Density and Outer Measure – Mathematics Stack Exchange

If $A subset mathbb{R}^{n}$ is Lebesgue measurable, then we see that $x in A$ is a point of density if
$$
lim_{r to 0} frac{mu(A cap B(x,r))}{mu(B(x,r))} =1
$$

The Lebesgue Density Theorem says that almost every point of $A$ is a point of density. Can I extend this notion of density to using the outer measure (for any subset of $mathbb{R}^{n}$) which I, a priori, don’t know is measurable? Does the Lebesgue Theorem apply in this case? My guess is yes, because the proof doesn’t really use the fact that $A$ is measurable. I want to use density to show that a set is measurable.

Tl;dr – obviously for measurable sets it makes no difference if I use “measure” or “outer measure”, but is there a notion of density for non-measurable sets (or sets not a priori measurable) using outer measure?

Inequality problem – Mathematics Stack Exchange

What are the values ​​of $B_1, B_2$, and $B_s$ so that the following inequality is satisfied with any value of $U_1$ and $U_2$? where $B_1, B_2, B_s, U_1$, and $U_2 in mathbb R^+$

$ B_1 U_1^2 + B_2 U_2^2 < B_s (U_1-U_2)^2 $

[ Mathematics ] Open Question : What is 230-220*0.5?

I was told that it is 5!

mathematics – 2d navigation for physics-based car

i’m making a game, one npc is a car with spear in head, he try to hit the enemy (yellow pot in image) with the spear point.

here is the promble ,the car is physics-based , it has veloctiy ,max angular velocity ,can turn around the car’s center, the npc can only change drive direction. since the positions of car and enemy can be very random (the enemy will not move once respawn), how can the car make the spear hit?

enter image description here

mathematics – Octagon border algorithm

I work on an open source game since 2 years and I’m very bad at math (it is not every time easy haha). My game permit to move a character on octagon. When character reach border coordinate (colored in yellow), I permit him to travel on a “new octagon”:

enter image description here

So, the algorithm goal is to know if an x, y tile coordinate is on a “yellow” tile, which direction is (North or North-Est or Est …) depending on map width and height.

I wrote this algorithm many times and in two different language, example with Rust:

pub fn get_corner(width: i16, height: i16, new_row_i: i16, new_col_i: i16) -> Option<CornerEnum> {
    let left_col_i_end = width / 3;
    let right_col_i_start = (width / 3) * 2;
    let top_row_i_end = height / 3;
    let bottom_row_i_start = (height / 3) * 2;
    let mut more = if new_row_i >= 0 { new_row_i } else { 0 };
    #(allow(unused_assignments))
    let mut right_col_i = 0;
    #(allow(unused_assignments))
    let mut left_col_i = 0;

    if new_row_i < top_row_i_end {
        right_col_i = right_col_i_start + more;
        left_col_i = left_col_i_end - more;
    } else {
        if new_row_i >= bottom_row_i_start {
            more = (height / 3) - (new_row_i - bottom_row_i_start + 1);
            more = if more >= 0 { more } else { 0 };
            right_col_i = right_col_i_start + more;
            left_col_i = left_col_i_end - more;
        } else {
            left_col_i = left_col_i_end;
            right_col_i = right_col_i_start;
        }
    }

    if new_col_i < left_col_i && new_row_i < top_row_i_end {
        return Some(CornerEnum::TopLeft);
    }
    if new_row_i < 0 && left_col_i <= new_col_i {
        return Some(CornerEnum::Top);
    }
    if new_col_i >= right_col_i && new_row_i < top_row_i_end {
        return Some(CornerEnum::TopRight);
    }
    if new_col_i > width - 1 && top_row_i_end <= new_row_i {
        return Some(CornerEnum::Right);
    }
    if new_col_i >= right_col_i && new_row_i >= bottom_row_i_start {
        return Some(CornerEnum::BottomRight);
    }
    if new_row_i > height - 1 && left_col_i_end <= new_col_i {
        return Some(CornerEnum::Bottom);
    }
    if new_col_i < left_col_i && new_row_i >= bottom_row_i_start {
        return Some(CornerEnum::BottomLeft);
    }
    if new_col_i < 0 && top_row_i_end <= new_row_i {
        return Some(CornerEnum::Left);
    }

    None
}

But it is not working well … I curse my math. I’m sure it’s not that complicated but i fail at each time in two years. So, i’m here to ask help, or for solution. That would be greatly appreciated!

