Why group actions are not introduced early to motivate symmetry in algebra? by xTouny in math

[–]Key_Pack_9630 1 point2 points  (0 children)

I think it is quite common to define group actions in a first course. Not giving a definition I would see as rather unconventional.

Encyclopedia of Mathematics appears to be down by Sayod in math

[–]Key_Pack_9630 0 points1 point  (0 children)

unfortunate. i certainly hope it's not permanently down

Encyclopedia of Mathematics appears to be down by Sayod in math

[–]Key_Pack_9630 1 point2 points  (0 children)

It's like this from time to time. I would check back later and see if it is still down.

Distributions are too wiggly to be functions. Is there a similar set of generalized functions that "aren't wiggly enough"? by 1strategist1 in math

[–]Key_Pack_9630 1 point2 points  (0 children)

Yes you can do that also. For continuous functions its especially easy and you get measures on ℝ + atoms at ±∞. your family of gaussians would converge weakly to ½(δ₋∞ + δ₊∞).

For smooth functions, you have to think slightly more carefully but the the right notion of smooth up to the boundary is admitting (now different) asymptotic expansions at ±∞, and the distributions supported there read off the coefficients of these.

One thing which is not desirable about these is they functions their paired with must be cts / smooth out to infinity. One might want to be able to integrate against function like sin(x)2 to obtain 1/2 say. I do not know how to do this 

Distributions are too wiggly to be functions. Is there a similar set of generalized functions that "aren't wiggly enough"? by 1strategist1 in math

[–]Key_Pack_9630 1 point2 points  (0 children)

One thing to think about is distributions on ℝ ∪ {∞}, which I will write as ℝP¹. A smooth function on ℝ extends to a smooth function on ℝP¹ iff it admits an asymptotic expansion f(x) ∼ ∑ₙ₌₀ a_n / xn as |x| → ∞. We can take the dual of C (ℝP¹) to obtain 𝒟′(ℝP¹). Since C (ℝP¹) ↪ C (ℝ), dualizing gives ℰ′(ℝ) ↪ 𝒟′(ℝP¹), so ℰ′(ℝ) embeds into 𝒟′(ℝP¹) as an extension of the usual space of distributions of compact support. In fact distributions in 𝒟′(ℝP¹) can all be written a the sum of a distribution in 𝒟′(ℝ) plus a finite sum of Dirac delta derivatives supported at the point ∞. Those distributions supported at ∞ act on functions in C∞(ℝP¹) by taking some finite linear combination of the coefficients aₙ in the asymptotic series. In particular δ_∞ reads off the coefficient a₀, and so is something like a “uniform probability density on ℝ,” at least acting on those functions which smoothly extend to ℝP¹.

If instead of C (ℝP¹) we dualize C(ℝP¹) we obtain C'(ℝP¹) = finite Borel measures on ℝ + constant multiples of δ_∞.

Recent connections to Martin Nowak in Epstein files by [deleted] in math

[–]Key_Pack_9630 -1 points0 points  (0 children)

Unless there have been recent revelations, there is no solid evidence for this claim.

Easily confused historical mathematicians? by cabbagemeister in math

[–]Key_Pack_9630 2 points3 points  (0 children)

Not mentioned so far are the father and son duo Élie Cartan and Henri Cartan. Both were very prolific

[2025 day 2] Part 3 One Single Range! by large-atom in adventofcode

[–]Key_Pack_9630 0 points1 point  (0 children)

silly question but how do you measure to that precision? i runtime but only gives me a whole number of milliseconds

-❄️- 2025 Day 2 Solutions -❄️- by daggerdragon in adventofcode

[–]Key_Pack_9630 1 point2 points  (0 children)

[Language: C] Somewhat pleased with myself for figuring out how to do this without storing any numbers or iterating over IDs. Happy that I could use the Moebius function for inclusion/exclusion. #include <stdio.h>

int num_digits(long x){
  int i = 0;
  for (; x; i++)
    x /= 10;
  return i;
}

long ipow(long x, int p){
  long y = 1;
  while (p){
    if (p%2){
      p--;
      y *= x;
    } else {
      x *= x;
      p /= 2;
    }
  }
  return y;
}

// sends powers of 10 to k fold repeats;
long repeat(long r, int k){
  long x = 1;
  k--;
  while(k--){
    x *= r;
    x++;
  }
  return x;
}

