• 44 Posts
  • 829 Comments
Joined 2 年前
Cake day: 2023年12月18日
  • General_Effort@lemmy.worldtoAsk Lemmy@lemmy.world•Are the USA heading towards civil war?
    0·
    4 天前

    This does not have the seeds of a civil war.

    Look back to 1861. The US was split between slave and free states. Both regions had different economic models that were in conflict. The slavers of the south controlled most of the nation’s capital. The slaves were worth more than all the railroads and foundries of the north. They had oligarchic control of the southern states.

    When Lincoln was elected, it was the first time that an outspoken abolitionist became president. It was clear, that the demographics of the US were changing in a way that would lose their power and status. So they started a war.

    Those weren’t the uneducated racists that you find waving confederate flags today.

    The US has seen presidential lawlessness before (notably Andrew “Trail of Tears” Jackson). It also has seen lawlessness by state politicians (eg Harry F. “Massive Resistance” Byrd or Governor Orval “Blood will run in the Streets” Faubus).

    The only way I could see a civil war is, if Trump directly threatened the Middle Class and/or the Wealthy Elites.

  • General_Effort@lemmy.worldOPtoscience@lemmy.world•Inside the Secret Meeting Where Mathematicians Struggled to Outsmart AIEnglish
    0·
    9 天前

    I’m too lazy to check it now, but here’s GPT-4o mini (via DDG). Maybe I’ll give Gemini a chance later, too.

    Elaborate the proof.

    Here is a high‐level outline of the proof that there are infinitely many primes in the arithmetic progression
     p ≡ a (mod m)
    whenever gcd(a, m) = 1. In your case a = 6 and m = 35, so gcd(6, 35) = 1, and the theorem then guarantees infinitely many primes ≡ 6 mod 35.

    1. Dirichlet characters
      • A Dirichlet character modulo m is a completely multiplicative arithmetic function
      χ : ℤ → ℂ such that
      – χ(n+m) = χ(n) for all n,
      – χ(n) = 0 whenever gcd(n, m) > 1,
      – χ(n) ≠ 0 (and in fact |χ(n)| = 1) whenever gcd(n, m)=1.
      • There are exactly φ(m) such characters (including the “principal character” χ₀, which is 1 on all n coprime to m, and 0 otherwise).

    2. Dirichlet L-series
      For each character χ mod m one forms the Dirichlet L-series
      L(s, χ) = ∑_{n=1}^∞ χ(n) n^(−s) , Re(s) > 1.
      This generalizes the Riemann zeta function (which is the special case χ = χ₀ with m=1).

    3. Euler product factorization
      Because χ is completely multiplicative, one shows for Re(s)>1:
      L(s, χ) = ∏_{p prime} (1 − χ(p) p(−s))(−1).
      This factorization links L(s, χ) to the distribution of primes in each residue class mod m.

    4. Nonvanishing at s = 1
      A key analytic fact (Dirichlet’s theorem) is that for every nonprincipal character χ (i.e. χ≠χ₀), the L-series L(s, χ) extends to a holomorphic, nonzero function at s=1. Meanwhile L(s, χ₀) has a simple pole at s=1, just like the Riemann zeta function ζ(s).

    5. Summing over characters
      Use the orthogonality relations of characters: for any integer n with gcd(n,m)=1,
      (1/φ(m)) ∑{χ mod m} χ(n) over all φ(m) characters = 1,
      and it’s 0 if n is not coprime to m.
      This lets you isolate those primes in a particular residue class a mod m by forming
      ∑      {χ mod m} χ(a)̄ · L(s,χ)
      (where χ(a)̄ is the complex conjugate of χ(a)). In the Euler product this picks out exactly the primes p with p≡a mod m.

    6. Logarithmic differentiation and Tauberian theorems
      One takes logarithms of the Euler products and looks at
      –(L′/L)(s,χ) = ∑{n=1}^∞ Λ(n) χ(n) n^(−s),
      where Λ is the von Mangoldt function. Summing these expressions over characters and using orthogonality isolates
      ∑      {p ≡ a (mod m)} p^(−s) ,
      plus lower‐order terms. One then applies complex‐analytic Tauberian theorems (or elementary partial summation arguments) to deduce that as x→∞
      #{ p ≤ x : p ≡ a (mod m) } ∼ (1/φ(m)) x / log x ,
      in particular infinitely many such primes exist.

    Conclusion
    Dirichlet’s theorem on arithmetic progressions (proven by exactly the steps sketched above) implies that for any m and any a with gcd(a,m) = 1, there are infinitely many primes p ≡ a mod m. Setting m = 35 and a = 6 shows there are infinitely many primes that leave remainder 6 upon division by 35.

  • General_Effort@lemmy.worldtomemes@lemmy.world•That would be quality
    2·
    10 天前

    Yes. That’s a good use of resources. Any country will do it that way, and has done it that way. The engineers develop new weapons. The workers build them. Entertainers do propaganda. Anyone else goes to the front.

    However, during Elvis Presley’s time to serve, there was no hot war being fought. I don’t think US conscripts were sent into battle between the end of the UN operation in Korea/the Korean War, and some time after the Gulf of Tonkin incident.

  • General_Effort@lemmy.worldOPtoscience@lemmy.world•Inside the Secret Meeting Where Mathematicians Struggled to Outsmart AIEnglish
    0·
    10 天前

    Not really. They are powered by language models, after all.

    Still, much of math is about manipulating symbols. So it is not mind-blowing that a certain understanding of math is there and can be improved on. But math also involves an understanding of space, or geometry. There I wouldn’t expect much.

    Anyway. My 2 top takeaways from the article:

    1. Mathematicians are hired to improve the major AI services.

    2. They are required to use E2EE (Signal) for communication so that their work doesn’t get picked up and used for LLM-training before the time is ready.

  • General_Effort@lemmy.worldOPtomemes@lemmy.world•I can ship that
    8·
    11 天前

    Musk’s daughter, maybe more estranged than his other kids, and Trump’s son. The idea is that the “feud” between them could be ended by a marriage of 2 of their children.

    Kinda like in Dune where the Bene Gesserit wanted to end the Atreides/Harkonnen feud by marriage and so ordered Jessica to only bear daughters.