\sqrt2}{2} + \big(-\tfrac{\sqrt2}{2}\big)
= -2\sqrt2 \)

so \(|S|=2\sqrt2\), violating the classical limit of 2.

10. Final takeaway

The visual/structural analogy is useful: both systems manage probability flow across allowed paths (integer-lattice paths vs. sequences of measurement outcomes). But Bell correlations are fundamentally incompatible with any classical Markov-chain (or local hidden-variable) representation because they violate inequalities (CHSH) that any such classical model must obey. That violation is the signature of quantum non-classicality (entanglement + interference + non-commuting observables).

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1. Recurrence

With the convention for and , the probability satisfies the recurrence

(This comes from conditioning on the value of the first draw: if the first draw is then we must subsequently hit .)

2. Generating function

Let . Summing the recurrence gives the closed form

Equivalently,

The coefficient of in this rational function equals .

3. Combinatorial (explicit) formula

Another useful representation counts ordered compositions of into parts each in .
If denotes the number of such compositions with exactly parts, then

By inclusion–exclusion one can write

so a fully explicit formula is

4. Remarks and small cases

• : for every (every draw is 1 so every positive integer is hit).
• : with ; values can be computed quickly from the recurrence.
• The generating function form is convenient for extracting asymptotics or for computing by partial fractions (roots of the denominator polynomial).

Recap (as before): draws are iid and uniform on . Let be the probability that some partial sum equals exactly .
With the convention for and ,

Numerical examples

Below are computed values of for and . (Remember .)

(each draw is 1 or 2 with probability )

n
0	1.000000000000
1	0.500000000000
2	0.750000000000
3	0.625000000000
4	0.687500000000
5	0.656250000000
6	0.671875000000
7	0.664062500000
8	0.667968750000
9	0.666015625000
10	0.666992187500
11	0.666503906250
12	0.666748046875

(each draw is 1, 2 or 3 with probability )

n
0	1.000000000000
1	0.333333333333
2	0.444444444444
3	0.592592592593
4	0.456790123457
5	0.497942386831
6	0.515775034294
7	0.490169181527
8	0.501295534217
9	0.502413250013
10	0.497959321919
11	0.500556035383
12	0.500309535772

(each draw is 1,2,3 or 4 with probability )

n
0	1.000000000000
1	0.250000000000
2	0.312500000000
3	0.390625000000
4	0.488281250000
5	0.360351562500
6	0.387939453125
7	0.406799316406
8	0.410842895508
9	0.391483306885
10	0.399266242981
11	0.402097940445
12	0.400922596455

Observations

• For the probabilities converge quickly to (the table shows values oscillating about that limit).
• For and the values for moderate tend to a limiting value (the limit exists and equals the reciprocal of the mean waiting-time to cross a large interval — this can be analyzed via renewal theory or by studying the generating function). Numerically you can see the values settle near about for and near for at the shown.

— End

— End

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The Reals in Are Not Countably Infinite

See also – Cardinality of the Rationals – Positive and Negative included

Claim

The interval is uncountable. Equivalently, there is no bijection
.

Proof (Cantor’s Diagonal Argument)

Suppose for contradiction that is countable. Then its elements can be listed as a sequence:

Write each in (a chosen) decimal expansion:

Convention: If a real admits two decimal expansions (e.g., ), choose the one
not ending with an infinite tail of 9s. This removes ambiguity.

Construct a new number by defining its digits along the diagonal:

By construction, differs from in the -th digit, so for all . This contradicts the assumption that lists all elements of .

Therefore, is uncountable.

Consequences

• has the same cardinality as (the continuum):

• The natural numbers are strictly “smaller” in cardinality:

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Also read ‘Cardinality of the Reals’

Do Negative Rationals Change the Cardinality?

Short answer: No. Adding negative rationals keeps the set countably infinite, the same cardinality as the integers .

1) Positive Rationals vs. Integers

The positive rationals can be listed (e.g., via Cantor’s diagonal), giving a bijection with :

2) Adding Negative Rationals

The nonzero rationals split as . The map is a bijection , so

The union of two countable sets is countable, hence

Adding the single element does not change cardinality.

3) Conclusion

Including negative rationals does not increase the cardinality; the rationals remain countably infinite, like the integers.

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The Consistency of the Axiom of Choice and the Generalized Continuum-Hypothesis with the Axioms of Set Theory

Background

• Set Theory and ZFC: The Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC) form the standard framework for modern set theory.
  • The Axiom of Choice (AC) states that for any collection of non-empty sets, there exists a choice function that selects one element from each set.
  • The Generalized Continuum Hypothesis (GCH) postulates that for any infinite cardinal κ, 2^κ = κ+ (the next cardinal).
• Hilbert’s Program: David Hilbert asked about the consistency of mathematics, including AC and GCH within ZFC.
• Motivation for Gödel’s Work: To show that AC and GCH are consistent with ZFC, assuming ZFC itself is consistent.