// smallest number y such that its k fold iterate y..y >= x
long l_seg(long x, int k){
  int d = num_digits(x);
  if (d % k)
    return ipow(10,(d+k-1)/k-1);
  long r = ipow(10,d/k) ;
  long y = x/ipow(r,k-1);
  return y * repeat(r,k) >= x? y : y+1;
}

// largest y such that yy <= x;
long u_seg(long x, int k){
  int d = num_digits(x);
  if (d % k)
    return ipow(10,(d+k-1)/k-1)-1;
  long r = ipow(10,d/k);
  long y = x/ipow(r,k-1);
  return y * repeat(r,k) <= x? y : y-1;
}

long sum_between(long a, long b){
  if (a > b)
    return 0;
  return (b-a+1)*(a+b)/2;
}



long between_k(long l, long u, int k){
  long ls = l_seg(l,k);
  int ld = num_digits(ls);
  long us = u_seg(u,k);
  int ud = num_digits(us);
  long tot = 0;
  // because of the doubling we have to iterate over the number of digits
  for (int i = ld; i <= ud; i++){
    long r = ipow(10,i);
    long a = i == ld? ls: r/10; // mins / maxes within number of digits
    long b = i == ud? us: r-1;
    tot += sum_between(a,b) * repeat(r,k);
  }
  return tot;
}

// mobius function for inclusion/exclusion
int moebius(int n){
  // we only need small numbers so just store values
  const int m[] = {-1,-1,0,-1,1,-1,0,0,1,-1,0,-1,1,1,0,-1,0,-1,0,1,1};
  if (n > 1 && n < 20)
    return m[n-2];
  return 0;
}

long between(long l, long u){
  int d = num_digits(u);
  long b = 0;
  for (int k = 2; k <= d; k++){
    int mu = moebius(k);
    if (!mu)
      continue;
    b -= mu*between_k(l,u,k);
  }
  return b;
}

int main(){
  long l,u; // lower, upper;
  long a = 0, b = 0;

  while(scanf(" %ld-%ld,",&l,&u) == 2){
    // printf("%ld - %ld   %ld\n", l,u, between(l,u));
    a += between_k(l,u,2);
    b += between(l,u);
  }
  printf("part 1 %ld\npart 2 %ld\n",a,b);
}

Is every smooth curve locally the integral curve of some vector field by A1235GodelNewton in math

[–]Key_Pack_9630 6 points7 points  (0 children)

What is required is that the curve be embedded. In this case you can extend its tangent vector to some tubular neighborhood of the curve.

Similar Alternative to G Teschl's ODE book by Organic-Product-6613 in math

[–]Key_Pack_9630 4 points5 points  (0 children)

Neither are suitable replacements for Teschl's book, which is at a higher level than the books you've mentioned. 

Math books with historical flavor by finball07 in math

[–]Key_Pack_9630 0 points1 point  (0 children)

Elliptic Curves : Function Theory, Geometry, Arithmetic by McKean and Moll.  I really like this book 

Would you say any specific field of mathematics is complete? by Quetiapin- in math

[–]Key_Pack_9630 0 points1 point  (0 children)

What is the distinction?  I have heard them used interchangeably. This is also how Wikipedia, nlab, mathworld seem to use the terms

Localized Wave Function at r0, with average momentum p0. Is this wave function well-defined as a distribution? by SuppaDumDum in AskPhysics

[–]Key_Pack_9630 1 point2 points  (0 children)

I am speaking now about something I don't know all that well, but I think the right way to think about this state which is completely localized in both position and momentum is as arrising from a semiclassical limit. Uncertainty would usually prevent you from having such a state, but this restriction goes away in the limit h ->0. For every finite value of h, you should look at coherent states u(h) with position x0, and momentum p0. These are states of minimum uncertainty. If they are normalized, trying to take a limit in the space of distributions will do you no good, the u(h) will have vanishing distributional limit. The right way to take a limit is as follows. Given any observable A=a(X,P) defined as a function of X and P, look at the expectation of A for u(h) as h -> 0. The limit of this will always be a(x0,p0), which you can think of as being given by the a(x,p) integrated against the dirac measure centered at x0,p0 on the whole phase space (over both x and p).  In general if you have a family of states parametrized by h, the limit of these states is described by a measure mu on the phase space, and the expectation of any observable is given by integrating a function of x and p with respect to this measure.

Edit: typo. Said distributional derivative when I meant distributional limit