Transcendental Number – Mathematics Stack Exchange

supposing a sum:
$sum _{n=1}^{infty }left(frac{1}{log _2left(nright)}right)$
Is it transcendental, and is there a subsequent general algorithm for proving transcendentity?
Furthermore, which $sum _{n=1}^{infty }left(frac{1}{log _kleft(nright)}right)$‘s k being any prime number are transcendental?

soft question – Mathematics based only on real numbers

I’m aware that >90% will outright reject this, so feel free to ignore it. I’d much appreciate those trying to figure out in which way this question (or rather its eventual answer) would make sense.

Is there a way to construct the foundations of mathematics from just the real numbers?
If not, why not?

Background: The real world (space, time) is arguably best described as a continuum. So it would make sense to take that as a foundation. Integers would only occur later e.g. as winding numbers, not as fundamental. All discrete things would be secondary. A continuous version of sets would probably be part of the foundation.

Rant: This construction from set theory to integers to rational numbers only to find that real numbers are then kind of problematic in the framework. I just can’t see any sense in that.

Proof of laminar matroid – Mathematics Stack Exchange

Let S be a ground set, and S be a laminar family of subsets of S, which means that for every two distinct subsets A,B ∈ S, we have either A ⊆ B or B ⊆ A or A ∩ B = ∅. For every A ∈ S, we are given a positive number k(A) ∈ Z. We define the group of subsets I as I = { X ⊆ S:|X ∩ A| ≤ k(A),∀A ∈ S}. I need to prove that (S;I) is a laminar matroid. I successfully proved that the empty set is independent and the hereditary property but I am struggling to prove the exchange property. Any direction on how to prove the exchange property in this situation? Thanks in advance.

recreational mathematics – Assigning keys to n people such that atleast any k people are required to open lock

Given the number of person(N) available and the number of locks(K) required to open a cell, give a specification of how to distribute the keys such that any K persons can open the locks, but no group of (k- 1) person can.

1<=N<=9
0<=k<=9

Ex-1 N=3 and k=1

((0),(0),(0))

Ex-2 For N=2 k=2

( (0), (1))

Ex-3 For N=3 and K=2

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

Ex-4 For N=3 and k=3

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

Ex-5 For N=5 and k=3

((0, 1, 2, 3, 4, 5), (0, 1, 2, 6, 7, 8), (0, 3, 4, 6, 7, 9), (1, 3, 5, 6, 8, 9), (2, 4, 5, 7, 8, 9))

lo.logic – Inverse Mathematics of Cousin's Lemma

This Normann and Sanders paper apparently caused quite a stir in the reverse mathematical community when it came out a couple of years ago. He says that Cousin's lemma, which is an extension of the Heine-Borel theorem, requires the full force of second-order arithmetic (SOA) to prove. He also says that this motto is useful for mathematically justifying the integral of Feynman's path. So this is an apparent counterexample for both

  • a) The inverse mathematical precept that classical analysis theorems can (usually?) be proved using one of SOA's "Big Five" subsystems, with the strength of the subsystem required to form a useful classification of such theorems; and
  • b) Solomon Feferman's argument that scientifically useful mathematics can generally be handled by relatively weak axioms, generally no stronger than PA / RCA0.

I don't exactly have a mathematical question about the Normann-Sanders article, but I would like to know if it has impacted the reverse mathematics program and what is its meaning. Could path integrals really require such powerful axioms?

Also, Cousins' motto is traditionally tested quite easily using the integrity property of real numbers. The problem is that the integrity property is a second order property of the reals (i.e. uses a set quantizer), and SOA is a First-Real theory, which has no sets of reals. However, in classic analysis, the integrity axiom actually refers to sets of reals, and this result shows that converting an integrity-based test to a first-order test is not that easy (I haven't read the document closely. And I have no idea at the moment how to prove Cousin's motto in SOA). Is that significant?

I can understand that the Peano axioms (second order) induction axiom naturally translates into the first order PA induction scheme, making induction tests work the same way as before. I'd be interested to know why parsing is identified with SOA rather than something that allows sets of reals (required for functions anyway) since there is no such direct translation of the integrity axiom. Analysis = SOA goes back a long way, since the Hilbert program was intended to test CON (SOA) once it was done with the consistency of arithmetic. Reverse mathematics came much later.

Thank you!