Gödel’s Results

• Inner Models and Constructibility: Gödel introduced the constructible universe L, a class of sets built in a step-by-step, definable manner.
• Axiom of Choice (AC): Gödel showed that AC holds within L.
• Generalized Continuum Hypothesis (GCH): Gödel proved that 2^κ = κ+ for all infinite cardinals κ in L.
• Consistency Proof: Gödel concluded that if ZFC is consistent, then ZFC + AC + GCH is also consistent.

Methodology

• The Constructible Universe: Gödel defined L as a hierarchy indexed by ordinals:
  • L0 contains all hereditarily finite sets.
  • Lα+1 includes subsets of Lα that are definable from parameters.
  • Lλ for limit ordinals λ is the union of all Lα for α < λ.
• Relative Consistency: Gödel showed that if ZFC is consistent, then so is ZFC + AC + GCH.

Impact and Subsequent Developments

• Completeness vs. Independence: Paul Cohen (1963) showed that AC and GCH are independent of ZFC, meaning they can neither be proved nor disproved from ZFC.
• Foundation of Modern Set Theory: Gödel’s methods laid the groundwork for later developments in set theory.
• Philosophical Implications: Gödel’s work highlighted the limitations of formal systems and the existence of different “universes” of set theory.

Conclusion

Gödel’s 1940 paper demonstrated the relative consistency of the Axiom of Choice and the Generalized Continuum Hypothesis with the axioms of set theory. This foundational work had a profound impact on set theory and mathematical logic, influencing both its philosophy and technical practice.

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We aim to construct a conformal equivalence f between the “angle”
A = { z ∈ ℂ | z ≠ 0, 0 < arg(z) < π/3 }
and the unit disk
𝔻 = { w ∈ ℂ | |w| < 1 }.

1. Map the angle to a horizontal strip

Use the logarithm map:

z → w₁ = log(z) = ln|z| + i·arg(z),

where z ∈ A. Under this map:

• The boundary ray arg(z) = 0 maps to the real axis (Im(w₁) = 0).
• The boundary ray arg(z) = π/3 maps to the line Im(w₁) = π/3.
• The region A maps to the horizontal strip
  S = { w₁ ∈ ℂ | 0 < Im(w₁) < π/3 }.

2. Map the strip to the upper half-plane

The exponential stretching map:

w₁ → w₂ = exp(πi·w₁ / 3)

transforms the strip S to the upper half-plane
H = { w₂ ∈ ℂ | Im(w₂) > 0 }.

3. Map the upper half-plane to the unit disk

The Möbius transformation:

w₂ → w₃ = (w₂ – i) / (w₂ + i)

is a conformal equivalence between the upper half-plane H and the unit disk 𝔻.

4. Combine the maps

The full conformal map f: A → 𝔻 is the composition:

f(z) = [(exp(πi·log(z) / 3) – i) / (exp(πi·log(z) / 3) + i)].

5. Simplified expression

Expanding the steps, the final form of f(z) is:

f(z) = [(exp(πi(ln|z| + i·arg(z)) / 3) – i) / (exp(πi(ln|z| + i·arg(z)) / 3) + i)].

“`

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Nowhere differentiable functions are functions that are continuous everywhere but do not have a well-defined derivative at any point. They exhibit erratic behavior, and although they can be integrated (as integration measures the area under the curve), they defy differentiation in the conventional sense.

1. Weierstrass Function

The Weierstrass function is one of the first discovered examples of a continuous, nowhere differentiable function. It is defined as:

    W(x) = ∑n=0∞ an cos(bn π x)

where:

• 0 < a < 1
• b is an odd integer such that ab > 1 + 3/2 π

This function is continuous but has no well-defined derivative at any point due to the rapid oscillations caused by the series.

Integration: Yes, the Weierstrass function is integrable since it’s continuous and bounded. The integral of the function over an interval exists and is well-defined, but the result might not be simple to compute due to its complex structure.

2. Cantor Function (Devil’s Staircase)

The Cantor function is another famous example of a function that is continuous but nondifferentiable almost everywhere. It’s defined on the unit interval [0, 1] using the Cantor set and is constructed by removing the middle thirds repeatedly from each remaining segment.

Integration: The Cantor function is not differentiable almost everywhere, but it is integrable. In fact, the integral of the Cantor function over the interval [0, 1] is equal to 0.5.

3. Brownian Motion (Wiener Process)

A Brownian motion path, denoted B(t), is a random process that is continuous almost surely but is nowhere differentiable with probability 1. This is commonly used in fields such as physics and finance for modeling stochastic processes.

Integration: Brownian motion is integrable in a stochastic sense (stochastic integrals), and techniques such as Itô calculus are used to handle such integrals. However, this is a special kind of integration designed to handle the irregularities of stochastic processes.

Can They Be Integrated?

Yes, most nowhere differentiable functions can be integrated, especially in the Riemann or Lebesgue sense, because integration is concerned with measuring the “area under the curve,” while differentiation is focused on the local rate of change, which is what these functions lack.

For example:

• The Weierstrass function is integrable over any interval due to its continuity.
• The Cantor function is also integrable, though its derivative is 0 almost everywhere.
• For Brownian motion, special methods (stochastic integrals) allow for meaningful integration.

In general, continuity guarantees integrability, but differentiability is not necessary for a function to be integrable.

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When discussing whether a function is analytic within or on the unit circle, we are referring to complex analysis, which deals with functions of a complex variable.

Analytic Within the Unit Circle

A function f(z) is said to be analytic within the unit circle if it is analytic (i.e., differentiable) at every point inside the unit circle. The unit circle is defined as the set of points z ∈ ℂ such that |z| = 1, and the region inside the unit circle is where |z| < 1.

• Analytic Function: A function is analytic at a point if it is differentiable not only at that point but also in a neighborhood around that point. More formally, a function f(z) is analytic if it has a Taylor series expansion that converges to f(z) in some neighborhood around the point of interest.
• Within the Unit Circle: The phrase “within the unit circle” means that the function is analytic at all points where |z| < 1. This includes all points strictly inside the circle but excludes the points on the boundary |z| = 1.

For example, the function f(z) = 1/(1 – z) is analytic within the unit circle because it can be expanded as a convergent power series:

f(z) = ∑n=0∞ zn   for   |z| < 1

This series converges for all z inside the unit circle, but the function becomes singular at z = 1 (on the unit circle), where it has a pole.

Analytic On the Unit Circle

A function f(z) is said to be analytic on the unit circle if it is analytic at every point on the unit circle, meaning |z| = 1. This is a stronger condition because the function must now be analytic at all points on the boundary of the disk defined by |z| = 1, not just within it.

• To be analytic on the unit circle, the function must:

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  ]]> https://stationarystates.com/pure-math/curves-over-finite-fields/?utm_source=rss&utm_medium=rss&utm_campaign=curves-over-finite-fields Tue, 08 Oct 2024 00:29:31 +0000 https://stationarystates.com/?p=642 Curves Over Finite Fields Curves over finite fields are algebraic curves defined by equations where the coefficients come from a finite field. A finite field is a field with a […]

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  ]]> Curves Over Finite Fields

  Curves over finite fields are algebraic curves defined by equations where the coefficients come from a finite field. A finite field is a field with a finite number of elements, often denoted as 𝔼_q, where q is a prime power (i.e., q = pⁿ for some prime p and positive integer n).

  Basic Concept

  An algebraic curve is given by a polynomial equation in two variables, say x and y, of the form f(x, y) = 0. When the coefficients of this polynomial come from a finite field, we call the curve a curve over a finite field.

  Example 1: Curve Over 𝔼₅

  Consider the field 𝔼₅ = {0, 1, 2, 3, 4}, which is the field of integers modulo 5. A simple curve over 𝔼₅ is given by the equation:

  y2 = x3 + 2x + 1 (mod 5)
  

  This is an example of an elliptic curve over the finite field 𝔼₅.

  To find the points on the curve, we substitute values of x from 𝔼₅ into the equation and check if the resulting value of y² has a solution in 𝔼₅.

  • For x = 0, y² = 1, so y = 1 or y = 4.
  • For x = 1, y² = 4, so y = 2 or y = 3.
  • For x = 2, y² = 4, so y = 2 or y = 3.
  • For x = 3, y² = 0, so y = 0.
  • For x = 4, y² = 4, so y = 2 or y = 3.

  Thus, the points on the curve are:

  (0, 1), (0, 4), (1, 2), (1, 3), (2, 2), (2, 3), (3, 0), (4, 2), (4, 3)
  

  Example 2: Line Over 𝔼₃

  Consider the field 𝔼₃ = {0, 1, 2}, and the line defined by the equation:

  y = 2x + 1 (mod 3)
  

  To find the points on this line, substitute values of x from 𝔼₃ into the equation:

  • For x = 0, y = 1.
  • For x = 1, y = 0.
  • For x = 2, y = 2 · 2 + 1 = 5 ≡ 2 (mod 3).

  Thus, the points on this line are:

  (0, 1), (1, 0), (2, 2)
  

  Key Concepts

  • Finite Fields: A finite field 𝔼_q contains q elements. For example, 𝔼₂ = {0, 1}, 𝔼₃ = {0, 1, 2}, etc.
  • Solutions Over Finite Fields: A curve over a finite field has a finite number of solutions (points), as both the x and y values are restricted to elements of the finite field.

  Applications

  Curves over finite fields have deep applications in number theory, coding theory, and cryptography. For instance:

  • Elliptic curves over finite fields are used in Elliptic Curve Cryptography (ECC), where the security of the cryptosystem relies on the difficulty of solving the elliptic curve discrete logarithm problem.
  • Error-correcting codes like Reed-Solomon codes use curves over finite fields for constructing robust codes to detect and correct errors in data transmission.

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